Package 'MPS'

Developed for computing pdf, cdf, quantile, random generation, drawing q-q plot, and estimating the parameters of 24 G-family of statistical distributions.

Description

Developed for computing the probability density function, computing the cumulative distribution function, computing the quantile function, random generation, and estimating the parameters of 24 G-family of statistical distributions via the maximum product spacing approach introduced in <https://www.jstor.org/stable/2345411>. These families are: beta G distribution due to Eugene et al. (2002), beta exponential G distribution due to Alzaatreh et al. (2013), beta extended G distribution due to Alzaatreh et al. (2013), exponentiated G distribution due to Gupta et al. (1998), exponentiated Kumaraswamy G distribution due to Lemonte et al. (2013), exponentiated exponential Poisson G distribution due to Ristic and Nadarajah (2014), exponentiated generalized G distribution due to Cordeiro et al. (2013), gamma type I G distribution due to Zografos and Balakrishnan (2009), gamma type II G distribution due to Ristic and Balakrishnan (2012), gamma uniform G distribution due to Torabi and Montazeri (2012), gamma-X generated of log-logistic family of G distribution due to Alzaatreh et al. (2013), gamma-X family of modified beta exponential G distribution due to Alzaatreh et al. (2013), geometric exponential Poisson G distribution due to Nadarajah et al. (2013), generalized beta G distribution due to Alexander et al. (2012), generalized transmuted G distribution due to Merovci et al. (2017), Kumaraswamy G distribution due to Cordeiro and Castro (2011), log gamma type I G distribution due to Amini et al. (2013), log gamma type II G distribution due to Amini et al. (2013), Marshall-Olkin G distribution due to Marshall and Olkin (1997), Marshall-Olkin Kumaraswamy G distribution due to Roshini and Thobias (2017), modified beta G distribution due to Nadarajah et al. (2013), odd log-logistic G distribution due to Gauss et al. (2017), truncated-exponential skew-symmetric G distribution due to Nadarajah et al. (2014), and Weibull G distribution due to Alzaatreh et al. (2013).

Details

Package: MPS

Type: Package

Version: 2.3.1

Date: 2019-09-04

License: GPL(>=2)

Author(s)

Mahdi Teimouri and Saralees Nadarajah

Maintainer: Mahdi Teimouri <[email protected]>

References

Alexander, C., Cordeiro, G. M., and Ortega, E. M. M. (2012). Generalized beta-generated distributions, Computational Statistics and Data Analysis, 56, 1880-1897.

Alzaatreh, A., Lee, C., and Famoye, F. (2013). A new method for generating families of continuous distributions, Metron, 71, 63-79.

Amini, M., MirMostafaee, S. M. T. K., and Ahmadi, J. (2013). Log-gamma-generated families of distributions, Statistics, 48 (4), 913-932.

Cheng, R. C. H. and Stephens, M. A. (1989). A goodness-of-fit test using Moran's statistic with estimated parameters, Biometrika, 76 (2), 385-392.

Cordeiro, G. M. and Castro, M. (2011). A new family of generalized distributions, Journal of Statistical Computation and Simulation, 81, 883-898.

Cordeiro, G. M., Ortega, E. M. M., and da Cunha, D. C. C. (2013). The exponentiated generalized class of distributions, Journal of Data Science, 11, 1-27.

Eugene, N., Lee, C., and Famoye, F. (2002). Beta-normal distribution and its applications, Communications in Statistics-Theory and Methods, 31, 497-512.

Gupta, R. C., Gupta, P. L., and Gupta, R. D. (1998). Modeling failure time data by Lehman alternatives, Communications in Statistics-Theory and Methods, 27, 887-904.

Gauss, M. C., Alizadeh, M., Ozel, G., Hosseini, B. Ortega, E. M. M., and Altunc, E. (2017). The generalized odd log-logistic family of distributions: properties, regression models and applications, Journal of Statistical Computation and Simulation, 87(5), 908-932.

Lemonte, A. J., Barreto-Souza, W., and Cordeiro, G. M. (2013). The exponentiated Kumaraswamy distribution and its log-transform, Brazilian Journal of Probability and Statistics, 27, 31-53.

Marshall, A. W. and Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families, Biometrika, 84, 641-652.

Merovcia, F., Alizadeh, M., Yousof, H. M., and Hamedani, G. G. (2017). The exponentiated transmuted-G family of distributions: Theory and applications, Communications in Statistics-Theory and Methods, 46(21), 10800-10822.

Nadarajah, S., Cancho, V. G., and Ortega, E. M. M. (2013). The geometric exponential Poisson distribution, Statistical Methods & Applications, 22, 355-380.

Nadarajah, S., Teimouri, M., and Shih, S. H. (2014). Modified beta distributions, Sankhya, 76 (1), 19-48.

Nadarajah, S., Nassiri, V., and Mohammadpour, A. (2014). Truncated-exponential skew-symmetric distributions, Statistics, 48 (4), 872-895.

Ristic, M. M. and Balakrishnan, N. (2012). The gamma exponentiated exponential distribution, Journal of Statistical Computation and Simulation, 82, 1191-1206.

Ristic, M. M. and Nadarajah, S. (2014). A new lifetime distribution, Journal of Statistical Computation and Simulation, 84 (1), 135-150.

Roshini, G. and Thobias, S. (2017). Marshall-Olkin Kumaraswamy Distribution, International Mathematical Forum, 12 (2), 47-69.

Torabi, H. and Montazeri, N. H. (2012). The gamma uniform distribution and its applications, Kybernetika, 48, 16-30.

Zografos, K. and Balakrishnan, N. (2009). On families of beta- and generalized gamma-generated distributions and associated inference, Statistical Methodology, 6, 344-362.

---

beta exponential G distribution

Description

Computes the pdf, cdf, quantile, and random numbers, draws the q-q plot, and estimates the parameters of the beta exponential G distribution. The General form for the probability density function (pdf) of the beta exponential G distribution due to Alzaatreh et al. (2013) is given by

$f(x,{\Theta}) = \frac{{d\,g(x-\mu,\theta ){{\left[ {1 - {{\left( {1 - G(x-\mu,\theta )} \right)}^d}} \right]}^{b - 1}}{{\left( {1 - G(x-\mu,\theta )} \right)}^{ad - 1}}}}{{B(a,b)}},$

where $\theta$ is the baseline family parameter vector. Also, a>0, b>0, d>0, and $\mu$ are the extra parameters induced to the baseline cumulative distribution function (cdf) G whose pdf is g. The general form for the cumulative distribution function (cdf) of the beta exponential G distribution is given by

$F(x,{\Theta}) = 1 - \frac{{\int_0^{{{\left( {1 - G(x-\mu,\theta )} \right)}^d}} {{y^{a - 1}}{{\left( {1 - y} \right)}^{b - 1}}} dy}}{{B(a,b)}}.$

Here, the baseline G refers to the cdf of famous families such as: Birnbaum-Saunders, Burr type XII, Exponential, Chen, Chisquare, F, Frechet, Gamma, Gompertz, Linear failure rate (lfr), Log-normal, Log-logistic, Lomax, Rayleigh, and Weibull. The parameter vector is $\Theta=(a,b,d,\theta,\mu)$ where $\theta$ is the baseline G family's parameter space. If $\theta$ consists of the shape and scale parameters, the last component of $\theta$ is the scale parameter (here, a, b, and d are the first, second, and the third shape parameters). Always, the location parameter $\mu$ is placed in the last component of $\Theta$.

Usage

dbetaexpg(mydata, g, param, location = TRUE, log=FALSE)
pbetaexpg(mydata, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
qbetaexpg(p, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
rbetaexpg(n, g, param, location = TRUE)
qqbetaexpg(mydata, g, location ="TRUE", method)
mpsbetaexpg(mydata, g, location = TRUE, method, sig.level)

Arguments

g	The name of family's pdf including: "birnbaum-saunders", "burrxii", "chisq", "chen", "exp", "f", "frechet", "gamma", "gompetrz", "lfr", "log-normal", "log-logistic", "lomax", "rayleigh", and "weibull".
p	a vector of value(s) between 0 and 1 at which the quantile needs to be computed.
n	number of realizations to be generated.
mydata	Vector of observations.
param	parameter vector $\Theta=(a,b,d,\theta,\mu)$
location	If FALSE, then the location parameter will be omitted.
log	If TRUE, then log(pdf) is returned.
log.p	If TRUE, then log(cdf) is returned and quantile is computed for exp(-p).
lower.tail	If FALSE, then 1-cdf is returned and quantile is computed for 1-p.
method	The used method for maximizing the sum of log-spacing function. It will be "BFGS", "CG", "L-BFGS-B", "Nelder-Mead", or "SANN".
sig.level	Significance level for the Chi-square goodness-of-fit test.

Details

It can be shown that the Moran's statistic follows a normal distribution. Also, a chi-square approximation exists for small samples whose mean and variance approximately are m(log(m)+0.57722)-0.5-1/(12m) and m($\pi^2$/6-1)-0.5-1/(6m), respectively, with m=n+1, see Cheng and Stephens (1989). So, a hypothesis tesing can be constructed based on a sample of n independent realizations at the given significance level, indicated in above as sig.level.

Value

1. A vector of the same length as mydata, giving the pdf values computed at mydata.

2. A vector of the same length as mydata, giving the cdf values computed at mydata.

3. A vector of the same length as p, giving the quantile values computed at p.

4. A vector of the same length as n, giving the random numbers realizations.

5. A sequence of goodness-of-fit statistics such as: Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn information criterion (HQIC), Cramer-von Misses statistic (CM), Anderson Darling statistic (AD), log-likelihood statistic (log), and Moran's statistic (M). The Kolmogorov-Smirnov (KS) test statistic and corresponding p-value. The Chi-square test statistic, critical upper tail Chi-square distribution, related p-value, and the convergence status.

Author(s)

Mahdi Teimouri

References

Cheng, R. C. H. and Stephens, M. A. (1989). A goodness-of-fit test using Moran's statistic with estimated parameters, Biometrika, 76 (2), 385-392.

Alzaatreh, A., Lee, C., and Famoye, F. (2013). A new method for generating families of continuous distributions, Metron, 71, 63-79.

Examples

mydata<-rweibull(100,shape=2,scale=2)+3
dbetaexpg(mydata, "weibull", c(1,1,1,2,2,3))
pbetaexpg(mydata, "weibull", c(1,1,1,2,2,3))
qbetaexpg(runif(100), "weibull", c(1,1,1,2,2,3))
rbetaexpg(100, "weibull", c(1,1,1,2,2,3))
qqbetaexpg(mydata, "weibull", TRUE, "Nelder-Mead")
mpsbetaexpg(mydata, "weibull", TRUE, "Nelder-Mead", 0.05)

---

beta G distribution

Description

Computes the pdf, cdf, quantile, and random numbers, draws the q-q plot, and estimates the parameters of the beta G distribution. General form for the probability density function (pdf) of beta G distribution due to Eugene et al. (2002) is given by

$f(x,{\Theta}) = \frac{{g(x-\mu,\theta ){{\left( {G(x-\mu,\theta )} \right)}^{a - 1}}{{\left( {1 - G(x-\mu,\theta )} \right)}^{b - 1}}}}{{B(a,b)}},$

where $\theta$ is the baseline family parameter vector. Also, a>0, b>0, and $\mu$ are the extra parameters induced to the baseline cumulative distribution function (cdf) G whose pdf is g. The general form for the cumulative distribution function (cdf) of the beta G distribution is given by

$F(x,{\Theta}) = \frac{{\int_0^{G(x-\mu,\theta )} {{y^{a - 1}}{{\left( {1 - y} \right)}^{b - 1}}} dy}}{{B(a,b)}}.$

Here, the baseline G refers to the cdf of famous families such as: Birnbaum-Saunders, Burr type XII, Exponential, Chen, Chisquare, F, Frechet, Gamma, Gompertz, Linear failure rate (lfr), Log-normal, Log-logistic, Lomax, Rayleigh, and Weibull. The parameter vector is $\Theta=(a,b,\theta,\mu)$ where $\theta$ is the baseline G family's parameter space. If $\theta$ consists of the shape and scale parameters, the last component of $\theta$ is the scale parameter (here, a and b are the first and second shape parameters). Always, the location parameter $\mu$ is placed in the last component of $\Theta$.

Usage

dbetag(mydata, g, param, location = TRUE, log=FALSE)
pbetag(mydata, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
qbetag(p, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
rbetag(n, g, param, location = TRUE)
qqbetag(mydata, g, location =TRUE, method)
mpsbetag(mydata, g, location = TRUE, method, sig.level)

Arguments

g	The name of family's pdf including: "birnbaum-saunders", "burrxii", "chisq", "chen", "exp", "f", "frechet", "gamma", "gompetrz", "lfr", "log-normal", "log-logistic", "lomax", "rayleigh", and "weibull".
p	a vector of value(s) between 0 and 1 at which the quantile needs to be computed.
n	number of realizations to be generated.
mydata	Vector of observations.
param	parameter vector $\Theta=(a,b,\theta,\mu)$
location	If FALSE, then the location parameter will be omitted.
log	If TRUE, then log(pdf) is returned.
log.p	If TRUE, then log(cdf) is returned and quantile is computed for exp(-p).
lower.tail	If FALSE, then 1-cdf is returned and quantile is computed for 1-p.
method	The used method for maximizing the sum of log-spacing function. It will be "BFGS", "CG", "L-BFGS-B", "Nelder-Mead", or "SANN".
sig.level	Significance level for the Chi-square goodness-of-fit test.

Details

It can be shown that the Moran's statistic follows a normal distribution. Also, a chi-square approximation exists for small samples whose mean and variance approximately are m(log(m)+0.57722)-0.5-1/(12m) and m($\pi^2$/6-1)-0.5-1/(6m), respectively, with m=n+1, see Cheng and Stephens (1989). So, a hypothesis tesing can be constructed based on a sample of n independent realizations at the given significance level, indicated in above as sig.level.

Value

1. A vector of the same length as mydata, giving the pdf values computed at mydata.

2. A vector of the same length as mydata, giving the cdf values computed at mydata.

3. A vector of the same length as p, giving the quantile values computed at p.

4. A vector of the same length as n, giving the random numbers realizations.

5. A sequence of goodness-of-fit statistics such as: Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn information criterion (HQIC), Cramer-von Misses statistic (CM), Anderson Darling statistic (AD), log-likelihood statistic (log), and Moran's statistic (M). The Kolmogorov-Smirnov (KS) test statistic and corresponding p-value. The Chi-square test statistic, critical upper tail Chi-square distribution, related p-value, and the convergence status.

Author(s)

Mahdi Teimouri

References

Cheng, R. C. H. and Stephens, M. A. (1989). A goodness-of-fit test using Moran's statistic with estimated parameters, Biometrika, 76 (2), 385-392.

Eugene, N., Lee, C., and Famoye, F. (2002). Beta-normal distribution and its applications, Communications in Statistics-Theory and Methods, 31, 497-512.

Examples

mydata<-rweibull(100,shape=2,scale=2)+3
dbetag(mydata, "weibull", c(1,1,2,2,3))
pbetag(mydata, "weibull", c(1,1,2,2,3))
qbetag(runif(100), "weibull", c(1,1,2,2,3))
rbetag(100, "weibull", c(1,1,2,2,3))
qqbetag(mydata, "weibull", TRUE, "Nelder-Mead")
mpsbetag(mydata, "weibull", TRUE, "Nelder-Mead", 0.05)

---

exponentiated exponential Poisson G distribution

Description

Computes the pdf, cdf, quantile, and random numbers, draws the q-q plot, and estimates the parameters of the exponentiated exponential Poisson G distribution. The general form for the probability density function (pdf) of the the exponentiated exponential Poisson G distribution due to Ristic and Nadarajah (2014) is given by

$f(x,{\Theta}) = \frac{{a\,b\,g(x-\mu,\theta ){{\left( {G(x-\mu,\theta )} \right)}^{a - 1}}{e^{ - b{{\left( {G(x-\mu,\theta )} \right)}^a}}}}}{{1 - {e^{ - b}}}},$

where $\theta$ is the baseline family parameter vector. Also, a>0, b>0, and $\mu$ are the extra parameters induced to the baseline cumulative distribution function (cdf) G whose pdf is g. The general form for the cumulative distribution function (cdf) of the exponentiated exponential Poisson G distribution is given by

$F(x,{\Theta}) = \frac{{1 - {e^{ - b{{\left( {G(x-\mu,\theta )} \right)}^a}}}}}{{1 - {e^{ - b}}}}.$

Here, the baseline G refers to the cdf of famous families such as: Birnbaum-Saunders, Burr type XII, Exponential, Chen, Chisquare, F, Frechet, Gamma, Gompertz, Linear failure rate (lfr), Log-normal, Log-logistic, Lomax, Rayleigh, and Weibull. The parameter vector is $\Theta=(a,b,\theta,\mu)$ where $\theta$ is the baseline G family's parameter space. If $\theta$ consists of the shape and scale parameters, the last component of $\theta$ is the scale parameter (here, a and b are the first and second shape parameters). Always, the location parameter $\mu$ is placed in the last component of $\Theta$.

Usage

dexpexppg(mydata, g, param, location = TRUE, log=FALSE)
pexpexppg(mydata, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
qexpexppg(p, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
rexpexppg(n, g, param, location = TRUE)
qqexpexppg(mydata, g, location = TRUE, method)
mpsexpexppg(mydata, g, location = TRUE, method, sig.level)

Arguments

g	The name of family's pdf including: "birnbaum-saunders", "burrxii", "chisq", "chen", "exp", "f", "frechet", "gamma", "gompetrz", "lfr", "log-normal", "log-logistic", "lomax", "rayleigh", and "weibull".
p	a vector of value(s) between 0 and 1 at which the quantile needs to be computed.
n	number of realizations to be generated.
mydata	Vector of observations.
param	parameter vector $\Theta=(a,b,\theta,\mu)$
location	If FALSE, then the location parameter will be omitted.
log	If TRUE, then log(pdf) is returned.
log.p	If TRUE, then log(cdf) is returned and quantile is computed for exp(-p).
lower.tail	If FALSE, then 1-cdf is returned and quantile is computed for 1-p.
method	The used method for maximizing the sum of log-spacing function. It will be "BFGS", "CG", "L-BFGS-B", "Nelder-Mead", or "SANN".
sig.level	Significance level for the Chi-square goodness-of-fit test.

Details

It can be shown that the Moran's statistic follows a normal distribution. Also, a chi-square approximation exists for small samples whose mean and variance approximately are m(log(m)+0.57722)-0.5-1/(12m) and m($\pi^2$/6-1)-0.5-1/(6m), respectively, with m=n+1, see Cheng and Stephens (1989). So, a hypothesis tesing can be constructed based on a sample of n independent realizations at the given significance level, indicated in above as sig.level.

Value

1. A vector of the same length as mydata, giving the pdf values computed at mydata.

2. A vector of the same length as mydata, giving the cdf values computed at mydata.

3. A vector of the same length as p, giving the quantile values computed at p.

4. A vector of the same length as n, giving the random numbers realizations.

5. A sequence of goodness-of-fit statistics such as: Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn information criterion (HQIC), Cramer-von Misses statistic (CM), Anderson Darling statistic (AD), log-likelihood statistic (log), and Moran's statistic (M). The Kolmogorov-Smirnov (KS) test statistic and corresponding p-value. The Chi-square test statistic, critical upper tail Chi-square distribution, related p-value, and the convergence status.

Author(s)

Mahdi Teimouri

References

Cheng, R. C. H. and Stephens, M. A. (1989). A goodness-of-fit test using Moran's statistic with estimated parameters, Biometrika, 76 (2), 385-392.

Ristic, M. M. and Nadarajah, S. (2014). A new lifetime distribution, Journal of Statistical Computation and Simulation, 84 (1), 135-150.

Examples

mydata<-rweibull(100,shape=2,scale=2)+3
dexpexppg(mydata, "weibull", c(1,1,2,2,3))
pexpexppg(mydata, "weibull", c(1,1,2,2,3))
qexpexppg(runif(100), "weibull", c(1,1,2,2,3))
rexpexppg(100, "weibull", c(1,1,2,2,3))
qqexpexppg(mydata, "weibull", location = TRUE, "Nelder-Mead")
mpsexpexppg(mydata, "weibull", TRUE, "Nelder-Mead", 0.05)

---

exponentiated G distribution

Description

Computes the pdf, cdf, quantile, and random numbers, draws the q-q plot, and estimates the parameters of the exponentiated G distribution. General form for the probability density function (pdf) of the exponentiated G distribution due to Gupta et al. (1998) is given by

$f(x,{\Theta}) = a\,g(x-\mu,\theta ){\left( {G(x-\mu,\theta )} \right)^{a - 1}},$

where $\theta$ is the baseline family parameter vector. Also, a>0 and $\mu$ are the extra parameters induced to the baseline cumulative distribution function (cdf) G whose pdf is g. The general form for the cumulative distribution function (cdf) of the exponentiated G distribution is given by

$F(x,{\Theta}) = \left( {G(x-\mu,\theta )} \right)^{a}.$

Here, the baseline G refers to the cdf of famous families such as: Birnbaum-Saunders, Burr type XII, Exponential, Chen, Chisquare, F, Frechet, Gamma, Gompertz, Linear failure rate (lfr), Log-normal, Log-logistic, Lomax, Rayleigh, and Weibull. The parameter vector is $\Theta=(a,\theta,\mu)$ where $\theta$ is the baseline G family's parameter space. If $\theta$ consists of the shape and scale parameters, the last component of $\theta$ is the scale parameter (here, a is the shape parameter). Always, the location parameter $\mu$ is placed in the last component of $\Theta$.

Usage

dexpg(mydata, g, param, location = TRUE, log=FALSE)
pexpg(mydata, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
qexpg(p, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
rexpg(n, g, param, location = TRUE)
qqexpg(mydata, g, location = TRUE, method)
mpsexpg(mydata, g, location = TRUE, method, sig.level)

Arguments

g	The name of family's pdf including: "birnbaum-saunders", "burrxii", "chisq", "chen", "exp", "f", "frechet", "gamma", "gompetrz", "lfr", "log-normal", "log-logistic", "lomax", "rayleigh", and "weibull".
p	a vector of value(s) between 0 and 1 at which the quantile needs to be computed.
n	number of realizations to be generated.
mydata	Vector of observations.
param	parameter vector $\Theta=(a,\theta,\mu)$
location	If FALSE, then the location parameter will be omitted.
log	If TRUE, then log(pdf) is returned.
log.p	If TRUE, then log(cdf) is returned and quantile is computed for exp(-p).
lower.tail	If FALSE, then 1-cdf is returned and quantile is computed for 1-p.
method	The used method for maximizing the sum of log-spacing function. It will be "BFGS", "CG", "L-BFGS-B", "Nelder-Mead", or "SANN".
sig.level	Significance level for the Chi-square goodness-of-fit test.

Details

It can be shown that the Moran's statistic follows a normal distribution. Also, a chi-square approximation exists for small samples whose mean and variance approximately are m(log(m)+0.57722)-0.5-1/(12m) and m($\pi^2$/6-1)-0.5-1/(6m), respectively, with m=n+1, see Cheng and Stephens (1989). So, a hypothesis tesing can be constructed based on a sample of n independent realizations at the given significance level, indicated in above as sig.level.

Value

1. A vector of the same length as mydata, giving the pdf values computed at mydata.

2. A vector of the same length as mydata, giving the cdf values computed at mydata.

3. A vector of the same length as p, giving the quantile values computed at p.

4. A vector of the same length as n, giving the random numbers realizations.

5. A sequence of goodness-of-fit statistics such as: Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn information criterion (HQIC), Cramer-von Misses statistic (CM), Anderson Darling statistic (AD), log-likelihood statistic (log), and Moran's statistic (M). The Kolmogorov-Smirnov (KS) test statistic and corresponding p-value. The Chi-square test statistic, critical upper tail Chi-square distribution, related p-value, and the convergence status.

Author(s)

Mahdi Teimouri

References

Cheng, R. C. H. and Stephens, M. A. (1989). A goodness-of-fit test using Moran's statistic with estimated parameters, Biometrika, 76 (2), 385-392.

Gupta, R. C., Gupta, P. L., and Gupta, R. D. (1998). Modeling failure time data by Lehman alternatives, Communications in Statistics-Theory and Methods, 27, 887-904.

Examples

mydata<-rweibull(100,shape=2,scale=2)+3
dexpg(mydata, "weibull", c(1,2,2,3))
pexpg(mydata, "weibull", c(1,2,2,3))
qexpg(runif(100), "weibull", c(1,2,2,3))
rexpg(100, "weibull", c(1,2,2,3))
qqexpg(mydata, "weibull", TRUE, "Nelder-Mead")
mpsexpg(mydata, "weibull", TRUE, "Nelder-Mead", 0.05)

---

exponentiated generalized G distribution

Description

Computes the pdf, cdf, quantile, and random numbers, draws the q-q plot, and estimates the parameters of the exponentiated generalized G distribution. The General form for the probability density function (pdf) of the exponentiated generalized G distribution due to Cordeiro et al. (2013) is given by

$f(x,{\Theta}) = a\,b\,g(x-\mu,\theta ){\left( {1 - G(x-\mu,\theta )} \right)^{a - 1}}{\left[ {1 - {{\left( {1 - G(x-\mu,\theta )} \right)}^a}} \right]^{b - 1}},$

where $\theta$ is the baseline family parameter vector. Also, a>0, b>0, and $\mu$ are the extra parameters induced to the baseline cumulative distribution function (cdf) G whose pdf is g. The general form for the cumulative distribution function (cdf) of the exponentiated generalized G distribution is given by

$F(x,{\Theta}) = {\left[ {1 - {{\left( {1 - G(x-\mu,\theta )} \right)}^a}} \right]^b}.$

Here, the baseline G refers to the cdf of famous families such as: Birnbaum-Saunders, Burr type XII, Exponential, Chen, Chisquare, F, Frechet, Gamma, Gompertz, Linear failure rate (lfr), Log-normal, Log-logistic, Lomax, Rayleigh, and Weibull. The parameter vector is $\Theta=(a,b,\theta,\mu)$ where $\theta$ is the baseline G family's parameter space. If $\theta$ consists of the shape and scale parameters, the last component of $\theta$ is the scale parameter (here, a and b are the first and second shape parameters). Always, the location parameter $\mu$ is placed in the last component of $\Theta$.

Usage

dexpgg(mydata, g, param, location = TRUE, log=FALSE)
pexpgg(mydata, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
qexpgg(p, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
rexpgg(n, g, param, location = TRUE)
qqexpgg(mydata, g, location = TRUE, method)
mpsexpgg(mydata, g, location = TRUE, method, sig.level)

Arguments

g	The name of family's pdf including: "birnbaum-saunders", "burrxii", "chisq", "chen", "exp", "f", "frechet", "gamma", "gompetrz", "lfr", "log-normal", "log-logistic", "lomax", "rayleigh", and "weibull".
p	a vector of value(s) between 0 and 1 at which the quantile needs to be computed.
n	number of realizations to be generated.
mydata	Vector of observations.
param	parameter vector $\Theta=(a,b,\theta,\mu)$
location	If FALSE, then the location parameter will be omitted.
log	If TRUE, then log(pdf) is returned.
log.p	If TRUE, then log(cdf) is returned and quantile is computed for exp(-p).
lower.tail	If FALSE, then 1-cdf is returned and quantile is computed for 1-p.
method	The used method for maximizing the sum of log-spacing function. It will be "BFGS", "CG", "L-BFGS-B", "Nelder-Mead", or "SANN".
sig.level	Significance level for the Chi-square goodness-of-fit test.

Details

It can be shown that the Moran's statistic follows a normal distribution. Also, a chi-square approximation exists for small samples whose mean and variance approximately are m(log(m)+0.57722)-0.5-1/(12m) and m($\pi^2$/6-1)-0.5-1/(6m), respectively, with m=n+1, see Cheng and Stephens (1989). So, a hypothesis tesing can be constructed based on a sample of n independent realizations at the given significance level, indicated in above as sig.level.

Value

1. A vector of the same length as mydata, giving the pdf values computed at mydata.

2. A vector of the same length as mydata, giving the cdf values computed at mydata.

3. A vector of the same length as p, giving the quantile values computed at p.

4. A vector of the same length as n, giving the random numbers realizations.

5. A sequence of goodness-of-fit statistics such as: Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn information criterion (HQIC), Cramer-von Misses statistic (CM), Anderson Darling statistic (AD), log-likelihood statistic (log), and Moran's statistic (M). The Kolmogorov-Smirnov (KS) test statistic and corresponding p-value. The Chi-square test statistic, critical upper tail Chi-square distribution, related p-value, and the convergence status.

Author(s)

Mahdi Teimouri

References

Cheng, R. C. H. and Stephens, M. A. (1989). A goodness-of-fit test using Moran's statistic with estimated parameters, Biometrika, 76 (2), 385-392.

Cordeiro, G. M., Ortega, E. M. M., and da Cunha, D. C. C. (2013). The exponentiated generalized class of distributions, Journal of Data Science, 11, 1-27.

Examples

mydata<-rweibull(100,shape=2,scale=2)+3
dexpgg(mydata, "weibull", c(1,1,2,2,3))
pexpgg(mydata, "weibull", c(1,1,2,2,3))
qexpgg(runif(100), "weibull", c(1,1,2,2,3))
rexpgg(100, "weibull", c(1,1,2,2,3))
qqexpgg(mydata, "weibull", TRUE, "Nelder-Mead")
mpsexpgg(mydata, "weibull", TRUE, "Nelder-Mead", 0.05)

---

exponentiated Kumaraswamy G distribution

Description

Computes the pdf, cdf, quantile, and random numbers, draws the q-q plot, and estimates the parameters of the exponentiated Kumaraswamy G distribution. The General form for the probability density function (pdf) of exponentiated Kumaraswamy G distribution due to Lemonte et al. (2013) is given by

$f(x,{\Theta}) = a\,b\,d\,g(x-\mu,\theta ){\left( {G(x-\mu,\theta )} \right)^{a - 1}}{\left[ {1 - {{\left( {G(x-\mu,\theta )} \right)}^a}} \right]^{b - 1}}{\left\{ {1 - {{\left[ {1 - {{\left( {G(x-\mu,\theta )} \right)}^a}} \right]}^b}} \right\}^{d - 1}},$

where $\theta$ is the baseline family parameter vector. Also, a>0, b>0, d>0, and $\mu$ are the extra parameters induced to the baseline cumulative distribution function (cdf) G whose pdf is g. The general form for the cumulative distribution function (cdf) of the exponentiated Kumaraswamy G distribution is given by

$F(x,{\Theta}) = {\left\{ {1 - {{\left[ {1 - {{\left( {G(x-\mu,\theta )} \right)}^a}} \right]}^b}} \right\}^d}.$

Here, the baseline G refers to the cdf of famous families such as: Birnbaum-Saunders, Burr type XII, Exponential, Chen, Chisquare, F, Frechet, Gamma, Gompertz, Linear failure rate (lfr), Log-normal, Log-logistic, Lomax, Rayleigh, and Weibull. The parameter vector is $\Theta=(a,b,d,\theta,\mu)$ where $\theta$ is the baseline G family's parameter space. If $\theta$ consists of the shape and scale parameters, the last component of $\theta$ is the scale parameter (here, a, b, and d are the first, second, and the third shape parameters). Always, the location parameter $\mu$ is placed in the last component of $\Theta$.

Usage

dexpkumg(mydata, g, param, location = TRUE, log=FALSE)
pexpkumg(mydata, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
qexpkumg(p, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
rexpkumg(n, g, param, location = TRUE)
qqexpkumg(mydata, g, location = TRUE, method)
mpsexpkumg(mydata, g, location = TRUE, method, sig.level)

Arguments

g	The name of family's pdf including: "birnbaum-saunders", "burrxii", "chisq", "chen", "exp", "f", "frechet", "gamma", "gompetrz", "lfr", "log-normal", "log-logistic", "lomax", "rayleigh", and "weibull".
p	a vector of value(s) between 0 and 1 at which the quantile needs to be computed.
n	number of realizations to be generated.
mydata	Vector of observations.
param	parameter vector $\Theta=(a,b,d,\theta,\mu)$
location	If FALSE, then the location parameter will be omitted.
log	If TRUE, then log(pdf) is returned.
log.p	If TRUE, then log(cdf) is returned and quantile is computed for exp(-p).
lower.tail	If FALSE, then 1-cdf is returned and quantile is computed for 1-p.
method	The used method for maximizing the sum of log-spacing function. It will be "BFGS", "CG", "L-BFGS-B", "Nelder-Mead", or "SANN".
sig.level	Significance level for the Chi-square goodness-of-fit test.

Details

It can be shown that the Moran's statistic follows a normal distribution. Also, a chi-square approximation exists for small samples whose mean and variance approximately are m(log(m)+0.57722)-0.5-1/(12m) and m($\pi^2$/6-1)-0.5-1/(6m), respectively, with m=n+1, see Cheng and Stephens (1989). So, a hypothesis tesing can be constructed based on a sample of n independent realizations at the given significance level, indicated in above as sig.level.

Value

1. A vector of the same length as mydata, giving the pdf values computed at mydata.

2. A vector of the same length as mydata, giving the cdf values computed at mydata.

3. A vector of the same length as p, giving the quantile values computed at p.

4. A vector of the same length as n, giving the random numbers realizations.

5. A sequence of goodness-of-fit statistics such as: Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn information criterion (HQIC), Cramer-von Misses statistic (CM), Anderson Darling statistic (AD), log-likelihood statistic (log), and Moran's statistic (M). The Kolmogorov-Smirnov (KS) test statistic and corresponding p-value. The Chi-square test statistic, critical upper tail Chi-square distribution, related p-value, and the convergence status.

Author(s)

Mahdi Teimouri

References

Cheng, R. C. H. and Stephens, M. A. (1989). A goodness-of-fit test using Moran's statistic with estimated parameters, Biometrika, 76 (2), 385-392.

Lemonte, A. J., Barreto-Souza, W., and Cordeiro, G. M. (2013). The exponentiated Kumaraswamy distribution and its log-transform, Brazilian Journal of Probability and Statistics, 27, 31-53.

Examples

mydata<-rweibull(100,shape=2,scale=2)+3
dexpkumg(mydata, "weibull", c(1,1,1,2,2,3))
pexpkumg(mydata, "weibull", c(1,1,1,2,2,3))
qexpkumg(runif(100), "weibull", c(1,1,1,2,2,3))
rexpkumg(100, "weibull", c(1,1,1,2,2,3))
qqexpkumg(mydata, "weibull", TRUE, "Nelder-Mead")
mpsexpkumg(mydata, "weibull", TRUE, "Nelder-Mead", 0.05)

---

gamma uniform G distribution

Description

Computes the pdf, cdf, quantile, and random numbers, draws the q-q plot, and estimates the parameters of the gamma uniform G distribution. General form for the probability density function (pdf) of the gamma uniform G distribution due to Torabi and Montazeri (2012) is given by

$f(x,{\Theta}) = \frac{{g(x-\mu,\theta )}}{{\Gamma (a){{\left( {1 - G(x-\mu,\theta )} \right)}^2}}}{\left( {\frac{{G(x-\mu,\theta )}}{{1 - G(x-\mu,\theta )}}} \right)^{a - 1}}{e^{ - \,\,\frac{{G(x-\mu,\theta )}}{{1 - G(x-\mu,\theta )}}}},$

where $\theta$ is the baseline family parameter vector. Also, a>0 and $\mu$ are the extra parameters induced to the baseline cumulative distribution function (cdf) G whose pdf is g. The general form for the cumulative distribution function (cdf) of the gamma uniform G distribution is given by

$F(x,{\Theta}) = \int_0^{\frac{{G(x-\mu,\theta )}}{{1 - G(x-\mu,\theta )}}} {\frac{{{y^{a - 1}}{e^{ - y}}}}{{\Gamma (a)}}dy}.$

Here, the baseline G refers to the cdf of famous families such as: Birnbaum-Saunders, Burr type XII, Exponential, Chen, Chisquare, F, Frechet, Gamma, Gompertz, Linear failure rate (lfr), Log-normal, Log-logistic, Lomax, Rayleigh, and Weibull. The parameter vector is $\Theta=(a,\theta,\mu)$ where $\theta$ is the baseline G family's parameter space. If $\theta$ consists of the shape and scale parameters, the last component of $\theta$ is the scale parameter (here, a is the shape parameter). Always, the location parameter $\mu$ is placed in the last component of $\Theta$.

Usage

dgammag(mydata, g, param, location = TRUE, log=FALSE)
pgammag(mydata, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
qgammag(p, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
rgammag(n, g, param, location = TRUE)
qqgammag(mydata, g, location = TRUE, method)
mpsgammag(mydata, g, location = TRUE, method, sig.level)

Arguments

g	The name of family's pdf including: "birnbaum-saunders", "burrxii", "chisq", "chen", "exp", "f", "frechet", "gamma", "gompetrz", "lfr", "log-normal", "log-logistic", "lomax", "rayleigh", and "weibull".
p	a vector of value(s) between 0 and 1 at which the quantile needs to be computed.
n	number of realizations to be generated.
mydata	Vector of observations.
param	parameter vector $\Theta=(a,\theta,\mu)$
location	If FALSE, then the location parameter will be omitted.
log	If TRUE, then log(pdf) is returned.
log.p	If TRUE, then log(cdf) is returned and quantile is computed for exp(-p).
lower.tail	If FALSE, then 1-cdf is returned and quantile is computed for 1-p.
method	The used method for maximizing the sum of log-spacing function. It will be "BFGS", "CG", "L-BFGS-B", "Nelder-Mead", or "SANN".
sig.level	Significance level for the Chi-square goodness-of-fit test.

Details

It can be shown that the Moran's statistic follows a normal distribution. Also, a chi-square approximation exists for small samples whose mean and variance approximately are m(log(m)+0.57722)-0.5-1/(12m) and m($\pi^2$/6-1)-0.5-1/(6m), respectively, with m=n+1, see Cheng and Stephens (1989). So, a hypothesis tesing can be constructed based on a sample of n independent realizations at the given significance level, indicated in above as sig.level.

Value

1. A vector of the same length as mydata, giving the pdf values computed at mydata.

2. A vector of the same length as mydata, giving the cdf values computed at mydata.

3. A vector of the same length as p, giving the quantile values computed at p.

4. A vector of the same length as n, giving the random numbers realizations.

5. A sequence of goodness-of-fit statistics such as: Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn information criterion (HQIC), Cramer-von Misses statistic (CM), Anderson Darling statistic (AD), log-likelihood statistic (log), and Moran's statistic (M). The Kolmogorov-Smirnov (KS) test statistic and corresponding p-value. The Chi-square test statistic, critical upper tail Chi-square distribution, related p-value, and the convergence status.

Author(s)

Mahdi Teimouri

References

Cheng, R. C. H. and Stephens, M. A. (1989). A goodness-of-fit test using Moran's statistic with estimated parameters, Biometrika, 76 (2), 385-392.

Torabi, H. and Montazeri, N. H. (2012). The gamma uniform distribution and its applications, Kybernetika, 48, 16-30.

Examples

mydata<-rweibull(100,shape=2,scale=2)+3
dgammag(mydata, "weibull", c(1,2,2,3))
pgammag(mydata, "weibull", c(1,2,2,3))
qgammag(runif(100), "weibull", c(1,2,2,3))
rgammag(100, "weibull", c(1,2,2,3))
qqgammag(mydata, "weibull", TRUE, "Nelder-Mead")
mpsgammag(mydata, "weibull", TRUE, "Nelder-Mead", 0.05)

---

gamma uniform type I G distribution

Description

Computes the pdf, cdf, quantile, and random numbers, draws the q-q plot, and estimates the parameters of the gamma uniform type I G distribution. General form for the probability density function (pdf) of the gamma uniform type I G distribution due to Zografos and Balakrishnan (2009) is given by

$f(x,{\Theta}) = \frac{{g(x-\mu,\theta )}}{{\Gamma \left( a \right)}}{\left[ { - \log \left( {1 - G(x-\mu,\theta )} \right)} \right]^{a - 1}},$

where $\theta$ is the baseline family parameter vector. Also, a>0 and $\mu$ are the extra parameters induced to the baseline cumulative distribution function (cdf) G whose pdf is g. The general form for the cumulative distribution function (cdf) of the gamma uniform type I G distribution is given by

$F(x,{\Theta}) = \int_0^{ - \log \left( {1 - G(x-\mu,\theta )} \right)} {\frac{{{y^{a - 1}}{e^{ - y}}}}{{\Gamma (a)}}dy}.$

Here, the baseline G refers to the cdf of famous families such as: Birnbaum-Saunders, Burr type XII, Exponential, Chen, Chisquare, F, Frechet, Gamma, Gompertz, Linear failure rate (lfr), Log-normal, Log-logistic, Lomax, Rayleigh, and Weibull. The parameter vector is $\Theta=(a,\theta,\mu)$ where $\theta$ is the baseline G family's parameter space. If $\theta$ consists of the shape and scale parameters, the last component of $\theta$ is the scale parameter (here, a is the shape parameter). Always, the location parameter $\mu$ is placed in the last component of $\Theta$.

Usage

dgammag1(mydata, g, param, location = TRUE, log=FALSE)
pgammag1(mydata, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
qgammag1(p, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
rgammag1(n, g, param, location = TRUE)
qqgammag1(mydata, g, location = TRUE, method)
mpsgammag1(mydata, g, location = TRUE, method, sig.level)

Arguments

g	The name of family's pdf including: "birnbaum-saunders", "burrxii", "chisq", "chen", "exp", "f", "frechet", "gamma", "gompetrz", "lfr", "log-normal", "log-logistic", "lomax", "rayleigh", and "weibull".
p	a vector of value(s) between 0 and 1 at which the quantile needs to be computed.
n	number of realizations to be generated.
mydata	Vector of observations.
param	parameter vector $\Theta=(a,\theta,\mu)$
location	If FALSE, then the location parameter will be omitted.
log	If TRUE, then log(pdf) is returned.
log.p	If TRUE, then log(cdf) is returned and quantile is computed for exp(-p).
lower.tail	If FALSE, then 1-cdf is returned and quantile is computed for 1-p.
method	The used method for maximizing the sum of log-spacing function. It will be "BFGS", "CG", "L-BFGS-B", "Nelder-Mead", or "SANN".
sig.level	Significance level for the Chi-square goodness-of-fit test.

Details

It can be shown that the Moran's statistic follows a normal distribution. Also, a chi-square approximation exists for small samples whose mean and variance approximately are m(log(m)+0.57722)-0.5-1/(12m) and m($\pi^2$/6-1)-0.5-1/(6m), respectively, with m=n+1, see Cheng and Stephens (1989). So, a hypothesis tesing can be constructed based on a sample of n independent realizations at the given significance level, indicated in above as sig.level.

Value

1. A vector of the same length as mydata, giving the pdf values computed at mydata.

2. A vector of the same length as mydata, giving the cdf values computed at mydata.

3. A vector of the same length as p, giving the quantile values computed at p.

4. A vector of the same length as n, giving the random numbers realizations.

5. A sequence of goodness-of-fit statistics such as: Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn information criterion (HQIC), Cramer-von Misses statistic (CM), Anderson Darling statistic (AD), log-likelihood statistic (log), and Moran's statistic (M). The Kolmogorov-Smirnov (KS) test statistic and corresponding p-value. The Chi-square test statistic, critical upper tail Chi-square distribution, related p-value, and the convergence status.

Author(s)

Mahdi Teimouri

References

Cheng, R. C. H. and Stephens, M. A. (1989). A goodness-of-fit test using Moran's statistic with estimated parameters, Biometrika, 76 (2), 385-392.

Zografos, K. and Balakrishnan, N. (2009). On families of beta- and generalized gamma-generated distributions and associated inference, Statistical Methodology, 6, 344-362.

Examples

mydata<-rweibull(100,shape=2,scale=2)+3
dgammag1(mydata, "weibull", c(1,2,2,3))
pgammag1(mydata, "weibull", c(1,2,2,3))
qgammag1(runif(100), "weibull", c(1,2,2,3))
rgammag1(100, "weibull", c(1,2,2,3))
qqgammag1(mydata, "weibull", TRUE, "Nelder-Mead")
mpsgammag1(mydata, "weibull", TRUE, "Nelder-Mead", 0.05)

---

gamma uniform type II G distribution

Description

Computes the pdf, cdf, quantile, and random numbers, draws the q-q plot, and estimates the parameters of the gamma uniform type II G distribution . General form for the probability density function (pdf) of the gamma uniform type II G distribution due to Ristic and Balakrishnan (2012) is given by

$f(x,{\Theta}) = \frac{{g(x-\mu,\theta )}}{{\Gamma \left( a \right)}}{\left[ { - \log \left( {G(x-\mu,\theta )} \right)} \right]^{a - 1}},$

where $\theta$ is the baseline family parameter vector. Also, a>0 and $\mu$ are the extra parameters induced to the baseline cumulative distribution function (cdf) G whose pdf is g. The general form for the cumulative distribution function (cdf) of the gamma uniform type II G distribution is given by

$F(x,{\Theta}) = \int_0^{ - \log \left( {G(x-\mu,\theta )} \right)} {\frac{{{y^{a - 1}}{e^{ - y}}}}{{\Gamma (a)}}dy}.$

Here, the baseline G refers to the cdf of famous families such as: Birnbaum-Saunders, Burr type XII, Exponential, Chen, Chisquare, F, Frechet, Gamma, Gompertz, Linear failure rate (lfr), Log-normal, Log-logistic, Lomax, Rayleigh, and Weibull. The parameter vector is $\Theta=(a,\theta,\mu)$ where $\theta$ is the baseline G family's parameter space. If $\theta$ consists of the shape and scale parameters, the last component of $\theta$ is the scale parameter (here, a is the shape parameter). Always, the location parameter $\mu$ is placed in the last component of $\Theta$.

Usage

dgammag2(mydata, g, param, location = TRUE, log=FALSE)
pgammag2(mydata, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
qgammag2(p, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
rgammag2(n, g, param, location = TRUE)
qqgammag2(mydata, g, location = TRUE, method)
mpsgammag2(mydata, g, location = TRUE, method, sig.level)

Arguments

g	The name of family's pdf including: "birnbaum-saunders", "burrxii", "chisq", "chen", "exp", "f", "frechet", "gamma", "gompetrz", "lfr", "log-normal", "log-logistic", "lomax", "rayleigh", and "weibull".
p	a vector of value(s) between 0 and 1 at which the quantile needs to be computed.
n	number of realizations to be generated.
mydata	Vector of observations.
param	parameter vector $\Theta=(a,\theta,\mu)$
location	If FALSE, then the location parameter will be omitted.
log	If TRUE, then log(pdf) is returned.
log.p	If TRUE, then log(cdf) is returned and quantile is computed for exp(-p).
lower.tail	If FALSE, then 1-cdf is returned and quantile is computed for 1-p.
method	The used method for maximizing the sum of log-spacing function. It will be "BFGS", "CG", "L-BFGS-B", "Nelder-Mead", or "SANN".
sig.level	Significance level for the Chi-square goodness-of-fit test.

Details

It can be shown that the Moran's statistic follows a normal distribution. Also, a chi-square approximation exists for small samples whose mean and variance approximately are m(log(m)+0.57722)-0.5-1/(12m) and m($\pi^2$/6-1)-0.5-1/(6m), respectively, with m=n+1, see Cheng and Stephens (1989). So, a hypothesis tesing can be constructed based on a sample of n independent realizations at the given significance level, indicated in above as sig.level.

Value

1. A vector of the same length as mydata, giving the pdf values computed at mydata.

2. A vector of the same length as mydata, giving the cdf values computed at mydata.

3. A vector of the same length as p, giving the quantile values computed at p.

4. A vector of the same length as n, giving the random numbers realizations.

5. A sequence of goodness-of-fit statistics such as: Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn information criterion (HQIC), Cramer-von Misses statistic (CM), Anderson Darling statistic (AD), log-likelihood statistic (log), and Moran's statistic (M). The Kolmogorov-Smirnov (KS) test statistic and corresponding p-value. The Chi-square test statistic, critical upper tail Chi-square distribution, related p-value, and the convergence status.

Author(s)

Mahdi Teimouri

References

Cheng, R. C. H. and Stephens, M. A. (1989). A goodness-of-fit test using Moran's statistic with estimated parameters, Biometrika, 76 (2), 385-392.

Ristic, M. M. and Balakrishnan, N. (2012). The gamma exponentiated exponential distribution, Journal of Statistical Computation and Simulation, 82, 1191-1206.

Examples

mydata<-rweibull(100,shape=2,scale=2)+3
dgammag2(mydata, "weibull", c(1,2,2,3))
pgammag2(mydata, "weibull", c(1,2,2,3))
qgammag2(runif(100), "weibull", c(1,2,2,3))
rgammag2(100, "weibull", c(1,2,2,3))
qqgammag2(mydata, "weibull", TRUE, "Nelder-Mead")
mpsgammag2(mydata, "weibull", TRUE, "Nelder-Mead", 0.05)

---

generalized beta G distribution

Description

Computes the pdf, cdf, quantile, and random numbers, draws the q-q plot, and estimates the parameters of the generalized beta G distribution. General form for the probability density function (pdf) of the generalized beta G distribution due to Alexander et al. (2012) is given by

$f(x,{\Theta}) = \frac{{d\,g(x-\mu,\theta ){{\left( {G(x-\mu,\theta )} \right)}^{ad - 1}}{{\left[ {1 - {{\left( {G(x-\mu,\theta )} \right)}^d}} \right]}^{b - 1}}}}{{B\left( {a,b} \right)}},$

where $\theta$ is the baseline family parameter vector. Also, a>0, b>0, d>0, and $\mu$ are the extra parameters induced to the baseline cumulative distribution function (cdf) G whose pdf is g. The general form for the cumulative distribution function (cdf) of the generalized beta G distribution is given by

$F(x,{\Theta}) = \frac{{\int_0^{{{\left( {G(x-\mu,\theta )} \right)}^d}} {{y^{a - 1}}{{\left( {1 - y} \right)}^{b - 1}}} dy}}{{B(a,b)}}.$

Here, the baseline G refers to the cdf of famous families such as: Birnbaum-Saunders, Burr type XII, Exponential, Chen, Chisquare, F, Frechet, Gamma, Gompertz, Linear failure rate (lfr), Log-normal, Log-logistic, Lomax, Rayleigh, and Weibull. The parameter vector is $\Theta=(a,b,d,\theta,\mu)$ where $\theta$ is the baseline G family's parameter space. If $\theta$ consists of the shape and scale parameters, the last component of $\theta$ is the scale parameter (here, a, b, and d are the first, second, and the third shape parameters). Always, the location parameter $\mu$ is placed in the last component of $\Theta$.

Usage

dgbetag(mydata, g, param, location = TRUE, log=FALSE)
pgbetag(mydata, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
qgbetag(p, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
rgbetag(n, g, param, location = TRUE)
qqgbetag(mydata, g, location = TRUE, method)
mpsgbetag(mydata, g, location = TRUE, method, sig.level)

Arguments

g	The name of family's pdf including: "birnbaum-saunders", "burrxii", "chisq", "chen", "exp", "f", "frechet", "gamma", "gompetrz", "lfr", "log-normal", "log-logistic", "lomax", "rayleigh", and "weibull".
p	a vector of value(s) between 0 and 1 at which the quantile needs to be computed.
n	number of realizations to be generated.
mydata	Vector of observations.
param	parameter vector $\Theta=(a,b,d,\theta,\mu)$
location	If FALSE, then the location parameter will be omitted.
log	If TRUE, then log(pdf) is returned.
log.p	If TRUE, then log(cdf) is returned and quantile is computed for exp(-p).
lower.tail	If FALSE, then 1-cdf is returned and quantile is computed for 1-p.
method	The used method for maximizing the sum of log-spacing function. It will be "BFGS", "CG", "L-BFGS-B", "Nelder-Mead", or "SANN".
sig.level	Significance level for the Chi-square goodness-of-fit test.

Details

It can be shown that the Moran's statistic follows a normal distribution. Also, a chi-square approximation exists for small samples whose mean and variance approximately are m(log(m)+0.57722)-0.5-1/(12*m) and m($\pi^2$/6-1)-0.5-1/(6m), respectively, with m=n+1, see Cheng and Stephens (1989). So, a hypothesis tesing can be constructed based on a sample of n independent realizations at the given significance level, indicated in above as sig.level.

Value

1. A vector of the same length as mydata, giving the pdf values computed at mydata.

2. A vector of the same length as mydata, giving the cdf values computed at mydata.

3. A vector of the same length as p, giving the quantile values computed at p.

4. A vector of the same length as n, giving the random numbers realizations.

5. A sequence of goodness-of-fit statistics such as: Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn information criterion (HQIC), Cramer-von Misses statistic (CM), Anderson Darling statistic (AD), log-likelihood statistic (log), and Moran's statistic (M). The Kolmogorov-Smirnov (KS) test statistic and corresponding p-value. The Chi-square test statistic, critical upper tail Chi-square distribution, related p-value, and the convergence status.

Author(s)

Mahdi Teimouri

References

Cheng, R. C. H. and Stephens, M. A. (1989). A goodness-of-fit test using Moran's statistic with estimated parameters, Biometrika, 76 (2), 385-392.

Alexander, C., Cordeiro, G. M., and Ortega, E. M. M. (2012). Generalized beta-generated distributions, Computational Statistics and Data Analysis, 56, 1880-1897.

Examples

mydata<-rweibull(100,shape=2,scale=2)+3
dgbetag(mydata, "weibull", c(1,1,1,2,2,3))
pgbetag(mydata, "weibull", c(1,1,1,2,2,3))
qgbetag(runif(100), "weibull", c(1,1,1,2,2,3))
rgbetag(100, "weibull", c(1,1,1,2,2,3))
qqgbetag(mydata, "weibull", TRUE, "Nelder-Mead")
mpsgbetag(mydata, "weibull", TRUE, "Nelder-Mead", 0.05)

---

geometric exponential Poisson G distribution

Description

Computes the pdf, cdf, quantile, and random numbers, draws the q-q plot, and estimates the parameters of the geometric exponential Poisson G distribution. General form for the probability density function (pdf) of the geometric exponential Poisson G distribution due to Nadarajah et al. (2013) is given by

$f(x,{\Theta}) = \frac{{a(1 - b)\,g(x-\mu,\theta )\left( {1 - {e^{ - a}}} \right){e^{ - a + a\,G(x-\mu,\theta )}}}}{{{{\left( {1 - {e^{ - a}} - b + b{e^{ - a + a\,G(x-\mu,\theta )}}} \right)}^2}}},$

where $\theta$ is the baseline family parameter vector. Also, a>0, 0<b<1, and $\mu$ are the extra parameters induced to the baseline cumulative distribution function (cdf) G whose pdf is g. The general form for the cumulative distribution function (cdf) of the geometric exponential Poisson G distribution is given by

$F(x,{\Theta}) = \frac{{{e^{ - a + aG(x-\mu,\theta )}} - {e^{ - a}}}}{{{{{1 - {e^{ - a}} - b + b{e^{ - a + aG(x-\mu,\theta )}}}}}}}.$

Here, the baseline G refers to the cdf of famous families such as: Birnbaum-Saunders, Burr type XII, Exponential, Chen, Chisquare, F, Frechet, Gamma, Gompertz, Linear failure rate (lfr), Log-normal, Log-logistic, Lomax, Rayleigh, and Weibull. The parameter vector is $\Theta=(a,b,\theta,\mu)$ where $\theta$ is the baseline G family's parameter space. If $\theta$ consists of the shape and scale parameters, the last component of $\theta$ is the scale parameter (here, a and b are the first and second shape parameters). Always, the location parameter $\mu$ is placed in the last component of $\Theta$.

Usage

dgexppg(mydata, g, param, location = TRUE, log=FALSE)
pgexppg(mydata, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
qgexppg(p, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
rgexppg(n, g, param, location = TRUE)
qqgexppg(mydata, g, location = TRUE, method)
mpsgexppg(mydata, g, location = TRUE, method, sig.level)

Arguments

g	The name of family's pdf including: "birnbaum-saunders", "burrxii", "chisq", "chen", "exp", "f", "frechet", "gamma", "gompetrz", "lfr", "log-normal", "log-logistic", "lomax", "rayleigh", and "weibull".
p	a vector of value(s) between 0 and 1 at which the quantile needs to be computed.
n	number of realizations to be generated.
mydata	Vector of observations.
param	parameter vector $\Theta=(a,b,\theta,\mu)$
location	If FALSE, then the location parameter will be omitted.
log	If TRUE, then log(pdf) is returned.
log.p	If TRUE, then log(cdf) is returned and quantile is computed for exp(-p).
lower.tail	If FALSE, then 1-cdf is returned and quantile is computed for 1-p.
method	The used method for maximizing the sum of log-spacing function. It will be "BFGS", "CG", "L-BFGS-B", "Nelder-Mead", or "SANN".
sig.level	Significance level for the Chi-square goodness-of-fit test.

Details

It can be shown that the Moran's statistic follows a normal distribution. Also, a chi-square approximation exists for small samples whose mean and variance approximately are m(log(m)+0.57722)-0.5-1/(12m) and m($\pi^2$/6-1)-0.5-1/(6m), respectively, with m=n+1, see Cheng and Stephens (1989). So, a hypothesis tesing can be constructed based on a sample of n independent realizations at the given significance level, indicated in above as sig.level.

Value

1. A vector of the same length as mydata, giving the pdf values computed at mydata.

2. A vector of the same length as mydata, giving the cdf values computed at mydata.

3. A vector of the same length as p, giving the quantile values computed at p.

4. A vector of the same length as n, giving the random numbers realizations.

5. A sequence of goodness-of-fit statistics such as: Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn information criterion (HQIC), Cramer-von Misses statistic (CM), Anderson Darling statistic (AD), log-likelihood statistic (log), and Moran's statistic (M). The Kolmogorov-Smirnov (KS) test statistic and corresponding p-value. The Chi-square test statistic, critical upper tail Chi-square distribution, related p-value, and the convergence status.

Author(s)

Mahdi Teimouri

References

Cheng, R. C. H. and Stephens, M. A. (1989). A goodness-of-fit test using Moran's statistic with estimated parameters, Biometrika, 76 (2), 385-392.

Nadarajah, S., Cancho, V. G., and Ortega, E. M. M. (2013). The geometric exponential Poisson distribution, Statistical Methods & Applications, 22, 355-380.

Examples

mydata<-rweibull(100,shape=2,scale=2)+3
dgexppg(mydata, "weibull", c(1,0.5,2,2,3))
pgexppg(mydata, "weibull", c(1,0.5,2,2,3))
qgexppg(runif(100), "weibull", c(1,0.5,2,2,3))
rgexppg(100, "weibull", c(1,0.5,2,2,3))
qqgexppg(mydata, "weibull", TRUE, "Nelder-Mead")
mpsgexppg(mydata, "weibull", TRUE, "Nelder-Mead", 0.05)

---

gamma-X family of modified beta exponential G distribution

Description

Computes the pdf, cdf, quantile, and random numbers, draws the q-q plot, and estimates the parameters of the gamma-X family of modified beta exponential G distribution. The General form for the probability density function (pdf) of the gamma-X family of the modified beta exponential G distribution due to Alzaatreh et al. (2013) is given by

$f(x,{\Theta}) = abg(x-\mu,\theta ){\left( {1 - G(x-\mu,\theta )} \right)^{ - 2}}{e^{ - b\frac{{G(x-\mu,\theta )}}{{1 - G(x-\mu,\theta )}}}}{\left[ {1 - {e^{ - b\frac{{G(x-\mu,\theta )}}{{1 - G(x-\mu,\theta )}}}}} \right]^{a - 1}},$

where $\theta$ is the baseline family parameter vector. Also, a>0, b>0, and $\mu$ are the extra parameters induced to the baseline cumulative distribution function (cdf) G whose pdf is g. The general form for the cumulative distribution function (cdf) of the gamma-X family of modified beta exponential G distribution is given by

$F(x,{\Theta}) = {\left( {1 - {e^{ - b\frac{{G(x-\mu,\theta )}}{{1 - G(x-\mu,\theta )}}}}} \right)^a}.$

Here, the baseline G refers to the cdf of famous families such as: Birnbaum-Saunders, Burr type XII, Exponential, Chen, Chisquare, F, Frechet, Gamma, Gompertz, Linear failure rate (lfr), Log-normal, Log-logistic, Lomax, Rayleigh, and Weibull. The parameter vector is $\Theta=(a,b,\theta,\mu)$ where $\theta$ is the baseline G family's parameter space. If $\theta$ consists of the shape and scale parameters, the last component of $\theta$ is the scale parameter (here, a and b are the first and second shape parameters). Always, the location parameter $\mu$ is placed in the last component of $\Theta$.

Usage

dgmbetaexpg(mydata, g, param, location = TRUE, log=FALSE)
pgmbetaexpg(mydata, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
qgmbetaexpg(p, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
rgmbetaexpg(n, g, param, location = TRUE)
qqgmbetaexpg(mydata, g, location = TRUE, method)
mpsgmbetaexpg(mydata, g, location = TRUE, method, sig.level)

Arguments

g	The name of family's pdf including: "birnbaum-saunders", "burrxii", "chisq", "chen", "exp", "f", "frechet", "gamma", "gompetrz", "lfr", "log-normal", "log-logistic", "lomax", "rayleigh", and "weibull".
p	a vector of value(s) between 0 and 1 at which the quantile needs to be computed.
n	number of realizations to be generated.
mydata	Vector of observations.
param	parameter vector $\Theta=(a,b,\theta,\mu)$
location	If FALSE, then the location parameter will be omitted.
log	If TRUE, then log(pdf) is returned.
log.p	If TRUE, then log(cdf) is returned and quantile is computed for exp(-p).
lower.tail	If FALSE, then 1-cdf is returned and quantile is computed for 1-p.
method	The used method for maximizing the sum of log-spacing function. It will be "BFGS", "CG", "L-BFGS-B", "Nelder-Mead", or "SANN".
sig.level	Significance level for the Chi-square goodness-of-fit test.

Details

It can be shown that the Moran's statistic follows a normal distribution. Also, a chi-square approximation exists for small samples whose mean and variance approximately are m(log(m)+0.57722)-0.5-1/(12m) and m($\pi^2$/6-1)-0.5-1/(6m), respectively, with m=n+1, see Cheng and Stephens (1989). So, a hypothesis tesing can be constructed based on a sample of n independent realizations at the given significance level, indicated in above as sig.level.

Value

1. A vector of the same length as mydata, giving the pdf values computed at mydata.

2. A vector of the same length as mydata, giving the cdf values computed at mydata.

3. A vector of the same length as p, giving the quantile values computed at p.

4. A vector of the same length as n, giving the random numbers realizations.

5. A sequence of goodness-of-fit statistics such as: Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn information criterion (HQIC), Cramer-von Misses statistic (CM), Anderson Darling statistic (AD), log-likelihood statistic (log), and Moran's statistic (M). The Kolmogorov-Smirnov (KS) test statistic and corresponding p-value. The Chi-square test statistic, critical upper tail Chi-square distribution, related p-value, and the convergence status.

Author(s)

Mahdi Teimouri

References

Cheng, R. C. H. and Stephens, M. A. (1989). A goodness-of-fit test using Moran's statistic with estimated parameters, Biometrika, 76 (2), 385-392.

Alzaatreh, A., Lee, C., and Famoye, F. (2013). A new method for generating families of continuous distributions, Metron, 71, 63-79.

Examples

mydata<-rweibull(100,shape=2,scale=2)+3
dgmbetaexpg(mydata, "weibull", c(1,1,2,2,3))
pgmbetaexpg(mydata, "weibull", c(1,1,2,2,3))
qgmbetaexpg(runif(100), "weibull", c(1,1,2,2,3))
rgmbetaexpg(100, "weibull", c(1,1,2,2,3))
qqgmbetaexpg(mydata, "weibull", TRUE, "Nelder-Mead")
mpsgmbetaexpg(mydata, "weibull", TRUE, "Nelder-Mead", 0.05)

---

exponentiated exponential Poisson G distribution

Description

Computes the pdf, cdf, quantile, and random numbers, draws the q-q plot, and estimates the parameters of the generalized transmuted G distribution. The general form for the probability density function (pdf) of the generalized transmuted G distribution due to Merovci et al. (2017) is given by

$f(x,{\Theta}) = a\,g(x-\mu,\theta ){\left( {G(x-\mu,\theta )} \right)^{a - 1}}\left[ {1 + b - 2bG(x-\mu,\theta )} \right]{\left[ {1 + b\left( {1 - G(x-\mu,\theta )} \right)} \right]^{a - 1}},$

where $\theta$ is the baseline family parameter vector. Also, a>0, b<1, and $\mu$ are the extra parameters induced to the baseline cumulative distribution function (cdf) G whose pdf is g. The general form for the cumulative distribution function (cdf) of the generalized transmuted G distribution distribution is given by

$F(x,{\Theta}) = {\left( {G(x-\mu,\theta )} \right)^a}{\left[ {1 + b\left( {1 - G(x-\mu,\theta )} \right)} \right]^a}.$

Here, the baseline G refers to the cdf of famous families such as: Birnbaum-Saunders, Burr type XII, Exponential, Chen, Chisquare, F, Frechet, Gamma, Gompertz, Linear failure rate (lfr), Log-normal, Log-logistic, Lomax, Rayleigh, and Weibull. The parameter vector is $\Theta=(a,b,\theta,\mu)$ where $\theta$ is the baseline G family's parameter space. If $\theta$ consists of the shape and scale parameters, the last component of $\theta$ is the scale parameter (here, a and b are the first and second shape parameters). Always, the location parameter $\mu$ is placed in the last component of $\Theta$.

Usage

dgtransg(mydata, g, param, location = TRUE, log=FALSE)
pgtransg(mydata, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
qgtransg(p, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
rgtransg(n, g, param, location = TRUE)
qqgtransg(mydata, g, location = TRUE, method)
mpsgtransg(mydata, g, location = TRUE, method, sig.level)

Arguments

g	The name of family's pdf including: "birnbaum-saunders", "burrxii", "chisq", "chen", "exp", "f", "frechet", "gamma", "gompetrz", "lfr", "log-normal", "log-logistic", "lomax", "rayleigh", and "weibull".
p	a vector of value(s) between 0 and 1 at which the quantile needs to be computed.
n	number of realizations to be generated.
mydata	Vector of observations.
param	parameter vector $\Theta=(a,b,\theta,\mu)$
location	If FALSE, then the location parameter will be omitted.
log	If TRUE, then log(pdf) is returned.
log.p	If TRUE, then log(cdf) is returned and quantile is computed for exp(-p).
lower.tail	If FALSE, then 1-cdf is returned and quantile is computed for 1-p.
method	The used method for maximizing the sum of log-spacing function. It will be "BFGS", "CG", "L-BFGS-B", "Nelder-Mead", or "SANN".
sig.level	Significance level for the Chi-square goodness-of-fit test.

Details

It can be shown that the Moran's statistic follows a normal distribution. Also, a chi-square approximation exists for small samples whose mean and variance approximately are m(log(m)+0.57722)-0.5-1/(12m) and m($\pi^2$/6-1)-0.5-1/(6m), respectively, with m=n+1, see Cheng and Stephens (1989). So, a hypothesis tesing can be constructed based on a sample of n independent realizations at the given significance level, indicated in above as sig.level.

Value

1. A vector of the same length as mydata, giving the pdf values computed at mydata.

2. A vector of the same length as mydata, giving the cdf values computed at mydata.

3. A vector of the same length as p, giving the quantile values computed at p.

4. A vector of the same length as n, giving the random numbers realizations.

5. A sequence of goodness-of-fit statistics such as: Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn information criterion (HQIC), Cramer-von Misses statistic (CM), Anderson Darling statistic (AD), log-likelihood statistic (log), and Moran's statistic (M). The Kolmogorov-Smirnov (KS) test statistic and corresponding p-value. The Chi-square test statistic, critical upper tail Chi-square distribution, related p-value, and the convergence status.

Author(s)

Mahdi Teimouri

References

Cheng, R. C. H. and Stephens, M. A. (1989). A goodness-of-fit test using Moran's statistic with estimated parameters, Biometrika, 76 (2), 385-392.

Merovcia, F., Alizadeh, M., Yousof, H. M., and Hamedani, G. G. (2017). The exponentiated transmuted-G family of distributions: Theory and applications, Communications in Statistics-Theory and Methods, 46(21), 10800-10822.

Examples

mydata<-rweibull(100,shape=2,scale=2)+3
dgtransg(mydata, "weibull", c(1,0.5,2,2,3))
pgtransg(mydata, "weibull", c(1,0.5,2,2,3))
qgtransg(runif(100), "weibull", c(1,0.5,2,2,3))
rgtransg(100, "weibull", c(1,0.5,2,2,3))
qqgtransg(mydata, "weibull", TRUE, "Nelder-Mead")
mpsgtransg(mydata, "weibull", TRUE, "Nelder-Mead", 0.05)

---

gamma-X generated of log-logistic-X familiy of G distribution

Description

Computes the pdf, cdf, quantile, and random numbers, draws the q-q plot, and estimates the parameters of the log-logistic-X familiy of G distribution. General form for the probability density function (pdf) of gamma-X generated of the log-logistic-X familiy of G distribution due to Alzaatreh et al. (2013) is given by

$f(x,{\Theta}) = \frac{{ag(x-\mu,\theta ){{\left[ { - \log \left( {1 - G(x-\mu,\theta )} \right)} \right]}^{ - a - 1}}}}{{\left( {1 - G(x,\theta )} \right){{\left\{ {1 + {{\left[ { - \log \left( {1 - G(x,\theta )} \right)} \right]}^a}} \right\}}^2}}},$

where $\theta$ is the baseline family parameter vector. Also, a>0 and $\mu$ are the extra parameters induced to the baseline cumulative distribution function (cdf) G whose pdf is g. It should be noted that here we set $W(G(x,\theta))=-log(1-G(x,\theta ))$. The general form for the cumulative distribution function (cdf) of the gamma-X generated of log-logistic familiy of G distribution is given by

$F(x,{\Theta}) = \frac{1}{{1 + {{\left[ { - \log \left( {1 - G(x,\theta )} \right)} \right]}^{-a}}}}.$

Here, the baseline G refers to the cdf of famous families such as: Birnbaum-Saunders, Burr type XII, Exponential, Chen, Chisquare, F, Frechet, Gamma, Gompertz, Linear failure rate (lfr), Log-normal, Log-logistic, Lomax, Rayleigh, and Weibull. The parameter vector is $\Theta=(a,\theta,\mu)$ where $\theta$ is the baseline G family's parameter space. If $\theta$ consists of the shape and scale parameters, the last component of $\theta$ is the scale parameter (here, a is the shape parameter). Always, the location parameter $\mu$ is placed in the last component of $\Theta$.

Usage

dgxlogisticg(mydata, g, param, location = TRUE, log=FALSE)
pgxlogisticg(mydata, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
qgxlogisticg(p, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
rgxlogisticg(n, g, param, location = TRUE)
qqgxlogisticg(mydata, g, location = TRUE, method)
mpsgxlogisticg(mydata, g, location = TRUE, method, sig.level)

Arguments

g	The name of family's pdf including: "birnbaum-saunders", "burrxii", "chisq", "chen", "exp", "f", "frechet", "gamma", "gompetrz", "lfr", "log-normal", "log-logistic", "lomax", "rayleigh", and "weibull".
p	a vector of value(s) between 0 and 1 at which the quantile needs to be computed.
n	number of realizations to be generated.
mydata	Vector of observations.
param	parameter vector $\Theta=(a,\theta,\mu)$
location	If FALSE, then the location parameter will be omitted.
log	If TRUE, then log(pdf) is returned.
log.p	If TRUE, then log(cdf) is returned and quantile is computed for exp(-p).
lower.tail	If FALSE, then 1-cdf is returned and quantile is computed for 1-p.
method	The used method for maximizing the sum of log-spacing function. It will be "BFGS", "CG", "L-BFGS-B", "Nelder-Mead", or "SANN".
sig.level	Significance level for the Chi-square goodness-of-fit test.

Details

It can be shown that the Moran's statistic follows a normal distribution. Also, a chi-square approximation exists for small samples whose mean and variance approximately are m(log(m)+0.57722)-0.5-1/(12m) and m($\pi^2$/6-1)-0.5-1/(6m), respectively, with m=n+1, see Cheng and Stephens (1989). So, a hypothesis tesing can be constructed based on a sample of n independent realizations at the given significance level, indicated in above as sig.level.

Value

1. A vector of the same length as mydata, giving the pdf values computed at mydata.

2. A vector of the same length as mydata, giving the cdf values computed at mydata.

3. A vector of the same length as p, giving the quantile values computed at p.

4. A vector of the same length as n, giving the random numbers realizations.

5. A sequence of goodness-of-fit statistics such as: Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn information criterion (HQIC), Cramer-von Misses statistic (CM), Anderson Darling statistic (AD), log-likelihood statistic (log), and Moran's statistic (M). The Kolmogorov-Smirnov (KS) test statistic and corresponding p-value. The Chi-square test statistic, critical upper tail Chi-square distribution, related p-value, and the convergence status.

Author(s)

Mahdi Teimouri

References

Cheng, R. C. H. and Stephens, M. A. (1989). A goodness-of-fit test using Moran's statistic with estimated parameters, Biometrika, 76 (2), 385-392.

Alzaatreh, A., Lee, C., and Famoye, F. (2013). A new method for generating families of continuous distributions, Metron, 71, 63-79.

Examples

mydata<-rweibull(100,shape=2,scale=2)+3
dgxlogisticg(mydata, "weibull", c(1,2,2,3))
pgxlogisticg(mydata, "weibull", c(1,2,2,3))
qgxlogisticg(runif(100), "weibull", c(1,2,2,3))
rgxlogisticg(100, "weibull", c(1,2,2,3))
qqgxlogisticg(mydata, "weibull", TRUE, "Nelder-Mead")
mpsgxlogisticg(mydata, "weibull", TRUE, "Nelder-Mead", 0.05)

---

Kumaraswamy G distribution

Description

Computes the pdf, cdf, quantile, and random numbers, draws the q-q plot, and estimates the parameters of the Kumaraswamy G distribution. General form for the probability density function (pdf) of the Kumaraswamy G distribution due to Cordeiro and Castro (2011) is given by

$f(x,{\Theta}) = a\,b\,g(x-\mu,\theta ){\left( {G(x-\mu,\theta )} \right)^{a - 1}}{\left[ {1 - {{\left( {G(x-\mu,\theta )} \right)}^a}} \right]^{b - 1}},$

where $\theta$ is the baseline family parameter vector. Also, a>0, b>0, and $\mu$ are the extra parameters induced to the baseline cumulative distribution function (cdf) G whose pdf is g. The general form for the cumulative distribution function (cdf) of the Kumaraswamy G distribution is given by

$F(x,{\Theta}) = 1 - {\left[ {1 - {{\left( {G(x-\mu,\theta )} \right)}^a}} \right]^b}.$

Here, the baseline G refers to the cdf of famous families such as: Birnbaum-Saunders, Burr type XII, Exponential, Chen, Chisquare, F, Frechet, Gamma, Gompertz, Linear failure rate (lfr), Log-normal, Log-logistic, Lomax, Rayleigh, and Weibull. The parameter vector is $\Theta=(a,b,\theta,\mu)$ where $\theta$ is the baseline G family's parameter space. If $\theta$ consists of the shape and scale parameters, the last component of $\theta$ is the scale parameter (here, a and b are the first and second shape parameters). Always, the location parameter $\mu$ is placed in the last component of $\Theta$.

Usage

dkumg(mydata, g, param, location = TRUE, log=FALSE)
pkumg(mydata, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
qkumg(p, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
rkumg(n, g, param, location = TRUE)
qqkumg(mydata, g, location = TRUE, method)
mpskumg(mydata, g, location = TRUE, method, sig.level)

Arguments

g	The name of family's pdf including: "birnbaum-saunders", "burrxii", "chisq", "chen", "exp", "f", "frechet", "gamma", "gompetrz", "lfr", "log-normal", "log-logistic", "lomax", "rayleigh", and "weibull".
p	a vector of value(s) between 0 and 1 at which the quantile needs to be computed.
n	number of realizations to be generated.
mydata	Vector of observations.
param	parameter vector $\Theta=(a,b,\theta,\mu)$
location	If FALSE, then the location parameter will be omitted.
log	If TRUE, then log(pdf) is returned.
log.p	If TRUE, then log(cdf) is returned and quantile is computed for exp(-p).
lower.tail	If FALSE, then 1-cdf is returned and quantile is computed for 1-p.
method	The used method for maximizing the sum of log-spacing function. It will be "BFGS", "CG", "L-BFGS-B", "Nelder-Mead", or "SANN".
sig.level	Significance level for the Chi-square goodness-of-fit test.

Details

It can be shown that the Moran's statistic follows a normal distribution. Also, a chi-square approximation exists for small samples whose mean and variance approximately are m(log(m)+0.57722)-0.5-1/(12m) and m($\pi^2$/6-1)-0.5-1/(6m), respectively, with m=n+1, see Cheng and Stephens (1989). So, a hypothesis tesing can be constructed based on a sample of n independent realizations at the given significance level, indicated in above as sig.level.

Value

1. A vector of the same length as mydata, giving the pdf values computed at mydata.

2. A vector of the same length as mydata, giving the cdf values computed at mydata.

3. A vector of the same length as p, giving the quantile values computed at p.

4. A vector of the same length as n, giving the random numbers realizations.

5. A sequence of goodness-of-fit statistics such as: Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn information criterion (HQIC), Cramer-von Misses statistic (CM), Anderson Darling statistic (AD), log-likelihood statistic (log), and Moran's statistic (M). The Kolmogorov-Smirnov (KS) test statistic and corresponding p-value. The Chi-square test statistic, critical upper tail Chi-square distribution, related p-value, and the convergence status.

Author(s)

Mahdi Teimouri

References

Cheng, R. C. H. and Stephens, M. A. (1989). A goodness-of-fit test using Moran's statistic with estimated parameters, Biometrika, 76 (2), 385-392.

Cordeiro, G. M. and Castro, M. (2011). A new family of generalized distributions, Journal of Statistical Computation and Simulation, 81, 883-898.

Examples

mydata<-rweibull(100,shape=2,scale=2)+3
dkumg(mydata, "weibull", c(1,1,2,2,3))
pkumg(mydata, "weibull", c(1,1,2,2,3))
qkumg(runif(100), "weibull", c(1,1,2,2,3))
rkumg(100, "weibull", c(1,1,2,2,3))
qqkumg(mydata, "weibull", TRUE, "Nelder-Mead")
mpskumg(mydata, "weibull", TRUE, "Nelder-Mead", 0.05)

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