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 [REMOVE_ME]f(x,Θ)=B(a,b)dg(x−μ,θ)[1−(1−G(x−μ,θ))d]b−1(1−G(x−μ,θ))ad−1,[REMOVEME2]
where θ is the baseline family parameter vector. Also, a>0, b>0, d>0, and μ 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 [REMOVE_ME]F(x,Θ)=1−B(a,b)∫0(1−G(x−μ,θ))dya−1(1−y)b−1dy.[REMOVEME2]
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 Θ=(a,b,d,θ,μ) where θ is the baseline G family's parameter space. If θ consists of the shape and scale parameters, the last component of θ is the scale parameter (here, a, b, and d are the first, second, and the third shape parameters). Always, the location parameter μ is placed in the last component of Θ.
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
where θ is the baseline family parameter vector. Also, a>0, b>0, d>0, and μ 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,Θ)=1−B(a,b)∫0(1−G(x−μ,θ))dya−1(1−y)b−1dy.
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 Θ=(a,b,d,θ,μ) where θ is the baseline G family's parameter space. If θ consists of the shape and scale parameters, the last component of θ is the scale parameter (here, a, b, and d are the first, second, and the third shape parameters). Always, the location parameter μ is placed in the last component of Θ.
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 Θ=(a,b,d,θ,μ)
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(π2/6-1)-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.
Returns
A vector of the same length as mydata, giving the pdf values computed at mydata.
A vector of the same length as mydata, giving the cdf values computed at mydata.
A vector of the same length as p, giving the quantile values computed at p.
A vector of the same length as n, giving the random numbers realizations.
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.
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.