# mog: Marshall-Olkin G distribution In MPS: Estimating Through the Maximum Product Spacing Approach

## Description

Computes the pdf, cdf, quantile, and random numbers, draws the q-q plot, and estimates the parameters of the Marshall-Olkin G distribution. General form for the probability density function (pdf) of the Marshall-Olkin G distribution due to Marshall and Olkin (1997) is given by

f(x,{Θ}) = \frac{{ag(x-μ,θ )}}{{{{≤ft[ {1 - ≤ft( {1 - a} \right)≤ft( {1 - G(x-μ,θ )} \right)} \right]}^2}}},

where θ is the baseline family parameter vector. Also, a>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 Marshall-Olkin G distribution is given by

F(x,{Θ}) = 1 - \frac{{a≤ft( {1 - G(x-μ,θ )} \right)}}{{≤ft[ {1 - ≤ft( {1 - a} \right)≤ft( {1 - G(x-μ,θ )} \right)} \right]}}.

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,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. Always, the location parameter μ is placed in the last component of Θ.

## Usage

 1 2 3 4 5 6 dmog(mydata, g, param, location = TRUE, log=FALSE) pmog(mydata, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE) qmog(p, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE) rmog(n, g, param, location = TRUE) qqmog(mydata, g, location = TRUE, method) mpsmog(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 Θ=(a,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(π^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.

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.

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.

## Examples

 1 2 3 4 5 6 7 mydata<-rweibull(100,shape=2,scale=2)+3 dmog(mydata, "weibull", c(0.5,2,2,3)) pmog(mydata, "weibull", c(0.5,2,2,3)) qmog(runif(100), "weibull", c(0.5,2,2,3)) rmog(100, "weibull", c(0.5,2,2,3)) qqmog(mydata, "weibull", TRUE, "Nelder-Mead") mpsmog(mydata, "weibull", TRUE, "Nelder-Mead", 0.05) 

MPS documentation built on Oct. 5, 2019, 1:04 a.m.