CWBio distribution {ComRiskModel} R Documentation

## Complementary Weibull binomial distribution

### Description

Evaluates the PDF, CDF, QF, random numbers and MLEs based on the complementary Weibull binomial (CWBio) distribution. The CDF of the complementary G binomial distribution is as follows:

 F(x)=\frac{\left[1-\lambda(1-G(x))\right]^{m}-(1-\lambda)^{m}}{1-(1-\lambda)^{m}};\qquad\lambda\in\left(0,1\right),\,m\geq1, 

where G(x) represents the baseline Weibull CDF, it is given by

 G(x)=1-\exp(-\alpha x^{\beta});\qquad\alpha,\beta>0. 

By setting G(x) in the above Equation, yields the CDF of the CWBio distribution.

### Usage

dCWBio(x, alpha, beta, m, lambda, log = FALSE)
pCWBio(x, alpha, beta, m, lambda, log.p = FALSE, lower.tail = TRUE)
qCWBio(p, alpha, beta, m, lambda, log.p = FALSE, lower.tail = TRUE)
rCWBio(n, alpha, beta, m, lambda)
mCWBio(x, alpha, beta, m, lambda, method="B")


### Arguments

 x A vector of (non-negative integer) quantiles. p A vector of probablities. n The number of random values to be generated under the CWBio. lambda The strictly positive parameter of the binomial distribution \lambda \in (0,1). m The positive parameter of the binomial distribution m \geq 1. alpha The strictly positive scale parameter of the baseline Weibull distribution (\alpha > 0). beta The strictly positive shape parameter of the baseline Weibull distribution (\beta > 0). lower.tail if FALSE then 1-F(x) are returned and quantiles are computed 1-p. log if TRUE, probabilities p are given as log(p). log.p if TRUE, probabilities p are given for exp(p). method the procedure for optimizing the log-likelihood function after setting the intial values of the parameters and data values for which the CWBio distribution is fitted. It could be "Nelder-Mead", "BFGS", "CG", "L-BFGS-B", or "SANN". "BFGS" is set as the default.

### Details

These functions allow for the evaluation of the PDF, CDF, QF, random numbers and MLEs of the unknown parameters with the standard error (SE) of the estimates of the CWBio distribution. Additionally, it offers goodness-of-fit statistics such as the AIC, BIC, -2L, A test, W test, Kolmogorov-Smirnov test, P-value, and convergence status.

### Value

dCWBio gives the (log) probability function. pCWBio gives the (log) distribution function. qCWBio gives the quantile function. rCWBio generates random values. mCWBio gives the estimated parameters along with SE and goodness-of-fit measures.

### Author(s)

Muhammad Imran and M.H Tahir.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com and M.H Tahir mht@iub.edu.pk.

### References

Tahir, M. H., & Cordeiro, G. M. (2016). Compounding of distributions: a survey and new generalized classes. Journal of Statistical Distributions and Applications, 3, 1-35.

Hallinan Jr, A. J. (1993). A review of the Weibull distribution. Journal of Quality Technology, 25(2), 85-93.

Rinne, H. (2008). The Weibull distribution: a handbook. CRC press.

pCExpGeo 
x<-rCWBio(20,2,1,2,0.2)