CB12NB function

Complementary Burr-12 negative binomial distribution

Evaluates the PDF, CDF, QF, random numbers and MLEs based on the complementary Burr-12 negative binomial (CB12NB) distribution. The CDF of the complementary G negative binomial distribution is as follows: [REMOVE_ME]$F(x)=\frac{\left(1-\lambda G(x)\right)^{-s}-1}{(1-\lambda)^{-s}-1};\qquad\lambda\in\left(0,1\right),s>0, [REMOVE_ME_2]$

where G(x) represents the baseline Burr-12 CDF, it is given by [REMOVE_ME]$G\left(x\right)=1-\left[1+\left(\frac{x}{a}\right)^{b}\right]^{-k};\qquad a,b,k>0. [REMOVE_ME_2]$

By setting G(x) in the above Equation, yields the CDF of the CB12NB distribution.

Description

Evaluates the PDF, CDF, QF, random numbers and MLEs based on the complementary Burr-12 negative binomial (CB12NB) distribution. The CDF of the complementary G negative binomial distribution is as follows:

where G(x) represents the baseline Burr-12 CDF, it is given by

By setting G(x) in the above Equation, yields the CDF of the CB12NB distribution.

dCB12NB(x, a, b, k, s, lambda, log = FALSE)
pCB12NB(x, a, b, k, s, lambda, log.p = FALSE, lower.tail = TRUE)
qCB12NB(p, a, b, k, s, lambda, log.p = FALSE, lower.tail = TRUE)
rCB12NB(n, a, b, k, s, lambda)
mCB12NB(x, a, b, k, s, lambda, method="B")

Arguments

• x: A vector of (non-negative integer) values.
• p: A vector of probablities.
• n: The number of random values to be generated under the CB12NB distribution.
• lambda: The strictly positive parameter of the negative binomial distribution $\lambda \in (0,1)$.
• s: The positive parameter of the negative binomial distribution $s > 0$.
• a: The strictly positive scale parameter of the baseline Burr-12 distribution ($a > 0$).
• b: The strictly positive shape parameter of the baseline Burr-12 distribution ($b > 0$).
• k: The strictly positive shape parameter of the baseline Burr-12 distribution ($k > 0$).
• 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 CB12NB 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 CB12NB 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.

Returns

dCB12NB gives the (log) probability function. pCB12NB gives the (log) distribution function. qCB12NB gives the quantile function. rCB12NB generates random values. mCB12NB gives the estimated parameters along with SE and goodness-of-fit measures.

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.

Zimmer, W. J., Keats, J. B., & Wang, F. K. (1998). The Burr XII distribution in reliability analysis. Journal of quality technology, 30(4), 386-394.

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 .

See Also

pCB12Geo

Examples

x<-data_actuarialm
rCB12NB(20,2,0.4,1.2,2,0.2)
dCB12NB(x,2,1,2,2,0.3)
pCB12NB(x,2,1,2,2,0.3)
qCB12NB(0.7,2,1,2,2,0.4)
mCB12NB(x, 2,1,0.2,0.2,0.4, method="B")