BellW distribution {ActuarialM} R Documentation

## Bell Weibull distribution

### Description

Computes the value at risk and expected shortfall based on the Bell Weibull (BellW) distribution. The CDF of the Bell G family is as follows:

 H(x)=\frac{1-\exp\left[-e^{\lambda}\left(1-e^{-\lambda K(x)}\right)\right]}{1-\exp\Bigl(1-e^{\lambda}\Bigr)};\qquad\lambda>0, 

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

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

By setting K(x) in the above Equation, yields the CDF of the BellW distribution. The following expression can be used to calculate the VaR:

VaR_{p}(X)=\left[\frac{-1}{\alpha}\ln\left(\frac{1}{\lambda}\left\{ \ln\left[\ln\left(1-p\left\{ 1-\exp(1-e^{\lambda})\right\} \right)+e^{\lambda}\right]\right\} \right)\right]^{1/\beta};\qquad p\in(0,1).

The ES can be computed from the following expression:

ES_{p}(X)=\frac{1}{p}\intop_{0}^{p}\left[\frac{-1}{\alpha}\ln\left(\frac{1}{\lambda}\left\{ \ln\left[\ln\left(1-z\left\{ 1-\exp(1-e^{\lambda})\right\} \right)+e^{\lambda}\right]\right\} \right)\right]^{1/\beta}dz. 

### Usage

vBellW(p, alpha, beta, lambda, log.p = FALSE, lower.tail = TRUE)
eBellW(p, alpha, beta, lambda)


### Arguments

 p A vector of probablities p \in (0,1). lambda The strictly positive parameter of the Bell G family of distributions \lambda > 0. 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-H(x) are returned and quantiles are computed for 1-p. log.p if TRUE then log(H(x)) are returned and quantiles are computed for exp(p).

### Details

The functions allow to compute the value at risk and the expected shortfall of the BellW distribution.

### Value

vBellW gives the values at risk. eBellW gives the expected shortfall.

### Author(s)

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

### References

Fayomi, A., Tahir, M. H., Algarni, A., Imran, M., & Jamal, F. (2022). A new useful exponential model with applications to quality control and actuarial data. Computational Intelligence and Neuroscience, 2022.

Alsadat, N., Imran, M., Tahir, M. H., Jamal, F., Ahmad, H., & Elgarhy, M. (2023). Compounded Bell-G class of statistical models with applications to COVID-19 and actuarial data. Open Physics, 21(1), 20220242.

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.

eBellEW, eBellE 
p=runif(10,min=0,max=1)