BellW distribution {ActuarialM}

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)

Muhammad Imran and M.H. Tahir.

R implementation and documentation: Muhammad Imran [e-mail] and M.H. Tahir [e-mail].

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.

See Also

eBellEW, eBellE

Examples

p=runif(10,min=0,max=1)
vBellW(p,1,2,1)
eBellW(p,1,2,1)