BellB12 distribution {ActuarialM}

Bell Burr-12 distribution

Description

Computes the value at risk and expected shortfall based on the Bell Burr-12 (BellB12) 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 Burr-12 CDF, it is given by

K\left(x\right)=1-\left[1+\left(\frac{x}{a}\right)^{b}\right]^{-k};\qquad a,b,k>0.

By setting K(x) in the above Equation, yields the CDF of the BellB12 distribution. The following expression can be used to calculate the VaR:

VaR_{p}(X)=a\left(\left[\left(\frac{1}{\lambda}\left[\ln\left(\left[\ln\left(1-p\left[1-\exp\Bigl(1-e^{\lambda}\Bigr)\right]\right)\right]+e^{\lambda}\right)\right]\right)\right]^{-1/k}-1\right)^{1/b},

where p \in (0,1). The ES can be computed from the following expression:

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

Usage

vBellB12(p, a, b, k, lambda, log.p = FALSE, lower.tail = TRUE)
eBellB12(p, a, b, k, lambda)

Arguments

p

A vector of probablities p \in (0,1).

lambda

The strictly positive parameter of the Bell G family (\lambda > 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-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 BellB12 distribution.

Value

vBellB12 gives the value at risk. eBellB12 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.

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.

See Also

eBellBX, eBellL

Examples

p=runif(10,min=0,max=1)
vBellB12(p,1,1,2,1.2)
eBellB12(p,1,1,2,1.2)