BellL distribution {ActuarialM}

Bell Lomax distribution

Description

Computes the value at risk and expected shortfall based on the Bell Lomax (BellL) 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 Lomax CDF, it is given by

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

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

VaR_{p}(X)=b\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)^{-1/q}-1\right],

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

ES_{p}(X)=\frac{b}{p}\intop_{0}^{p}\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)^{-1/q}-1\right]dz.

Usage

vBellL(p, b, q, lambda, log.p = FALSE, lower.tail = TRUE)
eBellL(p, b, q, lambda)

Arguments

p

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

lambda

The strictly positive parameter of the Bell G family (\lambda > 0).

b

The strictly positive scale parameter of the baseline Lomax distribution (b > 0).

q

The strictly positive shape parameter of the baseline Lomax distribution (q > 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 BellL distribution.

Value

vBellL gives the values at risk. eBellL 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.

Kleiber, C., & Kotz, S. (2003). Statistical size distributions in economics and actuarial sciences. John Wiley & Sons.

See Also

eBellBX, eBellB12

Examples

p=runif(10,min=0,max=1)
vBellL(p,1,1,2)
eBellL(p,1,1,2)