Approximate Finite Forms for the Cases Not Covered by the Finite Representation Approach

Abstract

In this chapter the authors set the guidelines to approach cases not covered by the finite form representations studied in the book, give new Mellin inversion formulas for both the p.d.f. and the c.d.f., and develop sharp upper bounds on the difference between the exact and approximate representations for the Meijer G functions as well as for the differences between the exact and approximate p.d.f.’s and c.d.f.’s of the product of independent Beta r.v.’s.

Appendix: Expressions for the Probability Density and Cumulative Distribution Functions of the GNIG Distribution

Let W be a r.v. with a GIG distribution of depth p, with rate parameters λ ₁, …, λ _p and shape parameters $r_1,\dots ,r_p\in \mathbb {N}$ and let W ^∗∼ Γ(r, λ ^∗), with $r\in {\mathbb R}^+\backslash {\mathbb N}$. Let further W and W ^∗ be two independent r.v.’s. Then the r.v.

$$\displaystyle \begin{aligned} Y=W+W^* \end{aligned}$$

has a Generalized Near-Integer Gamma (GNIG) distribution (Coelho, 2004) of depth p + 1, with rate parameters λ ₁, …, λ _p and λ ^∗ and corresponding shape parameters r ₁, …, r _p and r, with p.d.f.

$$\displaystyle \begin{aligned} \begin{array}{rcl} f_{Y}^{}(y) & = & \displaystyle f^{\scriptscriptstyle \,GNIG}\Big(y\,|\,\{r_j\}_{j=1:p},r;\{\lambda_j\}_{j=1:p},\lambda^*;p+1\Big)\\ & = & \displaystyle K(\lambda^*)^r\sum_{j=1}^p e^{-\lambda_jy}\sum_{k=1}^{r_j}\left\{c_{j,k}\frac{\varGamma(k)}{\varGamma(k+r)}y^{\,k+r-1}\,_1F_1\big(r,k+r,-(\lambda^*-\lambda_j)y\big)\right\} \end{array} \end{aligned} $$

and c.d.f.

$$\displaystyle \begin{aligned} \begin{array}{rcl} F_{Y}^{}(y) & = & \displaystyle F^{\scriptscriptstyle \,GNIG}\Big(y\,|\,\{r_j\}_{j=1:p},r;\{\lambda_j\}_{j=1:p},\lambda^*;p+1\Big)\\ & = & \displaystyle \frac{(\lambda^{*})^r\,y^r}{\varGamma(r+1)}\,_1F_1\big(r,r+1,-\lambda^{*}y\big)\\ & & \displaystyle \hskip .15cm -K(\lambda^{*})^r\sum_{j=1}^p e^{-\lambda_jy}\sum_{k=1}^{r_j}c^*_{j,k}\sum_{i=0}^{k-1}\left\{\frac{y^{\,r+i}\,\lambda_j^i}{\varGamma(r+1+i)}\,_1F_1\big(r,r+1+i,-(\lambda^{*}-\lambda_j)y\big)\right\} \end{array} \end{aligned} $$

for z > 0, where

$$\displaystyle \begin{aligned} K=\prod_{j=1}^p \lambda_j^{r_j}\,,~~~~~~c^*_{j,k}=\frac{c_{j,k}}{\lambda_j^k}\,\varGamma(k) \end{aligned}$$

and

$$\displaystyle \begin{aligned} _1F_1(a,b,y)=\sum_{i=0}^\infty \frac{\varGamma(a+i)}{\varGamma(b+i)}\,\frac{z^i}{i!} \end{aligned}$$

is the Kummer confluent hypergeometric function, which is a function that is quickly and correctly computed even in extended precision by software such as Mathematica^Ⓡ and Maxima.