Saralees Nadarajah

Samuel Kotz

RATIOS OF COMPONENTS FOR WASAN'S BIVARIATE INVERSE GAUSSIAN DISTRIBUTION by

Inverse Gaussian distributions arise naturally in stoehastic processes. In this note, the distribution of the ratio XjiX + Y) is derived when X and Y follow the bivariate inverse Gaussian distribution due to Professor M. T. Wasan [Skandinamsk Aktuarietidshrift, 1968, 69-96].

Tabulations of the associated percentage points and a program for generating them are also given. 1

INTRODUCTION

For given random variables X and Y, the distribution of the ratio X/(X + Y) is of interest in biological and physical sciences, eeonometrics and actuariat sciences. Examples incîude Mendelian inheritance ratios in genetics, mass to energy ratios in nuclear physies, target to control précipitation in meteorology, and inventory ratios in économies. Another important example is the stressstrength model in the context of reliability. It describes the life of a eomponent which has a random strength Y and is subjected to random stress X. The eomponent fails at the instant that the stress applied to it exceeds the strength and the eomponent will function satisfactorily whenever Y > X.

Thus, Pr(X < Y) is a measure of eomponent reliability. It has many applications especialîy in engineering concepts such as structures, détérioration of rocket motors, static fatigue of ceramic components, fatigue fallure of aircraft structures and the aging of concrète pressure vessels.

The distribution of the ratio X/{X + Y) has been studied by several authors especialîy when X and Y are independent random variables and corne from the saine family. For instance, see Marsaglia (1965) and [START_REF] Korhonen | The probability distribution of the ratio of the absolute values of two normal variables[END_REF] for normal family, Press (1969) for Student's t family, [START_REF] Basu | On the distribution of the ratio of two random variables having generalized life distributions[END_REF] for Weibull family, [START_REF] Shcolnick | On the ratio of independent stable random variables[END_REF] for stable family, Hawkins and Han (1986) for non-central chi-squared family, Provost (1989) for gamma family, and Pham-Gia (2000) for beta family.

In this note, we study the distribution of X/(X + Y) when X and Y are correlated inverse gaussian random variables with the joint pdf given by f(%, y) = s(ts)

;====== exp 2tt^/(x -|/)3y3 {x-y-{t-.s)}2 _ (y -s)2 2(x -y) 2y (1)
for x > y > 0 and t > s > 0. This distribution is due to Professor M. T. W&san and arises naturaüy in stochastic processes (see [START_REF] Wasan | On an inverse Gaussian process[END_REF]Wasan ( , 1969aWasan ( , 1969b)), [START_REF] Wasan | Tables of inverse Gaussian percentage points[END_REF], Wasan and Buckholtz (1973), [START_REF] Basu | On the first passage time processes of Brownian motion with positive drift[END_REF], and [START_REF] Buckholtz | Characterization of certain first passage processes[END_REF]). The aim of this paper is to dérivé an explicit expression for the pdf of W -X/(X -f F) and to provide tabulations of the associated percentage points obtained by means of intensive computing power. These values will be of use to the practitioners of the bivariate distribution.

The calculations of this note involve the modified Bessel function of the first kind defined by

"2\ k ru ™ 1 Ir2\K U{x) = 2</r {u +1) ^(y + i)kk\ (t) '
and the modified Bessel function of the third kind defined by

1.' ~h(x)} uK) " 2sin(i/7r)
We also need the following important îemma.

LEMMA 1 (Equation (3.471.9), Gradshteyn and Ryzhik, 2000) For fi > 0 and 7 > 0, J xv~x exp (-fi/xjx) dx 2 (fil'yY''2 Ku ^2^/fiÿJ ■ Purther properties of the above spécial fonctions can be found in Prudnikov et al. (1986) and Gradshteyn and Ryzhik (2000).

PDF

Theorem 1 dérivés the pdf of W -X/(X + F) when X and F are distributed according to (1).

THEOREM 1 If X and Y are jointly distributed according to (1) then

s(t -s) exp(t)\fw fw(w) ir(2w -1)(1 -w)\/(1 -w)(t -s)2 + (2w -l)s2 xK -w)(t -s)2 + (2Vf -l)s2 \l(2w -1)(1 -w) for 1/2 <«;<!. PROOF: From (1), the joint pdf of (R, W) = (X + F, X/R) becomes f(r, w) s(t -s) 2rrr2 \J(2w -1)3(1 -w)3 = exp (t -sf 2 2r(2w -1) 2r(l -w) (2)
Thus, the pdf of W can be written as

s(t -s) exp (t) fw(w) f -2 r exp (f -s) iw -iru -w)*3 jo s(t -s) exp{£) 2\/w(l -w)(2w -ï) 2 2r(2w -1)
2r(lw) dr 2tt\J(2w -1)3(1 -w)3 y/JT-w)^-s)^+ (2îf7-TJ? / y/wy/'(ï ' -'w)(t -8)2 + (2w -1 )s2

x K l y/(2w -l)(îw)

where the second step folîows by application of Lemma 1. The proof is complété.

À

[Figure 1 about here.j

Figure 1 illustrâtes the shape of the pdf (2) for selected values of t and s. The four curves in each plot correspond to selected values of sjt. The efîect of the parameters is évident.

PERCENTILES

In this section, we provide extensive tabulations of the percentiles of the distribution of W. These percentiles are computed numerically by solving the équation J,

s(t -s) exp(t)\/w 1/2 7r(2u; -1)(1 -w)\/(l -w)(t -s)2 + (2w -l)a ( y/Ü)y/{l -w)(t ~S)2 + {2lV -l)s y. K. *V y/(2w -1)(I -w) dw
Evidently, this involves computation of the Bessel fonctions and routines for this are widely available. We used the fonction BesseIK (•) in the algebraic manipulation package, MAPLE. The percentiles are given for q = 0.6, 0.7, 0.8, 0.9, 0.95, 0.99, t -1 and s/t -0.01,0.02,... ,0.50. We hope these mimbers wiîl be of use to the practitioners of the bivariate inverse Gaussian distribution. Siinilar tabulations couîd be easily derived for other values of t and s by using the Plots of the pdf of (2) for s/t = 0.2,0.4,0.6,0.8 and (a): t = 1; (b): t -2; (c): f = 5; and, (d): f = 10. In each plot, the curves from the right to the left correspond to increasing values of s/t.

  Figure1. Plots of the pdf of (2) for s/t = 0.2,0.4,0.6,0.8 and (a): t = 1; (b): t -2; (c): f = 5; and, (d): f = 10. In each plot, the curves from the right to the left correspond to increasing values of s/t.

APPENDIX

The following program in MAPLE can be used to generate tables similar to that presented in ff:=int(f ,w=0.5. .z) : pl:=fsolve(ff=0.6,z=0.5..1): p2:=fsolve(ff=0.7,z=0.5..1): p3:=fsolve(ff=0.8,z=0.5..1): p4:=fsolve(ff=0.9,z=0.5..1): p5:=fsolve(ff=0.96,z=0.5..1): p6:=fsolve(ff=0.99,z=0.5..1): print(s/t,pl,p2,p3,p4,p5,p6);