53
THE INVERSE GAUSSIANDISTRIBUTION
0.5
"0
1
2
1.5
Time t
Failure rate function of the inverse Gaussian distribution (p = 1, and h = I /4,
Fig. 2.26
enters in MTTF = p, as well as into var(T).
Notice that
11
location/scale parameters in the usual sense. Also notice that
1).
Hence (p,h) are nor
K;
A=-
(2.90)
K2
From (2.85) it follows that moments of arbitrarily high (positive) order exist for
IG(/L,A). From the result stated in Problem 2.41, it followsthat moments of arbitrarily
high "negative" order, also exist.
The failure rate function ofIG(p, h) is
(2.91)
for r > 0, 1-1 > 0, and h > 0. In Fig. 2.26 graphs of the failure rate function (2.91)
are given for p = I and selected values of h. Chhikara and Folks (1977, p. 155) have
shown that
(2.92)
Chhikara and Folks ( 1 989, Section 1.3) point out a surprising analogy between
sampling results for the inverse Gaussian distribution IG(p, A) and sampling results
known for the normal distribution X ( F ,s2). If one considers a random sample
T1 T2, .. . , T,, from the inverse Gaussian distribution IG(p, A):
1.
T = 1/ n ClJ=,
C'f=,( l/TJ -
1/T)
r and Cy=l 1(/
T, -
2. nh
3.
TJ is inverse Gaussian IG(p, nh).
1
isx2 distributed with ( n - 1) degrees of freedom.
IT)are independently distributed
54
FAILURE MODELS
4. The term in the exponent of the probability density function of IG(p, h) is
-1/2 times ax2 distributed variable.
The explanation of these and other analogies with the normal distribution has yet not
been revealed.
2.17
THE EXTREME VALUE DISTRIBUTIONS
Extreme value distributions play an important role in reliability analysis. They arise
in a natural way, for example, in the analysis of engineering systems, made up of n
identical items with a series structure, and also in the study of corrosion of metals, of
material strength, and of breakdown of dielectrics.
Let T I ,T2, ...,T,, be independent, identically distributed random variables (not
necessarily life lengths) with a continuous distribution function F T ( ~ )for
, the sake
of simplicity assumed to be strictly increasing for FF1(0) < t < FFI (1). Then
T(I) = min{Tl, T2,..., 7',,} = U,
(2.93)
T(,,)
max{T I ,T2, ..., T, J = V,,
=
(2.94)
are called the extreme values.
The distribution functions of U,, and V,, are easily expressed by FT(.) in the
following way (e.g., see CramCr 1946, p. 370; Mann et al. 1974, p. 102).
Fu,,(u)
Fv,,(~)
1 - (1 - FT(u))" = L,,(u)
=
F T ( u ) ~= H n ( u )
=
(2.95)
(2.96)
Despite the simplicity of (2.95) and (2.96), these formulas are usually not easy to
work with. If F T ( ~ ) ,say, represents a normal distribution, one is led to work with
powers of FT(?),which may be cumbersome.
However, in many practical applications to reliability problems, n is very large.
Hence, one is lead to look for asymptotictechniques, which under general conditions
on FT ( t )may lead to simple representations of Fun(u) and Fv,,(u).
CramCr (1946) suggested the following approach: Introduce
(2.97)
where U,, is defined as in (2.93). Then for y 1 0,
pr(y,
I y)
=
P [I.;.(u,)I
'1
(31
(31
(:))I
=
P [U,, I F;'
=
FU,,
=
1-[I
=
l - ( l - y
[FFI
n
- FT (FF'
(2.98)
55
THE EXTREME VALUE DlSTRlBUTlONS
Asn + 0
Pr(Y, 5 y) + 1 -
for y > 0
(2.99)
Since the right-hand side of (2.99) expresses a distribution f~nction,~continuous for
y > 0, this implies that Y, converges in distribution to a random variable Y , with
distribution function:
F y ( y ) = 1 - e-'
for y >
(2.100)
0
Hence it follows from (2.95) that the sequence of random variables U,, converges in
distribution to the random variable FF'(Y/n).
(2.101)
Similarly,let
where V,, is defined in (2.94). By an analogous argument it can be shown that for
2
> 0.
Pr(Zn 5 z ) = 1 -
(2.103)
which implies that
V,,4 FYI (1 -
f)
(2.104)
for z > 0
(2.105)
where 2 has distribution function
Pr(Z 5 z ) = 1 - e-'
It is to be expected that the limit distribution of U, and V,,will depend on the type
of distribution Fr(.).However, it turns out that there are only three possible types of
limiting distributions for the minimum extreme U,,, and only three possible types of
limiting distributions for the maximum extreme V,,.
For a comprehensive discussion of the application of extreme value theory to
reliability analysis, the reader is referred to Mann et al.
(1974, p.
106), Nelson
(1982, p. 39). Lawless (1982, p. 169), and Johnson and Kotz (1970, Vol. I, Section
2.1). Here we will content ourselves with mentioning three of the possible types of
limiting distributions and indicate areas where they are applied.
3The exponential distribution with parameter
i,=
1