Biometrika (1992), 79, 2, pp. 416-9
Printed in Great Britain
A new test for mean residual life times
BY IBRAHIM A. AHMAD
Division of Statistics, Northern Illinois University, DeKalb, Illinois 60115, U.S.A.
SUMMARY
The mean residual life at age t is the expected value of the remaining life beyond that age. A
new (/-statistic is derived for testing constant versus decreasing mean residual life. The new test
is easier to compute and performs better for several alternatives than previous tests.
1. INTRODUCTION
Let X be a nonnegative absolutely continuous random variable representing the life of an item
with distribution function F(x) ( x ? 0 ) . Let F(x) = l - F ( x ) and define the mean residual life
function as
-J:
F(u)du/F(x)
(11)
for F(x) > 0, x 5* 0.
An important class of distributions in reliability and life testing studies is the class where n(t)
is decreasing; see Bryson & Siddiqi (1969), Bryson (1974), Blakema & De Haan (1974), Hall &
Wellner (1981), Hollander & Proschan (1975), Langenberger & Srinivasan (1979), Yang (1978)
and Ahmad (1983a, b) among others for discussions, properties, estimates and tests of fi(t).
Bryson (1974) and Hollander & Proschan (1975) discuss testing hypothesis Ho that /i(f) is
constant against the alternative that /x(t) is strictly decreasing. The latter authors obtain a test
based on an L-statistic and discuss its properties. In contrast, in § 2 we present a test statistic
based on a (/-statistic, which is simpler to compute and has higher asymptotic Pitman relative
efficiency for several alternatives, including three alternatives given by Hollander & Proschan
(1975).
2. TESTING FOR DECREASING MEAN RESIDUAL LIFE
2 1 . The U-statistic test
The test presented here depends on a simple observation. If n{x) is decreasing and if /i(x) is
differentiate, then d/u(x)/dx«£0 for x 3=0. But
d»(x)/dx = n'(x) = {/(x),(x) - F\x))/ F\x),
(21)
where v(x) = J F(u)du, with the integral over (x, °o), and/denotes the probability density function
corresponding to F. Thus fi(x) is nonincreasing if and only if/(x)c(x)« F2(x). Hence, a measure
of the deviation from the null hypothesis Ho that fj,(x) is constant is
SF=\ {F\x)-f(x)V(x)}dx= [ {2xF(x)-*(x)}dF(x),
(2-2)
Jo
Jo
where the second equality is obtained by integration by parts assuming that xF 2 (x)->0 as x-»oo.
Many life distributions, e.g. those with bounded densities, satisfy this assumption including those
we list in our efficiency comparisons.
Downloaded from http://biomet.oxfordjournals.org/ at New York University on February 6, 2016
Some key words: Decreasing then increasing mean residual life; Exponential distribution; Hypothesis testing;
Mean residual life.
Miscellanea
417
Let Xx,..., Xn be a random sample from F, let Fn(x) denote the empirical distribution, and
estimate v{x) by
x),
(2-3)
where I(a) = 0 if a =£ 0 and I(a) = 1 if a > 0. We estimate 5 F in (2-2) by
8F= [ {2xFn(x)-i5n(x)}dFn(x) = /.-2 f f (3*,-X))/(X)-X,).
(2-4)
/-u-i
Jo
If we define <^(X,, X2) = (3X, - X 2 )/(X 2 - Xx), then the symmetrized [/-statistic form of 5 F is
given by
(2-5)
2-2. Asymptotic properties
Using results of Hoeffding and others (Serfling, 1980) we have the following.
2-1. (i) As n-»oo, Un converges to SF with probability one.
(ii) As n-»oo, n^(Un — SF) is asymptotically normal with mean 0 and variance
THEOREM
^(3-X1) + 2XlF(Xl)-2\
udF(u)\.
Jx,
J
(iii) If £x = a2 above and
then
var (Un)= 2
(iv) Under Ho, SF = 0 and n*Un is asymptotically normal with mean 0 and variance o-% = \.
(v) If F is continuous with decreasing mean residual life then the Un test is consistent.
Proof, (i) Follows from Theorem A of Serfling (1980, p. 190), while (ii) results from applying
Theorem A of Serfling (1980, p. 192). Parts (iii) and (iv) follow easily by standard calculations.
To prove part (v),firstlet d(x) = xF(x) - v(x). Since F has decreasing mean residual life, d(x) & 0
and since it is not exponential, d(x)> 0 for at least one value of x Let XQ be such that d(x^) > 0
and let
x, = inf {x|x =s Xo, F(x)
Since F is continuous, so is d(x). Thus
d(x.) = x,F(x,) - *(*,) * XoF(xo) - v(xo) = d(Xo) > 0.
(2-7)
Hence there is a real number S > 0 such that d(x, - S) > 0 and F(x,) - F(x, - 5) > 0 since x, is a
point of decrease. We therefore see that SF > 0 and thus Un is consistent.
D
2-3. Asymptotic relative efficiency comparison
The test presented by Hollander & Proschan (1975) is based on the measure of departure from
Ho given by
-JL
F(s)F(t){n(s)-p(t)}dF(s)dF(t).
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Our test rejects Ho for large values of Un.
418
I B R A H I M A.
AHMAD
The test statistic is Vj = A n /X, where An is the L-statistic obtained from A by replacing the
distribution and residual functions by estimates Fn and vn.
We compare the Un and V$ tests by Pitman asymptotic relative efficiency. Consider testing Ho
that F is exponential versus H, that F = FBm, where 6n = 60+kn~^ and F^ is exponential. The
Pitman asymptotic relative efficiency is
e(T l f T2) = M(6)/crl(0)}/{(i'2(0)/cr2(e)},
(2-8)
where, for i = 1, 2, fi\(8) and o-,(8), respectively, denote the derivative of the limit of E(T,) and
the limit of var(T,) as n->oo evaluated at the null hypotheses. The following three distributions
are used for comparison by Hollander & Proschan (1975):
linear failure rate distribution with
F 2 (x) = e x p [ - { x + 0 ( x + e * Weibull distribution with
F 3 (x) = exp(-x 1 ')
(flsl.xsO).
For these three alternatives, exponentiality occurs when 6 = 0,0 = 0 and 0 = 1 respectively. Direct
calculations give the following asymptotic relative efficiencies:
e(Un, V*) = 2-63,
e(Un, V*) = 4-2,
e(t/ B , V*) = 1-43.
The Un test also compares favourably with tests in the literature for the smaller class of increasing
hazard rate as well as for the class of new better than used. To illustrate the point we select the
tests of Proschan & Pyke (1967), Bickel & Doksum (1969), and Ahmad (1975) for the former
class, and Hollander & Proschan (1972) for the latter class. These tests are defined as follows.
(i) Proschan & Pyke test:
where V0=l if D,ssD, with D, = (n-i+l)(Xin-X(t-V)
order statistics.
and X ( 1 ) « . . . « X ( n ) are the sample
(ii) Bickel & Doksum test:
I
i—i y— l
IJ/I
/
/-i
with D, as in (i)
(iii) Ahmad test:
Zn = {4/n(n - l)(n - 2 ) ( n - 3 ) } £ c /{min {Xu,Xh), (
where / ( a , b) = 1 if a > b and is 0 otherwise, and 2 C extends over all combinations 1 « ia « n,
a = 1,2,3,4 such that (I'I , i2) # (i 3 , i4) and i, < i 2 , i3 < i4.
(iv) Hollander & Proschan test:
where / ( a , b) is as in (iii) and 2,. extends over all combinations 1 « ia « n, a = 1,2,3 such that
'i + «2. 'i + '3 and i2 < i3.
Direct calculations of the asymptotic relative efficiencies of the Um test with respect to these
four tests are given in Table 1.
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Makeham distribution with
Miscellanea
419
Table 1. Asymptotic relative efficiencies of Un for
Proschan & Pike test, Bickel & Doksum test, Ahmad
test and Hollander & Proschan test
Test distribution
F,
F2
F3
wn
zn
31754
2-3816
4-3027
40472
4-4055
3-3041
4-2655
4-6628
0-6670
0-5003
10333
1-4037
J.
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[Received January 1991. Revised September 1991]
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