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Journal of Statistical Planning and Inference 37 (1993) 327-337
North-Holland
327
Estimation of the quantile function of residual life time distribution
Khursheed Alam and K.B. Kulasekera
Department of Mathematical Sciences, Clemson Unicersity, Clemson, SC, USA
Received 14 February 1992; revised manuscript received 9 September 1992
Abstract
The estimation of the quantile function of the residual life time distribution is of interest in life testing. Since
the sample quantile function (empirical estimator) of the residual life is discontinuous, we smooth the
function by convolving it with a kernel function. The smoothed function is called a kernel-type estimator.
We study the asymptotic properties of the kernel-type estimator and the empirical estimator. Specifically,
we consider the convergence in mean squared error. Empirical results are given on the small sample
properties of the two estimators, based on a Monte Carlo simulation study.
AMS Subject Classification: 62605, 62N05.
Key words and phrases: Life testing; residual life; quantile function; kernel estimation
1. Introduction
The study of the residual life time distribution arises naturally in life testing.
A parameter of the distribution of particular interest is the mean residual life time. In
some situations it is more appropriate to consider the median residual life time and
more generally, the p-quantile residual life time. Schmittlein and Morrison (1981) have
given an example of the application of the median residual life time in the study of the
duration of strikes in industrial companies in the United States. Gerchak (1984) gives
an example of the p-percentile residual life time in the study of time-to-recidivism of
parolees in the State of Illinois. Of course, the underlying life time distribution can be
known from the knowledge of the p-percentile function for two or more values of
p (Arnold and Brackett, 1983). In this paper we consider the problem of estimating the
p-quantile function of the residual life time distribution from the sample data on the
life time distribution.
Correspondence to: Dr. K.B. Kulasekera, Dept. of Mathematical Sciences, College of Sciences, Clemson University, Martin Hall, Clemson, SC 29634.1907, USA.
037%3758/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved.
328 K. Alam, K.B. KulasekeralEstimation of the quantile function
Let F denote a life time distribution on (0, co) and Q denote the corresponding
quantile function, given by
Q(x)=F-‘(x)
=inf(t>O: F(t)>x).
The (1 -p)-quantile residual life time function is given by
R,(t)=Q[l--p(l-F((t))]-tt, t>O.
We shall suppress the suffix p and write R(t) for RJt). Let
t(l)<...<+,)
denote the ordered observations in a sample from the distribution F. The empirical
version of R(t) is given by
R,(t)=Q,Cl-p(l--F,(t))l-t, (1.1)
where F,(t) denotes the sample distribution function and Q,(y) denotes the sample
quantile function, given by
i-l i Qn(Y)=t(i)t n<Y Gn.
We extend the definition of R(t) and R,(t) over (-co, 0), letting R(u) = Q( 1 -p) - u and
R,(u)=Q,(l -p)-u for ub0.
The function R,(t) is piece-wise linear with jump discontinuities. The function can
be smoothed by convolving it with a kernel function. The smoothed function, called
a kernel-type estimator, is given by
s cc K”(t) = R,(u)K((u - WA) $> (1.2) -CC
where K denotes the kernel and A = A(n) denotes the bandwidth. We shall assume that
K(u) is a probability density function, centered at the origin. The extent of smoothness
is determined by the value of 1. The larger the value of I, the more smooth is the
function &(t). The behavior of the function depends also on the form of the kernel
function K. But this dependence is generally not very crucial.
Csiirgii (1987) derived the asymptotic distribution of R,(t). In this paper we derive
the asymptotic distribution of &(n(t) and the rate of convergence in mean squared error
(MSE) of R,(t) and k,(t). Empirical results are given on the small sample properties of
the two estimates, based on a Monte Carlo study.
Kernel-type estimators of the quantile function of the life time distribution have
been considered by various authors. Falk (1984) has studied the relative deficiency of
the sample quantile function with respect to the kernel type estimators. Padget (1986)
K. Alum, K.B. KulasekeraiEstimation qf the quantile function 329
has studied a Kernel-type estimator from right-censored data, using the product limit
estimator of the life time distribution, due to Kaplan and Meier (1958). Padget and
Thombs (1989) have considered alternative estimators from right censored data.
Chung (1989) has given confidence bands for the percentile residual life time from
right censored data. Feng and Kulasekera (199 1) have studied a smooth version of the
empirical estimator.
The asymptotic distributions of R,(t) and R”,(t) are given in Section 2. In Section 3
we derive the asymptotic convergence in mean squared error of the quantile functions.
The derivation is based on a result due to Alam and Kulasekera (1992) on the
asymptotic convergence of an order statistic. This result is reproduced in Appendix. In
Section 4 we give empirical results to compare the performance of R,(t) and I?“(t).
2. Distribution of R,(t) and k,,(t)
C&go (1987) has considered in length the asymptotic properties of R,(t), as n-03.
He has derived the asymptotic distribution, viewing R,,(t) as (i) a stochastic process in
t for fixed p(O<p< l), (ii) a stochastic process in p for fixed t >O and (iii) a two
parameter process in (p, t). His results provide also asymptotically distribution-free
confidence bands for R,(t). He has given the following results.
Let ,f(t) denote the density function of the life time distribution,
h(r) = Cf(W + Ql - ‘>
and
G(t)=BCl -p(l -W))l-PBCWI,
where B is a Brownian bridge on (0, 1). For a fixed t > 0, G(t) is a normally distributed
with mean 0 and variance v(t)=p(l --p)(l -F(t)). Let t>O and O<p< 1 be fixed and
let f(Q(.)) be positive and continuous at 1 --p(l -F(t)). Then r,(t) is asymptotically
normally distributed with mean 0 and variance v(r).
In addition, let
Assumption 1. (1) f(t)>0 on (Q(l -p),co) and for some y>O
sup x’Q(1 -P)
F(x)(l -F(x))!$# X
(2) f(t) is ultimately nonincreasing in t, as t-co.
Theorem 2.1 (C&go). If Assumption 1 holds then
a.s.
SUP 1r.WWI = o(h,), o<t<m
330 K. Alum, K.B. KulasekeralEstimation of the quantile function
where
6 = log log n 1’4 n
[ 1 n (log n)l’Z,
as n+oo.
We derive the asymptotic distribution of the kernel-type estimator &(t) from
Theorem 2.1 as follows. Let
and [S
m
G(t) = -Co
G(u)h(u)K (y)F],/h(t)
~“;l(t)=~C~“(t)--R”(t)llh(t)
= (2.1)
Here we extend the definition of G(t) over (- co, 0) by letting G(u) = B(l -p) for u < 0.
Suppose that
From (2.1) (2.2) and Theorem 2.1 we have
Theorem 2.2. If (2.2) and Assumption 1 hold then F”(t) is asymptotically distributed
as G”(t). Moreover, as n+oo, jf~(t)-6(t)la~ O(6,).
From the distributional property of the Brownian bridge we have that
ECG(~)G(~)I=PU-P)+(P+P~)(F(~)~F(~))
-P(F(u)fN-PU-F(w)))+(F(w)N-P(l--F(u)))1
= cl(u, w), say. Therefore
E[G(t)]2= co is s m p(u, w) h(u)h(w)K[(u-t)/LJ’C[(w-t)/L] 7 h2(t) -00 -m
= VA(t), say. (2.3)
K. Alam, K.B. KulasekeralEstimation of the quantile function 331
It follows from Theorem 2.2 that f”(t) is asymptotically normally distributed with
mean 0 and variance vi(t). On the other hand, r,(t) is asymptotically normally
distributed with mean 0 and variance v(t), by Theorem 1 of CsijrgB. Therefore
denoting the ratio of the asymptotic variances may be considered as a measure of the
asymptotic relative efficiency of R,(t) and l?“(t).
3. Convergence in Mean Square Error of R,,(t) and l?“(t)
Let q=l-p(l-F(t)),q,=l-p(l-F”(t)) and
T,= sup ~{(Q'(q))-1(Qn(qn)-Q(qn))+(Fn(Q(qn))-q4n)j o<t<m
= sup {r,(t)--J;;(Q'(q))-'(Q(q")-Q(q))+(F,(Q(q,))-q,)}. (3.1) o-=t<m
From Theorem Al of Appendix we deduce that E 1 T,( =o(n- ‘j4) and ET: = o(n- “‘) as 11+co, if condition (i) and (ii) of the theorem are satisfied for x >Q(l -p). Now
nEC(Q'(q))-'(Q(qn)-Q(q))-{(Fn(Q(qn))-4.)'1
=~EC(Q'(q))-'~Q(q,)-QQ(q)~2+~~~(Q(q~))-q~~21 =pZF(t)(l -F(t))+p(l -p)(l -F(t))+o(n_“2)
=pq(l -F(t))+o(n-1’2).
Hence, as n+ cc
where
Let
sup [E(r”(t))* -v*(t)] = o(n- 114), (3.2) O~t<m
v*(t)=pq(l -F(t)). (3.3)
H(u, ~)=C~(u)~(w)lC~--F(u)~(w)lh(u)h(w)K((u--t)/~)K((w--)/~).
Define
v:(t)=(&>‘j;a j_:W, w)dudw+vl(t).
As we derived (3.2) we can show that, as n-+oo,
sup [E(FJt))2-v);(t)]=o(n-“4), o<t<m
when the conditions given for (2.3) and (3.2) are satisfied.
(3.4)
332 K. Alam, K.B. KulasekeralEstimation of the quantile function
To summarize. we have shown that
Theorem 3.1. If conditions (i) and (ii) of Theorem Al and the relation (2.2) are satisjied,
then
and
sup [E(m(t))‘-v*(t)] =o(n- 1’4) o<tico
sup [E(f,,(t))‘-v?(t)] =O(IC”~), oitcm
as n-+cc.
We have that
m-RR(W Is m [R(u)-R(t)]K 7 $ 0 ( )I
(u-t) K /1 X (by condition (ii) Appendix) 2 (“-‘)“” < A2 yo2,
where a2= ST, u2K(u)du. It is assumed that j:, uK(u)du=O. Therefore
nMSE(R”,(t))=nE{k,,(t)-R(t)}2
znE{&(t)--R”(t)J2,
as n--+m and 1-O such that n&+0. Let vX(t)=(avn*(t)/a~)(~=o and let
? (t)=nlMSE(R.(t))--SE(~“,(t))l n
~CWI’
denote the normalized difference between MSE(R,(t)) and MSE(%(t)). From
Theorem 3.1 we have
Corollary 3.1. As n-03 and A-+0 such that A$-+0 then q”(t)-vg(t)+O.
4. Empirical results
Corollary 3.1 gives the asymptotic value of the normalized difference between the
mean squared error of the kernel-type estimator R,,(t) and the empirical estimator
R,(t), as n+cc and A+O. We have carried out a Monte Carlo study to examine the
difference for moderate values of n and A>O. We have drawn the graphs of m(t)
shown below, using 500 simulations for n = 200 and 1=0.1 and 0.25, when the
K. Alam, K.B. Kulasekera /Estimation of the quantile ,function 333
Exponential Distribution 1=0. I
0.05 -
0 0.5 1 1.5 2 2.5
Fig. 1
0.35 1
0.3 -
0.25
0.2 Weibull Disnibution 0.15 ko.l, a+os
17”(t)
0.1
il-\
-
0.05
0 0.5 1 1.5 2 2.5
Fig. 2.
0.35
0.3 -
0.25 -
Weibull Distribution
o~2~Q&Lj/ ;
MI.!, a=lS
0.15
vnft) 0.1
0.05
OO 0.5 1 1.5 2 2.5
Fig. 3.
underlying life time distribution is (i) exponential and (ii) Weibull with shape para-
meter values a=05 and 1.5 and scale parameter 1, using uniform distribution on
[ - 1, l] for the kernel. Similar graphs were drawn for n = 50 and 100. The three
graphs were seen to be close together, which indicates rapid convergence. It is seen
334 K. Alam, K.B. KulasekeralEstimation of the quantile function
Exponential Distribution 1=0.25
0.05 -
0.5 1.5 2 2.5
Fig. 4.
0.35, I
0.3
Weibull Distribution Lo,25 ) cx=o. 5
0 0.5 1 1.5 2 2.5
Fig. 5.
o-3 //
0.05
0 1 0.5 1 1.5 2 2.5
Fig. 6.
from the figures that m(t) is monotonically decreasing to 0, as t increases from 0 to
a value close to 2.5. The monotonicity of q”(t) is essentially due to the normalizing
factor h(t). In all the cases we have considered here, we observe that R”,(t) has smaller
mean squared error compared to R,(t).
K. Alum, K.B. Kulasekera/Estimation of the quantile function 335
5. Conclusion
In this paper we have shown the asymptotic convergence of the quantile function of
the residual life time distribution based on the empirical estimator R,(t) and its
smoothed version the kernel type estimator I?“(t), as n+co. We have given also the
rate of convergence with respect to the mean squared error. The empirical results we
have given for some special cases, show interestingly that the kernel type estimator,
besides being smoother, has also smaller mean squared error than the empirical
estimator for a moderate values of the sample size. The problem of selecting the value
of A (bandwidth) has not been considered in this paper. This topic will be considered in
a subsequent paper.
Appendix: Convergence in mean of an order statistic
Let X1, . . . . X, be a sample from a distribution F on the real line and let XCmj denote
the m-th order statistic. Let q = m/(n + l), F(c) = q, Z,(q) be the number of observations
in the sample less than or equal to [ and
X,*,=i- Z&k *q
*f(i) ’ 64.1)
wheref(x) denotes the density function of the distribution. Bahadur (1966) showed
that if m and n increase together, maintaining the relation
m
then
qz- n+l’
4, = XC,,,) - X,,,
is ~(n-~/~(log #“(log log n)‘14) almost surely. Subsequently Kiefer (1967) derived the
exact order of R, and showed that n3’“f([)Rn converges in law to a distribution with
mean 0 and variance (2p4/7~)~‘~ where p= l-q.
When F is a uniform distribution on (0,l) we denote XC,,,) by UC,,,) and R, by S,.
Duttweiler (1973, Theorem 1) has shown that
E(S,)2q2/n) s [ 1 112
<ne312.
It follows from (A.l) and (A.2) that
64.2)
(nE(U~,,-q)Z-q(l-q)(~ 4q(;-q) ( 1
112 +(4q(l -q))2n-1’4
(A.3)
336 K. Alam, K.B. Kulasekera/Estimation of the quantile function
and 114
<n-1/4 64.4)
where e,(q) is given by (A.8) below.
For arbitrary F when the density exists in a neighborhood of [ and f ([)>O, he has
shown (Theorem 2) that
[E(R )*]“2=IZ-3/4f-1([) 2q(l -q) ( > 1’4
n 7[:
+ o(n - l+ a/2), 64.5)
for any 6 > 0, where the remainder term depends on the value of q. It follows from (A.l)
and (A.3) that, given q
n(f(i)Y E(XC,~-lY-q(l -q)=o(n-‘44) (‘4.6)
and
(A.7)
where
e,(q)=n- ?JFJZ,-Hnq)
<(q(l -q)y’*. (‘4.8)
The limiting value of e,(q) is equal to (2/n)q(l -q))r’*, as n-+co.
Here we generalize the results given by (A.2) through (A.4). We derive a correspond-
ing result for an arbitrary distribution F, satisfying some regularity conditions given
below.
Let A denote the support of the distribution F. We shall assume that A is a finite or
an infinite interval and that f(x)>0 and f’(x) exists for all XEA. In addition, we
introduce the following conditions:
(i) Y(x)CF(x)(l -Fb41Y<~,
(ii) 2 If’Wl
F*(x)(l -F(x)) f3(X) <a>
for all XEA, where CI, /I and y are positive constants. Alternatively, the conditions (i)
and (ii) are given by
(i) (~(1 - u))’ d aQ’(u), (ii) Cu(l-41’ IQ”b)l G,
for O<u < 1, where Q= F-’ denotes the quantile function. The conditions (i) and (ii)
are satisfied by the distributions we come across in life testing, such as the exponential
distribution, and more generally, the Weibull and gamma distributions.
Let 6’=(3 +4y)/4(1 +y). We have that (Alam and Kulasekera, 1992)
K. Alam, K.B. KulasekeralEstimation of the quantile function 337
Theorem Al. If conditions (i) and (ii) are satisfied then
&f(OElKlC(l +(W2)O-‘14 and
n(_f([)2ER,Zd2(1 +(7~2~2/8)d,)n-“Z
for (m-4)(n-m-3)>(n+l)n6’, where c,+l and d,-+l, as n-m.
Corollary A2. Zf conditions (i) and (ii) are satisfied then
IJ;If(C)ElX ~,,-iI-en(q)I~(l+(aB/2)c,)n-“4 and
Inf2(1)E(X~,)-i)‘-qq(l -q)I62(1 +(7x2P2/8)d,)n-“2
+(1+(~$/2)c,)n-~‘~
$0~ (m-4)(n-rn-3)>(n+l)n”.
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