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Simultaneous upper confidence boundsfor distances from the best two-parameterexponentlal distributionHubert J. Chen a & Kanlaya Vanichbuncha ba Department of Statistics , University of Georgia , Athens, Georgia, 30602, U.S.A.b Academic Department , Chulalongkom University , Bangkok, 10500, ThailandPublished online: 27 Jun 2007.

To cite this article: Hubert J. Chen & Kanlaya Vanichbuncha (1989) Simultaneous upper confidence bounds fordistances from the best two-parameter exponentlal distribution, Communications in Statistics - Theory and Methods,18:8, 3019-3031

To link to this article: http://dx.doi.org/10.1080/03610928908830074

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COMMUN. STATIST.-THEORY METH., 1 8 ( 8 ) , 3019-3031 (1989)

SIMULTANEOUS UPPER CONFIDENCE BOUNDS FOR DISTANCES FROM THE BEST TWO-PARAMETER EXPONENTLAL DISTRIBUTION

Hubert J. Chen Kanlaya Vanichbuncha

Department of Statistics University of Georgia Athens, Georgia 30602 U.S.A.

Academic Department Chulalongkom University Bangkok 10500 Thailand

Key Words and Phrases: best population; confidence bounds; guaranteed life span; ranking and selection; type I1 censored data.

ABSTRACT

Consider k independent exponential distributions possibly with different

location parameters and a common scale parameter. If the best population is

defined to be the one having the largest mean or equivalently having the larg-

est location parameter, we then derive a set of simultaneous upper confidence

bounds for all distances of the means from the largest one. These bounds not

only can serve as confidence intervals for d l distances from the largest pararn-

eter but they also can be used to identify the best population. Relationships to

ranking and selection procedures are pointed out. Cases in which scale

parameters are known or unknown and samples are complete or type I1 cen-

sored are considered. Tables to implement this procedure are given.

Copyright @ 1989 by Marcel Dekker, Inc.

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1. INTRODUCTION

CHEN AND VANICHBUNCHA

In life testing a problem of frequent interest is the identification of the

best one and/or good ones of k two-parameter exponential populations with

guaranteed life spans (also referred to as location parameters) and a common

standard deviation, where the "best" population is defined to be the one which

has the largest guaranteed life span and the "good ones" are those within a

small distance from the best. In this exponential distribution case, identifying

the best population is equivalent to identifying the population which has the

largest mean. In this paper we apply Hsu's (1981) theory, the simultaneous

upper confidence bounds for all distance from the best, to the two-parameter

exponential distribution. Because these confidence bounds have nonnegative

values, the goodness of these populations from the best can be assessed by

ordering the bounds: a smaller interval corresponds a good population. So, a

reasonable rule to select a single best population is to select the one with the

smallest upper confidence bound.

Following the guide line of Hsu (1981) we state the relationship between

these simultaneous upper confidence bounds and those obtained by use of the

indifference zone selection procedure of Bechhofer (1954) and by use of the

subset selection procedure of Gupta (1965) for the two-parameter exponential

distribution.

Let q, i = 1, ..., k, represent the two-parameter exponential population

having a common standard deviation a and location parameter Bi (also

referred to as guaranteed life span) with the probability density function

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SIMULTANEOUS UPPER CONFIDENCE BOUNDS 3021

Denote the exponential population (1.1) by E(ei, a). For convenience, let rc(k)

be associated with the unknown "best" population which has the largest 8-

value, eIk]. In the case where more than one population has a %value which

is tied with the largest, then exactly one of these tied populations is defined to

be the "best" population according to some fixed rule. Let Yi be the smallest

order statistic based on a random sample (TI, ..., X,,,) of size n drawn from

population xi, i = 1, ..., k. Then the distribution of Yi is again exponential,

E(ei, o h ) . We shall derive loop*% (I/k < P* < 1) simultaneous upper

confidence bounds for all distances of Bi from eLk], eIk] - el, ..., eF1 - Bk.

Simultaneous confidence bounds are considered for both the case where

the common standard deviation is known and the case where it is unknown,

and the samples are complete or type I1 censored. Tables necessary to imple-

ment this procedure are also computed.

2. COMPLETE DATA AND KNOWN COMMON STANDARD DEVIATION

In the case where the common standard deviation is known, we wish to

find a set of confidence intervals, Ii = [O, Ci], with Ci 2 0 for Btk1 - ei, i = 1 ,

..., k, such that the probability that the confidence interval Ii covers eIk1 - 8;

for all i is at least P*, i.e.,

P(eFI - ei E I;, i = 1, ..., k) 2 P*

for some specified value of P*(l/k < P* < 1).

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3022 CHEN AND VANICHBUNCHA

Theorem 1 :

A set of loop*% (l/k < P* < 1) simultaneous upper confidence intervals

for

e[k] - 01, ..., @&] - $

is given by

[O, C1], ..., [O, Cklt (2.2)

where Ci = Max(Ma,xYj - Yi + dk,p*o/n, 0 ) and d = dk.P1 > 0 is the solution J h

of the equation,

e d ( l - (1 - e-d)k)/k = P*.

Proof:

Let Y(k) be associated with OF] and d = dk,P+ Then

P(Ci 2 - ei, i = 1, ..., k )

= P(Max{Ma,xYj - Yi + d o h , 01 2 - ei, i = 1, .in

r P(Y,, - Yi + d o h 2 e[k] - ei for all i ;t (k); 0 = BF1 - ei, i = (k))

= P(Y(,) - etkI 2 Yi - ei - d o h for all i s (k))

= P ( Z k 2 Z i - d , i = 1, ..., k - I ) ,

where Zi = n(Yi - Oi)/o, i = 1, ..., k - 1, and Zk = n(Y(k) - eF1)/O, are

independent and identically distributed as E(0, I), which completes the proof

by a straightforward integration and setting the last equation equal to P*.

If 8&] - 8k-11 2 dk,p,o/n, then the occurrence of the event

(Y,,, - 2 yi - ei - dkp.o/n for all i t (k)) implies Y(k) = Y[k]. It fol-

lows that the nominal confidence coefficient 100P*% is attained whenever P*

2 l/k and n 2 [dk,p*~/detk] - e[k-ll)l. (Note: pb] = &k).)

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SIMULTANEOUS UPPER CONFIDENCE BOUNDS 3023

Since the left hand side of equation (2.3) is increasing in d for d > 0, it

is clear that there is a unique positive solution for P* > l/k. Using equation

(2.3) we can easily obtain the numerical solution for d by a pocket calculator.

Partial tables of the vales of d may be found in Raghavachari and Starr (1970)

under a different form. For completeness with (3.3) when u goes to infinity

the values of d are given in the last row of Tables 1-3 for P* = .90, .95 and

.99 and k = 2(1)10. For any solution d its calculated probability away from

P* is with .00005.

Analogous to normal distribution case discussed by Hsu (1981) the

simultaneous upper confidence intervals for distances from the best for all

exponential distributions and the solution to equation (2.3) not only can

guarantee that the probability of correct selection for selecting the single best

exponential population over the preference zone is greater than P* as

developed by Barr and Rizvi (1966) and Raghavachari and Starr (1970), but

can also assert that the selected subset containing the best exponential popula-

tion cmies the probability of correct selection over the general parameter

space being at least P* in a subset selection as developed by Gupta (1965).

3. COMPLETE DATA AND UNKNOWN STANDARD DEVIATION

Since the common standard deviation, o, is unknown, we estimate o by

where Xij is the j" observation from the i" population, j = 1, ..., n, i = 1, ..., k

and u = k(n - 1). The estimator 6 is the minimum variance unbiased estima-

tor of o and is independent of Yi, i = 1, ..., k, (see, e.g., Tanis (1964)). It is

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3024 CHEN AND VANICHBUNCHA

well-known that T = 810 is a X2(2u)12u (chi-square with 22, d.f. divided by

2u) random variable independent of the Yi's (see e.g., Lawless (1982)).

Theorem 2:

A set of lOOP*% simultaneous upper confidence intervals for

CIi = Max(MaxYj - Yi + ~ , ~ * , , , ~ l n , 0) jti

and q = Q,P*,~ is the solution of

P ( Z i S Z k + q T , i = 1, ..., k - 1 ) =P*.

Proof:

2 P(Y(k) - Yi + q%/n 2 - Oi for all i # (k))

= P{Y(,, - elk] 2 Yi - Oi - q%ln for all i + (k) )

= P(Z(k) 2 Zi - qT for all i # (k))

= P(Zk 5 Z i - qT, i = 1, ..., k - l ) ,

where the Z's are defined as in Theorem 1. The proof is completed by setting

the last probability equal to P*. The percentage points q can be numerically

calculated by the following equation obtained by a straightforward calculation

from (3.2).

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SIMULTANEOUS UPPER CONFIDENCE BOUNDS

TABLE I Values of q for Simultaneous Upper Confidence Intervals for All Distances from the Best Exponential Population, P* = .90

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CHEN AND VANICHBUNCHA

TABLE II

Values of q for Simultaneous Upper Confidence Intervals for All Distances from the Best Exponential Population, P* = .95

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SIMULTANEOUS UPPER CONFIDENCE BOUNDS

TABLE 111

Values of q for Simultaneous Upper Confidence Intervals for All Distances from the Best Exponential Population, P* = .99

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3028 CHEN AND VANICHBUNCHA

p* = C (-1j+' k](l + (j - l )q /~)-~/ lc . (3.3) j=1

A partial table of q values in a difference form was computed by Desu,

Narula and Villarreal (1977) in their two-stage selection problem for the case

P* = .95, n = 2(1)30 and k = 2(1)6. For our interval estimation purpose we

have calculated a more complete table by using Newton's iteration method.

Tables 1, 2, and 3 give values of q for P* = .90, .95, and .99; u = 2(1)30, 40,

60, 80, 120, -; and k = 2(1)10. The absolute difference between computed

probability (3.3) and P* is less than .00005 for any solution of q. Another

complete table was calculated by Vanichbuncha (1986).

4. TYPE I1 CENSORED DATA

Experiments involving type I1 censoring are often used, for example, in

life testing; a total of n items are placed on test, but instead of continuing

until all n items have failed, the test is terminated at the time of the rfh item

failure for the ith population, i = 1, ..., k. Such a test can save time and

money, since it could take a very long time for all items to fail in some

instances. With type I1 censoring the number of observations ri from popu-

lation xi is determined in advance. Formally, the data consist of the ri smal-

lest lifetimes yl s . . . < Xi,, 1 I ri 2 n, for some ri 2 2, out of a random

sample of n lifetimes Y1, ...,&, from the population n,. Here we require the

sample size n to be the same for all populations, but the preassigned number

of failures, ri, may be different.

Let Y1, ..., Yk be the first order statistics based on type II consored data

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SIMULTANEOUS UPPER CONFIDENCE BOUNDS 3029

from two-parameter exponential distributions. Then the distribution of

Yi, i = 1, ..., k, is the same as E (ei, oh).

When the common standard deviation is known, the set of simultaneous

confidence intervals for OBI - €4, i = 1, ..., k, is exactly the same as in

Theorem 1. This is because the sufficient statistic is the minimum failure time

of a sample, whether we censor or not is irrelevant.

When the common standard deviation, a , is unknown, we estimate o by

It is well known (see, e.g., Lawless (1982)) that Yi and %* are stochasti-

cally independent, and T* = %*/a has a x2(2u*)/2u* distribution.

When the common standard deviation is unknown, the theory of sirnul-

taneous confidence bounds for all distances from the best for type II censored

data applies with & replaced by %* and v replaced by V* in Section 3. The

percentage points of q are given in Tables 1-3.

FUTURE RESEARCH

The following related problems have not been considered in this

manuscript and they should merit future research. These problems are: (1)

multiple comparisons with the best exponential population for unequal sample

sizes (For normal distribution case, see Hsu (1984a).); (2) multiple comparis-

ons between each treatment and the true best of the other treatments,

ei - max ej, for exponential distributions (For normal distribution case, see jti

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3030 CHEN AND VANICHBUNCHA

Hsu (1984b).); and (3) the asymptotic relative efficiency (ARE) between

parametric procedure in exponential distributions and nonparametric procedure

(For normal case, see Hsu (1981).).

ACKNOWLEDGEMENT

The first author's research is supported by PHs Grant Number

2ROlCA40702-02 awarded by the National Cancer Institute, DHHS. The

authors which to thank the referees for their helpful comments and suggestions

on the earlier version of this paper and Mrs. Gayle Rodriguez for her typing

the manuscript.

BIBLIOGRAPHY

Barr, D. R., and Rizvi, M. H. (1966). An introduction to ranking and selec- tion procedures. J. Amer. Statist. Assoc. 61, 640-645.

Bechhofer, R. E. (1954). A single-sample multiple decision procedure for ranking means of normal populations with known variances. Ann. Math. Statist. 25, 16-39.

Desu, M. M., Narula, S. C., and Villarreal, B. (1977). A two-stage procedure for selecting the best of k exponential distributions. Commun. Statist- Theor. Meth. A6(12), 1223-1230.

Gupta, S. S . (1965). On some multiple decision (selection and ranking) rules. Technometrics. 7, 225-245.

Hsu, J. C. (1981). Simultaneous confidence intervals for all distances from the best. Ann. of Statist. 9, 1026-1034.

Hsu, J. C. (1984a). Ranking and selection and multiple comparisons with the best. Chapter 3, Design of Experiments: Ranking and Selection. (T. J . Santner and A. C . Tamhane, editors). Marcel Dekker, New York.

Hsu, J. C. (1984b). Constrained simultaneous confidence intervals for multi- ple comparisons with the best. Ann. Statist. 12, 1136-1 144.

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SIMULTANEOUS UPPER CONFIDENCE BOUNDS 3031

Lawless, J. F. (1982). Statistical Models and Methods for Lifetime Data. John Wiley & Sons, New York.

Raghavachari, M. and Starr, N. (1970). Selection problems for some terminal distributions. Metron. 28, 185- 197.

Tanis, E. A. (1964). Linear forms in the order statistics from an exponential distribution. Ann. Math. Statist. 35 , 270-276.

Vanichbuncha, K. (1986). Multiple Comparisons with the best population. Ph. D. dissertation, Department of Statistics, The University of Georgia, Athens, Georgia.

Received M u c h 1987; Revbed May 1989.

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