CHAPTER 3

Inference Procedures forExponential Distributions

The exponential distribution occupies an important position in lifetime

distribution work, and an entire chapter will be devoted to a discussion of

statistical procedures for it. (The Weibull distribution will be given similarly

favored treatment in Chapter 4.) Historically the exponential distribution

was the first lifetime model for which statistical methods were extensively

developed. Early work by Sukhatme (1937) and later work by Epstein and

Sobel (1953, 1954, 1955) and Epstein (1954, 1960a) gave numerous results

and populartzed the exponential as a lifetime distribution, especially in the

area of industrial life testing. Many authors have contributed to the statisti-

cal methodology of the distribution. The lengthy bibliographies of

Mendenhall (1958), Govindarajulu (1964), and Johnson and Kotz (1970,

Ch. l8) give some idea of the very large number of papers in this area, even

up to 1970.This chapter covers basic inference procedures for the exponential distri-

bution. Estimation and significance tests for single samples are considered

first, with separate treatments for Type II and Type I censoring. The

comparison of distributions is the topic of Section 3.3. In Section 3.4 we

examine life test acceptance procedures and discuss the planning of life test

experiments. Although the results in these sections are specific to the

exponential distribution, the discussion illustrates several general points

about inference with censored data in parametric models. Methods for the

two-parameter exponential distribution are considered in Section 3.5, where

a threshold parameter is included in the model. Section 3.6 concludes with

some cautionary remarks about the sensitivity of these procedures to

departures from the exponential model.

100

SI )'

3.1 SINGLT .

The exponenr

in which case ̂function.

With complc: ,and well knou nfor Type II cen.I f 1, , . . . , t , is a :based on this i:

(For notational c, :representing rand, :the random variabr.--sometimes using lt,'.,.the realized value:maximizing (3.1.2 r. .

easily seen that f i.standard exponenti.r.with index paramc-rc:

Similar results htrl.:first r observations ;s ize n. From (1.4. I ) r i : -

y1 l.

T;-7F il

I

t l ': -

(n- ,

If we let

Lawless, JF (1982), Statistical Models and Methods for Lifetime Data, Wiley, p. 100-104.

: -lres for::ibutions

SINGLE SAMPLES: CoMPLETE oR TYPE II CENSoRED DATA 101

3.T SINGLE SAMPLES: COMPLETE OR TYPE II CENSORED DATA

The exponential model will be written with its p.d.f. in the form

f ( t ;0) :0-1e-t /e r>o (3.1.1)

in which case 0 is the mean of the distribution and tr:d-ris the hazardfunction.

With complete (i.e., uncensored) samples inference procedures are simpleand well known. Although these can be given as a special case of the resultsfor Type II censored data, let us review them briefly by way of introduction.I f / , , . . . , tn is a random sample f rom (3.1.1), then the l ikel ihood funct ionbased on this is

inportant Position in lifetime

be devoted to a discussion of.:nbution will be given similarlY

'' the exPonential distribution

-:rcal methods were extensivelY

,nd later work bY EPstein and

- 1960a) gave numerous results

: distribution, esPeciallY in the

.rve contributed to the statisti-'. lerrgthy bibliograPhies of

,nd Johnson and Kotz (1970,

-e r of papers in this area' even

,.res for the exPonential distri-

-ingle samPles are considered

rnd TyPe I censoring. The.':ction 3.3. In Section 3.4 we

.CUSS the Planning of life test.ections ate sPecific to the

.:retes several general Points::ric models. Methods for the

.'..idered in Section 3.5, where

,'. Section 3.6 concludes with

.rt) ' of these Procedures to

r-#;rj'.-oL-lf we let

L(0): , ! , f ( r , :0) : f i .^p

t ( i l+ (n- r ) tsy

- $ r \2.0 I '

1: l I

( , ! , , , , , + (n- r)tr , t) t t l (3 r 3)

(3.r .2)

(For notational convenience we will not adhere completely to the habit ofrepresenting random variables by capital letters and the realized values ofthe random variables by lower case letters. No ambiguity should result fromsometimes using lower case letters to represent random variables as well asthe realized values of the random variables.) The m.l.e. of 0, obtained bymaximizing (3.1.2), is easily found to be 0-T/n, where T-2t,. It is alsoeasily seen that Z is sufficient for 0, and since the l, /0's are independentstandard exponential variates, T/0 has a one-parameter gamma distributionwith index parameter n. Equivalently,2T/0-Xb,, (see Section 1.3.4).

Similar results hold for Type II censored sampling. Suppose that only thefirst r observations /11; 1trD< . . .

srze n. From (1.4.1) the jo int p.d. f . of /11; , . . . , / , . . , is

#i( ,u, ; s- ' |1 '1/o )(e-""r0 ) ' - '

r - \ ll : I

IO2 INFERENCE PROCEDURES FOR EXPONENTIAL DISTRIBUTIONS

the likelihood function can be taken to be

SIN(; :

Since T:2',

Corollary 3.1.1

has a distr ibut i , :

3.1.1 Tests and t

Tests and conf ic-quanr i ty 2T70 :conf idence interr .

:

where x2ea,, is rh-

is the I -c conf id.

Example 3.1.1lifetimes from &fl r-

Hence n:12, r :b.hours. To obtain. l, :f ind by using tablc'Pr(6.91<xl ,ur <28.sdence interval f.or 0. ':

confidence interval I

Tests or interval c-s imi lar ly found, s in. . -dence intervals and : -rate A- l /0, (2) the .

S(ro):exp(- to/0). , .

L(o)- #e-r /o (3.1.4)

dropping the constant term ,,./(n-r)! in (3.1.3). Clearly T is sufficient for0 and the m.l.e. is d: T/r. Note that the definition of T agrees with theearlier definition T-2t, in the case in which the data are uncensored.

Remark T is sometimes referred to as the "total observed lifetime," orthe "total time on test," since it is the total of the observed lifetimes for all nindividuals. This statistic is central to inference for the exponential distribu-tion.

The distribution of Z is easily found: make the change of variables

Wr:ntr ,

w,:(n- i - t t ) ( r1 iy - t t , - r l ) i :2, . . . , r . (3.1.5)

Since

t<, ,1(n-r) t , ,1

and the Jacobian is

o(Wt, . . . ,W,)o(11rr . , . / . ,J

the jo int p.d. f . of Wr, . . . ,W, is found from (3.1.3) to be

We have proved the following result:

rHEonEu 3.1.1 Let ts1, . . . , t ( , \ be the f i rst r ordered obseroat ions of arandom sample of size n from the exponential distribution (3.1.1). Then thequantit ies Wr,...,W, giaen by (3.1.5) are independent and identically distrib-uted, also wi th p.d. f . (3.1.1).

r

:2 w,l : l

r

r - li : I

71 l.: -(n- r ) l

#.^o (- , t ; ) w, >o

:BUTIONS

(3.1 .4)

i: sufficient for

-isrees with thercensored.

:d lifetime," or:times for all nrintial distribu-

, . i r iables

(3.1 .5 )

'ir.r of afhen the' distrib-

SINGLE SAMPLES: coMPLETE oR TYPE II CENSoRED DATA 103

Since T:2i:1W1, we also immediately have

Corollary 3.1.1 Under the conditions of Theorem 3.1.1,

r

T- > /1; ; * (n- r ) t1,1l : I

has a distribution given by 2T/0-X?zn.

3.1.1 Tests and Confidence Intervals

Tests and confidence intervals for 0 are easily obtained using the pivotalquantity 2f /e. For example, to obtain an equitailed, two-sided l-aconfidence interval for 0, we take

where x'co,, is the pth quantile of xb,t.Then

2T^27,<rs. .

X(zr l . t -a/2 X-12r1,"72

is the I - a confidence interval for 0.

Example 3.1.1 The first 8 observations in a random sample of 12lifetimes from an assumed exponential distribution are, in hours

31, 58, 151,185, 300, 470,497,673.

Hence n:12, r :8, and Z-5063. The m.l .e. for 0 ts 0:5063/8:632.9hours. To obtain, for example, a two-sided .95 confidence interval for 0, wefind by using tables of the 12 distribution that Pr(6.91<27/0<28.8;-Pr(6.91<X?,ul<28.8)- .95, which gives Qf /28.8,27/6.91) as a .95 conf i -dence interval for 0. For the sample observed, T:5063, and the reahzed .95confidence interval for 0 is therefore i351.6. 1465.4).

Tests or interval estimates for other characteristics of the distribution aresimilarly found, since these are simple functions of 0. ln particular, confi-dence intervals and tests are readily obtained for (l) the constant hazardrate l,-1/0, (2) the survivor (i.e., reliability) function at time ro, given byS(lo)-exp( -to/0), and (3) the pth quantile of the distribution, given by

Y,(xbo,o/24T =r 'o. , \ -a/2)- t -o,