Optimum two-sided confidence interval for the location parameter of the exponential distribution

Rehabd:ty Engmeermg 13 (1985) 1-10

Optimum Two-sided Confidence Interval for the Location Parameter of the Exponential Distribution Dimitri Kececioglu Professor of Aerospace and Mechamcal Engineering, The Umverslty of Arizona, Tucson, Arizona 85721, USA

and

Dingjun Li China Aviation Research Institute for Standardization, Beljmg, People's Repubhc of China (Recewed 7 February 1985)

ABSTRACT The purpose o[ this paper is to gwe the opttmum two-stded confidence :nterval./or the location parameter oj the two-parameter exponenttal dtstrtbutlon when the test isJailure termmated. The optimum confidence interval ts defined A method oJ calculating the mterval ts derived and Is illustrated by an example

1 INTRODUCTION Sinha and Kale (ref. 1, p. 113) gave the general idea for obtaining the optimum, two-sided confidence intervals for the location parameter, ),, of the exponential distribution, but they did not give the result. They suggested that tables, such as those obtained by Tate and Klett 2 for the opttmum two-sided confidence interval for the variance, tr2, of the normal distribution, should be prepared to solve this problem. The purpose of this paper is to derive the optimum two-sided 1

Rehabihty Engmeermg 0143-8174/85/$03 30 © Elsevier Applied Science Pubhshers Ltd, England, 1985. Printed m Great Bmain

2

Dtmttrt Kece~ u~glu, Dmglun Lt

confidence interval of 7 for a failure-terminated test, and to show that ~t ~s not necessary to prepare such tables, because the confidence interval can be calculated in a closed form.

2

CALCULATION OF THE TWO PARAMETERS

Consider the two-parameter exponential distribution whose probablhty density funcUon (pdf) is gwen by j(x,-;,0)=oexp

-

,

x>7,

-~c<7<~c.,

0>0

(1)

Let X~lI < x~zI < < x~,l be the ordered observaUons of a sample of size n from this distribution in a failure-terminated test. It is well known that the maximum likehhood estimators (MLE) of 7 and 0 are given (ref 4, pp. 162-7) by = xll j (2) and 0 = Sr/r

(2')

Sr = ~-" ( x , ~ - xll~) + (n - r)(xlr ~- x ~ )

(3)

where

1=1

3

OPTIMUM CONFIDENCE

INTERVAL

Assume that j ( x , O) is a pdf with 0, the unknown parameter. A n y two functions of observations, TL(x l, x 2, ., x.) and T u ( x j, x 2 . . . . x . ) are said to be the lower and upper confidence limits of 0 at the confidence level 1 - - ~ , If P[Te(xl,

x 2,

,.x.)<_O<_Te(x

1 , x 2,

, x.)]=l-~

(4)

then the interval [ T e ( x ~, x 2, , x . ) , T u ( x 1, x 2 . . . . x.)] is called the confidence interval of 0 at the confidence level (1 - ~) Furthermore, if T~L(XI, X 2 , ,

,

x.)

and

T~v(x 1, x2,_

, x,,)

Optimum two.sided confidence interval of 7

3

satisfy eqn (4), and have the property that T* = ~ ( x , ,

x2 . . . . . x.) - ~ ( x l ,

x2 . . . .

x.)

is the minimum value of all possible values of T, where T = T u ( x , , x 2. . . . .

x.) - TL(Xl, x 2....

x.)

which is equal to the difference between the upper and lower confidence limits of 0, then the interval [ ~ ( x ~ , x2 . . . .

x.), ~ ( X l , x~ . . . .

x.)]

is the optimum confidence interval of 0 at the confidence level (1 - ~).

4

DERIVATION OF THE CONFIDENCE INTERVALS

Consider the random variable T_n(x,1,-,)Sr

_2n(x,,,-'~)/2S,o

(5)

It can be shown (ref. 4, p. 165) that U = 2n(x~l) - ~) 0

(6)

is x2-distributed with two degrees of freedom and V = 2S, 0

(7)

is x2-distributed with ( 2 r - 2 ) degrees of freedom, and they are independent. From eqns (6) and (7), since U and V are x2-distributed, the random variable (ref. 3, p. 323) F -

U/2 = n(x~l) - ~) V / 2 ( r - l) S r / ( r - 1)

(8)

Is F-distributedwith [ 2 , 2 ( r - 1)] degrees of freedom. Then, for any two positive numbers !1 and /2, where l~ < 12 P 11 -< S ~ -

1) < 12 = F2 21r-1~(/2) - F2 2~r-1)(ll)

4

Dtmltrl Kece~loglu, Dmglun Lt

where F2.2~r 1) ( ) stands for the cdf of the F-distribution with [ 2 ; 2 ( r - i)] degrees of freedom If the values of l I and l 2 a r e selected such that F2

21r-1~(/2) -

F2 2~,- ll(ll) = 1 -- ~

(9)

and Sr n(r - 1 )

( 1 2 - ll) is the m i n i m u m

(10)

then according to the definition given in Sectmn 3, the interval x,~)

t sr

n ( r - - 1)'

x,,,

tl& -I

n ( r - - 1)J

(11)

is the o p t i m u m confidence interval of )' at the confidence level (1 - ~) Now the problem becomes how to obtain the values of l~ and l 2 which minimize { S J [ n ( r - I)]}(/z - ll) and also satisfy the constraint o f e q n (9) This is a conditional extreme value problem. But the Lagrangian multiplier method is not appropriate here since the extreme value would be reached at the boundary. F r o m eqn (10), it can be seen that if (12 - l~ ) reaches its m i n i m u m value, then { S j [ n ( r - 1)]}(l 2 - l 1) also reaches its m i n i m u m value_ If/3 is a positive value such that O_
(12)

then the value of l I should be chosen such that i'j2,2~,_ ,l(x) dx

(13)

wherej2.2~,- 1~(x) is the p d f of the F-distribution with [2; 2(r - 1)] degrees of freedom. Since l~ and 12 should satisfy eqn (9), 12 should satisfy

f :~J2,2~r- l)(x) dx ~---0t -- fl

(14)

2

For the F - d l s t n b u t m n with [2, 2(r - 1)] degrees of freedom, cdf reduce to J2,2~,-1~( x ) =

1+

r--I

ItS

p d f and

(15)

Opt|mum two-stded confidence interval of 7

5

and F2,2~,-1)( x ) = l -

(16)

l+r_l

Substitution of eqn (15) into eqns (13) and (14) yields 1+

r-I

dx=fl

(17)

dx=~-fl

(18)

and ,

1+

r-I

Equation (17) may be put m the form (r - 1) f l + u,/it- 1)1

d,

u-r du = fl

where x u = 1 +-r-1

du 1 dx r - I

du =

1 dx r-1

Therefore

rl _ 1 ul _rll +tll/lr( r - 1) 1 l+u':r-l)| u - r d u = dl

__ ~Ar-- lll l +|l~'(r-1

1

(19)

or

1-

l+r_

1

l+r_

1

(20)

Similarly, eqn (18) yields =c~-fl

(21)

6

Dtrnttrt Kececwglu, Dmglun Lt

Solving eqns (20) and (21) for l I and l 2, respectively, yields

]1 = [(1 -- f l ) J , I , - , ) _

1](r-- l)

(22)

l 2 = [(6-- f l ) l / ~ l - r ~

1](r - 1)

(23)

and

Consequently

z(fl) -----l 2 -- /1 ---- (r -- 1)[(~

- / 3 ) 1,,(1 -r) __

(1 -/3)1/(1-r}t]

(24)

Taking the first derlvaUve of z(/3) with respect to fl yields d[z(fl)] _ (6 -/3),,.~1-,)_ (1 -/3y/~1 - ~

d/3

(25)

Since r > 1, 0

d/3

0 for any/3

(26)

It means that the function z(/3) = 12 - ll

IS an increasing function with regard to/3 Consequently, when/3 = 0, z(/3) reaches its minimum value. F r o m eqn (14), since/3 = 0 and J2,2(,- 1)(X) > 0

then Il = 0

(27)

and eqn (15) becomes

f


1)(x)

dx = ~

(28)

2

therefore

t

I +

12 t l -r r-1

= ~

(29)

Solving eqn (29) yields / 2 = [c~l ' l l - ~ - 1 ] ( r - 1)

(30)

Optimum two-sided confidence interval of 7

7

Substitution of eqns (27) and (30) into eqn (11) yields { [ x'l'

S, n ( r - 1 ) ( r - 1 ) ( ~ t 1/(1- r ' - l ) l , x " ) t

(31)

Simplification of eqn (31) yields the optimum confidence interval of 7 at the confidence level (1 - 0t), or I x
(32)

n

5

EXAMPLE

Assume that the failure times, x, of a unit follow the two-parameter exponential distribution. Six such umts are tested, the test is terminated after the fifth failure, and the following times-to-failure are obtained: x<~ = 155.5, xt2 ~= 166.8, xt31 = 180.0, xt41 = 195.0, and x~5~= 225-0h. Find the optimum confidence interval of the location parameter, 7, at the confidence level of 0.90. Table 1 gives the plotting posit,ons for these data using the concept of median ranks (MRs). The probability plot of these data is shown in F~g. 1. The curve in Fig. 1 represents the plot of the original data, x~jj vs the median ranks on which the estimate of the non-zero location parameter, ~, is indicated along the x-axis as ~;= 150 h. The data given in column 3 of Table 1 are the modified data, xtj~- 150, and the straight line in Fig. 1 represents the plot of the modified data vs the same MRs. This straight line shows that the data came from an exponentially distributed population with ~ = 150 h. TABLE 1 Time-to-Failure Data for S~x Units Tested Simultaneously and the Test Terminated after the Fifth Umt Failed

j

xu) (h)

Medtan rank

xu~ - 150 (h)

1 2 3 4 5

155 5 166 8 180-0 195 0 225-0

10 910 26 445 42 141 57 859 73 555

5 5 16 8 30-0 45 0 75 0

(%)

8

Dimttrz Kececloglu, Dmglun LI

g Oo 999~v~

o

99-

" 0

90 -

°~

-

700

~x~O6

//

60-

/// //

\ M R v s [~(j,-150~}

"iI"

4O

R vs

x(j)

"~ u~ 3O 8g

2

O.

i

10

I I

5 I 2

I 3

I I I Illl 4 5 678910

I 2

I 3

I 4

I I I111 5 6 7aQlO

1

~,

I

2

x'~O

I

I

3

4

I

I I 111

5 6 789J10

x 1"o2

x ( j ) - 7 (h) Fig. i.

Welbull probablhty paper plot of exponentially distributed time-to-failure data

Substituting a = 0.10, r = 5, n = 6, x(l ) = 155 5, into eqn (3), i e Sr = ~

(x(,) - x ( 1 ) ) + ( n - r ) ( x ( , ) - x ( 1 ) )

(3)

/=l

gives Sr = 2 1 4 . 3 F r o m eqn (32) Tt,=155-5

214-3 ~(0.101(1-5)_1)

or

T L = 127 7 h and T u =x(1 ) = 155 5 h C o n s e q u e n t l y , the o p t i m u m confidence interval for 7 at a 90 7o confidence level is (127.7h, 155 5h).

Optimum two-sidedconfidencemterval of ~ 6

9

S E L E C T I O N OF ! 1 A N D l 2

From the derivation m Section 4, it can be seen that for values of 11 and 12 which satisfy the constraint eqn (9), the expression

[

11S,] n(r- 1)

12St

x~l) n(r-

1) 'x~ll

(33)

always gives the confidence interval of y at the confidence level (1 - 0t). Consequently, based on different criteria of choosing the values of 11 and 1z a different confidence interval of g at the same confidence level will result. The optimum confidence interval is the shortest confidence interval among all confidence intervals at the same confidence level. Another commonly used criterion is the 'equal-tads' criterion which chooses the values of 11 and 12 in such a way that

J2.2 ,- 1)(x) dx =

(34)

and

o~,

2,2~,-l)(x) dx = 2

(35)

Since fl = or/2, eqns (22) and (23) respectwely are gwen by 11=

1--

-1

(r-I)

(36)

and /2=

-1

(r-I)

(37)

Substitution of eqns (36) and (37) into eqn (33) yields the following equal-tails confidence interval:

(38) For the example given m Section 5, the equal-tails confidence interval at the same confidence level of 90 ~o is (115.7, 155.0). Comparing this result with the result given in Section 5, it can be seen that the optimum confidence interval Is much shorter than the equal-tails

10

Dtmttrt Kecectoglu, Dmglun Lt

confidence mterval. This is due to the asymmetry of the F-distribution with ( 2 , 2 ( r - l)) degrees of freedom.

REFERENCES 1 Smha, S. K. and Kale, B. K Ltje Testing and Rehabdtty Esttmation, John Wiley & Sons, New York, 1980. 2. Tate, R. F and Klett, G W. Optimum confidence interval for the variance of a normal distribution, J. Am. Star Assoc, 54 (1959), pp 674-82. 3 Mann, N. R , Schafer, R. E and Smgpurwalla, N. D Methods/or Stattsttcal Analysts oj Rehabtlity and LtJe Data, John Wdey & Sons, New York, 1974 4 Barn, L J, Stattstical Analysis o/ Reliabdity and Life-Testing Models-Theory and Methods, Marcel Dekker, New York, 1978