i ' f f /~

E L S E V I E R Statistics & Probability Letters 31 (1996) 31-39

STATISTICS& PROBABILITY

LETTERS

Conditional inference procedures for the Laplace distribution based on Type-II right censored samples

Aaron Childs, N. Balakrishnan Department of Mathematics and Statistics, McMaster University, Hamilton, Ont., Canada LSS 4K1

Received December 1995; revised February 1996

Abstract

In this paper we develop procedures for obtaining confidence intervals for the parameters of a Laplace distribution as well as upper and lower 7 probability tolerance intervals for a proportion fl, given a Type-II right censored sample from the Laplace distribution. The intervals are obtained by conditioning on the observed values of the ancillary statistics. The intervals are exact, and will generalize the work of Kappeuman (1975, 1977) who considered the full sample case.

Keywords." Ancillary statistics; Tolerance intervals; Confidence intervals; Order statistics; Laplace distribution; Type-II right censoring; Maximum likelihood estimators

I. Introduction

Let Xl:n ~X2:n ~ "'" ~Xr:n be the order statistics from a Type-II right censored sample from a Laplace (or double exponential) distribution with probabil i ty density function

f ( x ) = l e -bx-°l/~, - o < ~ < x < c < z , - o < z < 0 < o < z , a > 0 (1.1)

and cumulative distribution function

{ ½e (x-°)/a, x <~ O, F(x) = . (1.2)

1 - ~ e - ( x - ° ) / ~ , x > O.

Balakrishnan and Cutler (1995) recently found that i f r > n/2 then the MLEs for 0 and cr are given by

~Xm+l:2m+l i f n is odd, n = 2m + 1,

t) = [, any value in [Xra:2m,Xm+l:2ra] i f n is even, n = 2m, (1.3)

0167-7152/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved PII S0167-7152(96)00010-7

32 A. Childs, N. Balakrishnan I Statistics & Probability Letters 31 (1996) 31-39

and m

r i = m + 2 i=1 d

6 =

1 (n r)X~:n + X/:n X,':n F

i = m + l

and if r <~ n/2 then the MLEs are

and

if n i s o d d , n = 2 m + l ,

if n is even, n = 2m,

(1.4)

0 = Xr:. + ~ In n/2 (1.5) r

- r - 1 ) X r : n - :n r

(1.6)

In this paper (Section 2) we will use the above MLEs to derive the marginal conditional distribution of the pivotal quantities v = ((~ - 0)/8 and w = 6/a given the ancillary statistics

u i - ~ , i = 1,2 . . . . . r. (1.7)

These conditional distributions will then be used to obtain conditional confidence intervals for 0 and a. These results will generalize the full sample results of Kappenman (1975).

In Section 3 we will similarly derive conditional tolerance intervals. We will show how to find q such that the intervals ( 0 - qt , oo) and ( - o c , 0 - qS) are lower and upper ~, probability conditional tolerance intervals for proportion fl (1>0.5) respectively. By definition, this is equivalent to finding q such that

P f(x;O,a)dx~fllUl,U2 . . . . . ur = ~ (1.8) --qd

and

qd

P f(x;O,a)dx>~fllul,u2 . . . . . u, = y (1.9)

respectively. These results will generalize the results of Kappenman (1977) who considered the full sample case.

Finally in Section 4 we will give an illustrative example. We should mention here that unconditional tolerance intervals for the full sample case have been discussed

by Shyu and Owen (1986a, b), and that conditional inference in general has been discussed in detail by Lawless (1982) for a variety of models under different sampling situations.

2. Conditional confidence intervals

Based on a Type-II right-censored sample, Xl:n ~<X2:n ~< " ' " ~ X r : n , for any member of a general location- scale family

A. Childs, N. Balakrishnan I Statistics & Probability Letters 31 (1996) 31-39 33

the joint conditional probability density function of the pivotal quantities v and w, given ul, u2 . . . . . u~, is given by

kl) r-1 g ( (u i + v)w) (1 - - G((ur + I))W)) n-r, --00 < V < 00, W > O, (2.1)

where k is the constant of proportionality; for example, see Lawless (1982) for details. For the Laplace distribution (where 9(x)= ½e-lxl), using (1.1) and (1.2), (2.1) gives the joint conditional density function of v and w, given Ul,U2,-.-,Ur, to be

{kl~ r-1 e -w ~-~=1 I0+~,1 { 1 - !aw(v+u~)l.n-r 2 " J

f(v, w lul,u2 .. . . . ur) = kwr-le-wE'% Iv+"d{le-wfv+"r)} "-r,

where the constant of proportionality k is derived as follows:

l _ [ ° ° [ -"r ~e } dv dw - Jo J-oo wr-le-wE~=' Iv+u'l{1 -- 1 W(V+Ur) n--r

1 [ ° ° [ °° + 2---Jo"-r j_ur wr-le-w~%' Iv+Ude-W(n-~Xv+ur) dvdw

= Z n - - r (--1) j : ~ :-u~ r-- l --W[--~--~; ,"i] wj(l~q-lgr) j=o J 2J Jo J-oo TM e = e dvdw

l f o ~ / - - I t J r ' r u j ..~ ~ W r - 1 e-W[( -2j)v+ Y]i=)+ , i - Y~i=, u,] e--W(n--rXv+ur ) dv d w

.i -Uj+l

.r() Z n - - r (--1)) : ~ [--"r r - - " " • -~" W 1 eW[(r+J)v+JUr-}-~i=l ui] dv dw j=o J 2J Jo J - ~

1 r - 1 o o --uj

+ 2-'~JT_rj~=o f 0 /_uj+t wr-le-w[(n-2j)v+cr'"(j)]dOdw

where we set uo -- -oo , and we define

t

Cs, n(t) = (n - $)u r "Jr- ui -- Z ui" i = t + l i=1

N o w , i f n is odd, then (2.3) becomes

1 : E ( n - - r ) (--1) j fo °~ j=o J 2J-(7 q--)) w~-2e-'[cr"rfr)]dw

l { E l f o ° ° W r - 2 e - w [ C r , , ( J ) - ( n - 2 j ) U : ] d w +2-;=;-" j=o(2:-

- - 0 ( 3 < V ~ --Ur, w > O ,

--Ur ~ l) ~ O0, w > O ,

(2.2)

(2.3)

(2.4)

34 A. Childs, N. Balakrishnan I Statistics & Probability Letters 31 (1996) 31-39

} r--I 1 wr-2 e -w[G'"(j)-(n-2j)uj+d dw + Z ( n -

j=0

_ F ( r - 1 ) ~ - - ~ f n - r ' ~ (-1) j [G72~(r)~7--1Z.., ~, J j = o J 2 J ( ~ )

+ / ' ( ; ~ 1) ~ [Cr, n(j) - (n - 2J)Uj] l - r - [Cr, n ( j ) - (n - 2 j )uj+l] l - r

j=o (2j -- n)

=iF(r-- l ){ 1 ( ~ ( n - r ) (-1) j [G'2r(r)]~-I \j=o j 2J'-(; q--))

,) - - + 2n-r(n- 2r)

1 ~ [Cr, n(j) - (n - 2j)uj] 1-r

-+- 2n_r_-------- ~ ~-~ 7 n ) ~ 7 ~ j=l

(2.5)

If n is even, then (2.3) becomes

1 = E ( n _ r ) ( _ 1 ) j foo j=0 J 2J(r +j)Jo wr-Ze-WtCr'2a~)ldw

+ ~ (2 j - j#m

r-1

j=O j#m

1 ~ OO - - ~ l wr -2e -w[G'"(j)-(n-2j)uj+~] dw

(n - 2j)Jo

+ H (1 - 6r, j)(Um+l urn) j= l

F ( r - 1 ) ~ - ~ ( n - r ) ( - 1 ) j [G-~-2~(r)] r- ' j 2J~+-j)

j=O

+ F~n~_r l )~ [Cr, n ( j ) - ( n - 2j)uj]l-r-[Cr, n ( j ) - ( n - 2j)uj+l] 1-r

j=O j•m

+ f i ( 1 -- 6 ..~(Um+l ~um)F(r) j=l r,j, 2n_r[Cr, n(m)] r ,

(2j -- n)

(2.6)

where (~ i , j is the Kronecker delta.

A. Childs, N. Balakrishnan I Statistics & Probability Letters 31 (1996) 31-39 35

From (2.2), upon integrating out w, we derive the conditional density function of the pivotal quantity v to be

. - r n -- r ( - - 1 ) j kF(r)j~o ( j )2J[C2j, j(r--~-~_(j+r)v]r i f - o o < v < - u r ,

g(VIUl,U2 . . . . . Ur) =

kF(r) r if - Ur <<. v < oo, 2"-" [)-'~=, Iv + uil + (n - r)(v + ur)]

where k is given in (2.5) or (2.6). Conditional 100(1 - ~)% confidence intervals for the parameter 0 can be produced by finding two constants d and e such that

and

P ( v > e l u l , u 2 . . . . . Ur ) = ~ /2

P(v < d lUl,U2 . . . . . Ur) = ~ /2 .

The 100(1 - g)% conditional confidence interval for 0 is then (0 - ed, 0 - d#) with 0 and ~ given in (1.3)-(1.6).

For n odd, we similarly obtain the conditional density function of w to be

w r-2 ~_, 1 2)e_W[C,.,(j)_(n_2j)uA} + 2 "- ' - - - - -T ( 2 j - n)(n - 2j +

j= l

with k given by (2.5). And for n even we get,

I-

0(W[Ul,U2 . . . . . Ur) = k I E ( I ' I - F ) ( - - 1 ) J Wr--2e--WCr,2r(r) L :=° :

1 . . . . ( e--W[Cr.,,(J)--(n--2j)ujl __ e--w[C,.,,(J)--(n--2y)uj+d ) + 2"-r j=0 ( 2 j -- n )

jgm

I] + 1 - I (1 - &,. / ) (u , .+l - Um)wr- - l e -''cr''(m) ,

j= l

where k is given in (2.6). Conditional 100(1 - e ) % confidence intervals for the parameter a can be produced by finding two constants d and e such that

P( w > e lul,u2,...,Ur) = ~/2

and

P(w < d lul,u2 . . . . . u,) = or~2.

The 100(1 - g ) % conditional confidence interval for a is then (8/e,8/d) where d is given in (1.4) or (1.6).

36 A. Childs, AT. Balakrishnan I Statistics & Probability Letters 31 (1996) 31-39

Note that all of the results o f Kappenman (1975) may be derived simply by setting r = n in the above results.

3. Conditional tolerance intervals

To find a lower 7 probability conditional tolerance interval for proportion fl o f the form ( t ~ - q~, c~) we first evaluate the integral in (1.8) to get

F P v - q < - - ul,u2 . . . . . ur = 7 (3.1) 1_ w

where v = (0 - 0)/~, w = #/~ and d = In[2(1 - fl)]. We now write (3.1) as

~0 oo fq+d/w

J_~ f(v, wlul,u2 . . . . . u r ) d v d w = 7 .

Let h (~<r) be the smallest positive integer such that

Q(h) = f (v , w lul,u2 . . . . . u r )dvdw < 7"

I f there is no such value of h, i.e. if

f O ° ° / _ - u ~ f ( v , W l . . . . . ur )dvdw > ,

then we set Q(h) =_ O. Otherwise Q(h) may be evaluated explicitly in a manner very similar to that of the normalizing constant k derived in the previous section. It turns out that if n is odd then

k I ' ( r - 1 ) ~ - ~ ( n - r ) (--1) j Q(h) = [C--~2~(~]-~_ 1 j=0 J 2J(r + j )

kr(r - 1) ~ [C~,n(j) -- (n -- 2j)uj] 1-r -- [Cr, n( j ) - (n - 2j)uj+l] l-r

+ ~ - - 7 ~ (2j - n) (3.2)

j=h

and if n is even then

k F ( r - 1 ) ~ - ~ ( n - r ) ( - 1 ) J O(h) = [Cr,2r-------(rS]r-Z1 j=0 J 2J(r -q-j)

kV(r - 1) ~ [ C r , n ( j ) - (n - 2 j ) u j ] 1 - r - [Cr, n( j ) - (n - 2 j ) u j + l ] 1 - r

+ 2-~--7 ~ (2j -- n) j=h j¢m

+ 1-I (1 - ~ , j ) (1 - 6h i) k(um+l um)l"(r)

j=l i=m+l , 2n_r[Cr, n(m)]r , (3.3)

where Cs, n(t) is given in (2.4). Finally, we need to find a value of q such that

p + Q(h) = ~ (3.4)

A. Childs. N. Balakrishnan / Statistics & Probability Letters 31 (1996) 31-39 37

where

fcx~ fq+d/w Jo J-oo f(V'WlUl'U2 ... . . ur)dvdw if Q(h)-O,

P = foo fq+d/w Jo J--uh f(t)'WlUl'U2' ""Ur)dl)dw otherwise.

In the second case, an explicit expression for p may be derived using the second case in (2.2) as follows,

k f~ fq+d/w P= 2--~'ST-r JO J-uh wr-l e-WEiL' Iv+ude-W(n-r)(V+Ur) dvdw

_ _ k f ~ fq+a/w 2n-r Jo J-uh wr-le -w[(r-2(h-1))v+Ei~=h u~-E'h=-]~ "de-W(n-r)(v+=r) dv dw

_ k foo fq+d/Wwr_le_wt(n_2(h_l))v+Cr,.(h_l)ldvdw. (3.5) 2";--r Jo J-~h

Now if h - 1 ¢ n/2 then (3.5) becomes

. . . .

P = 2"-r[2(h-- 1) - n] _-v w~ 2e w[(. 2(h 1))(q+d/w)+Cr.(h 1)] dw

_ fo°°Wr-2e-W[-(n-2(h-l))uh+C.,.(h-1)] dw)

n r k F ( r - 1) { e -[n-2(h-l)]d = 2 - [ --~-- 1 ) - n ] [Cr, n (h- 1 ) - ~ Z 2 - ~ - 1))q] ~-~

1 } (3.6) [Cr, n(h - 1 ) - (n - - -2 (h- l ))Uh] r -1 '

and if h - 1 = n/2 (= m) then from (3.5) we get

kd f~wr-2e-wCr'"(m) dw k(q + uh) f°°wr_le_WC~,,(m) dw + P - ~U-7 Jo 2._---7 jo

= k(q + uh)F(r) kdF(r - 1 ) + (3.7) 2n-r[Cr, n(m)] r 2n-r[Cr, n(m)] r-l"

But if Q ( h ) - O, then we use the first case in (2.2) to get

too rq+d/w r p = [ / kwr-le-WE,=,lv+uil{1 _12eW(V+u)~ }n--r d/)dw

dO J -- cx~

=n-r ( n - r ) k ( - 1 ) J f°°[q+d/Wwr-lew[(r+j)v+jur+~,~=lU~]dl)d j=~ j 2J JO JO

1.~--, n - r e(~+J~(-1) j ~ k / ~ ( r -

j=o J 2J(r + j ) [ ~ - - ~ +j)ql r-~" (3.8)

38 A. Childs, N. Balakrishnan / Statistics & Probability Letters 31 (1996) 31-39

Use of the expression for p in (3.6) allows us to solve equation (3.4) explicitly to get for h - 1 ~ n/2,

q = - 1 )

n - 2(h - 1) (3.9)

where

a = 2 n - r [ 2 ( h - 1 ) - n ] [ 7 - Q ( h ) ]

kF(r - 1 )

b = [ C r . . ( h - 1 ) - ( n - 2 ( h - 1 ) ) u h ]

and

C = e -[n-2(h-1)]d.

For h - 1 = n/2, we use equation (3.7) in (3.4) to get

[7 - Q(h)][Cr, n(m)] r2n-r Cr,,(m)d q = kF(r) ( r - 1) Uh. (3.10)

But if Q(h) = 0 then we must use the expression for p in equation (3.8). As a result, (3.4) cannot be solved explicitly, and therefore numerical techniques will have to be employed.

To summarize the procedure for finding a lower ~ probability conditional tolerance interval for proportion fl of the form ( 0 - q d , c~), we start by calculating the MLEs 0 and 6 according to (1 .3)- (1 .6)and the ancillary statistics ui given in (1.7). We then calculate k using (2.5) if n is odd and (2.6) if n is even. Then we find, by successively computing Q(r), Q ( r - 1), Q ( r - 2) . . . . . or by trial and error, the smallest positive integer h (~<r) such that Q(h)< ~ where Q(h) is given in (3.2) if n is odd and (3.3) if n is even. If there is no such value of h then we must numerically solve for q in the equation p = y where p is given in (3.8). If there is such a value of h then q is given by (3.9) if h ~ n/2 + 1 and (3.10) if h = n/2 + 1.

The procedure for deriving an upper y conditional tolerance interval for proportion ~ of the form ( -c~ , 0 - q6) is obtained simply replacing d by - d and y by 1 - 7 in the above procedure.

Finally, we note that by setting r = n in all of the above results, we obtain a procedure equivalent to that of Kappenman (1977).

4. An example

Kappenman (1975, 1977) considers the following data,

1.96, 1.96, 3.60, 3.80, 4.79, 5.66, 5.76, 5.78, 6.27, 6.30, 6.76, 7.65,

7.84, 7.99, 8.51, 9.18, 10.13, 10.24, 10.25, 10.43, 11.45, 11.48, 11.75,

11.81, 12.34, 12.78, 13.06, 13.29, 13.98, 14.18, 14.40, 16.22, 17.06

which are assumed to represent a random sample of size 33 from a Laplace distribution. Based on this data, he found the 95% conditional confidence interval for 0 to be

8.99 < 0 < 12.41

and the 95% confidence interval for a to be

2.49 < a < 4.97.

He found that a lower ~ = 0.9 probability tolerance interval for proportion fl = 0.8 is [4.96, cx~].

A. Childs, N. Balakrishnan I Statistics & Probability Letters 31 (1996) 31-39 39

Now suppose that the largest 10 observations are censored. In this case, we use the method described in Section 2, with r = 23, to obtain the 95% conditional confidence interval for 0. It turns out to be

7.69 < 0 < 11.40

which is actually a 96.6% conditional confidence interval if the entire sample is used. The 95% conditional confidence interval for cr is

2.73 < cr < 6.31

which is actually a 91.5% conditional confidence interval if the entire sample is used. Now, the unconditional confidence intervals for 0 and a are obtained by simulating the distribution o f the

pivotal quantities (0 - 0) /d and d /a respectively, and they turn out to be (based on 5000 simulations)

8.47 < 0 < 11.79

and

2.76 < a < 6.46.

How much confidence do we have that the above intervals contain the true value o f 0 and ¢r if we condition on the ancillary statistics? The probability that the above intervals actually contain the true values of 0 and tr, respectively, when we condition on the ancillary statistics are 87% and 95%, respectively.

It is interesting to note here that Kappenman (1975) similarly found, for the full sample, that the uncon- ditional confidence interval for tr had the same conditional confidence level (95%), while the unconditional confidence interval for 0 had a lower (77%) conditional confidence level.

To find conditional tolerance intervals, for the censored sample, we take r = 23, n = 33, y --- 0.9, and fl = 0.8 in the procedure described in Section 3. We find from (2.5) that k = 3.3 x 1013 and from (3.2) that h = 15 is the smallest positive integer such that Q(h) < 0.9 (Q(15) = 0.868). Then (3.9) gives q = 1.57. Since 0 = 10.13 and t~ = 3.882 for the censored sample (using (1.3) and (1.4)), we conclude that a lower 0.9 probability tolerance interval for proportion 0.8 is [4.03, c~].

Acknowledgements

The authors would like to thank the Natural Sciences and Engineering Research Council o f Canada for supporting this research.

References

Balakrishnan, N. and C.D. Cutler (1995), Maximum likelihood estimation of Laplace parameters based on Type-II censored samples, in: H.N. Nagaraja, P.K. Sen and D.F. Morrison, eds., Statistical Theory and Applications: Papers in Honor o f Herbert A. David (Springer, New York) pp. 145-151.

Kappenman R.F. (1975), Conditional confidence intervals for double exponential distribution parameters, Technometrics 17, 233-235. Kappenman R.F. (1977), Tolerance intervals for the double exponential distribution, J. Amer. Statist. Assoc. 72, 908-909. Lawless, J.F. (1982), Statistical Models and Methods for Lifetime Data (Wiley, New York). Shyu, J.C. and D.B. Owen (1986a), One-sided tolerance intervals for the two-parameter double exponential distribution, Comm. Statist.

Simulation Comput. 15, 101-119. Shyu, J.C. and D.B. Owen (1986b), Two-sided tolerance intervals for the two-parameter double exponential distribution, Comm. Statist.

Simulation Comput. 15, 479-495.