- Email: [email protected]
BLUEs of location and scale parameters of laplace distribution based on type-II censored samples and associated inference
Microelectron.Reliab.,Vol. 36, No. 3, pp. 371-374, 1996 Elsevier ScienceLtd
Pergamon
Printed in Great Britain
0026-2714/96 $15.00+.00 0026-2714(95)00072-0
BLUEs OF LOCATION AND SCALE PARAMETERS OF LAPLACE DISTRIBUTION BASED ON TYPE-II CENSORED SAMPLES AND ASSOCIATED INFERENCE N. B A L A K R I S H N A N and M. P. C H A N D R A M O U L E E S W A R A N McMaster University, Hamilton, Ontario, Canada L8S 4K1 R. S. AMBAGASPITIYA University of Calgary, Calgary, Alberta, Canada T2N 1N4
(Received for publication 24 March 1995) Abstract--In this paper, we first present the Best Linear Unbiased Estimators (BLUEs) of the location and scale parameters, # and a, of the Laplace distribution based on Type-II censored samples. We next present some simulated percentage points for three pivotal quantities which will enable one to construct confidence intervals or carry out tests of hypotheses for the parameters # and or. Finally, we present an example to illustrate the methods of inference developed in this paper.
1. I N T R O D U C T I O N Let X,+l: N ~< X,+2:N ~ < ' " ~< X~-~:N be a doubly Type-II censored sample available from the Laplace (or double exponential) population with p.d.f.
f(x;/~, -~
1
~r) = _ e-I(x-~/~l, 2a <#<
~,a>0.
(1)
Here, out of N items placed on a life-testing experiment, the smallest r and the largest s items have been censored. The most common situation in a life-testing problem is, of course, a Type-II rightcensored sample (with r = 0 and s > 0) as the experimenter will often terminate the experiment as soon as a certain number of items have failed instead of waiting for all the items to fail; for example, see Cohen and Whitten [1] and Balakrishnan and Cohen [2]. It should be mentioned here that Govindarajulu [3] derived explicit expressions for the single and product moments of order statistics from the standard double exponential distribution, in terms of the corresponding quantities from the standard exponential distribution. By making use of these expressions, Govindarajulu [4] derived the BLUEs of the parameters/~ and a based on symmetrically Type-II censored samples (with r = s ) for N up to 20. R a g h u n a n d a n a n and Srinivasan [5] derived simplified linear estimators of /~ and a in this situation. Ali et al. [6] discussed the estimation of quantiles based on optimally selected order statistics. Shyu and Owen I-7, 8] constructed the one-sided and twosided tolerance intervals based on complete samples. Balakrishnan and Ambagaspitiya [9] discussed the 371
robustness features of various linear estimators of # and a when a single scale-outlier is present in the sample. In this paper, we first derive the BLUEs of p and a, and present the tables of coefficients and variances and covariances of these estimators for N = 20 for the case of right-censored samples (with r = 0 and s = 0 ( 2 ) N - 2). Next, we present some percentage points of three pivotal quantities determined by Monte Carlo simulations. These pivotal quantities, based on the BLUEs of/1 and a, will enable one to construct confidence intervals and carry out tests of hypotheses for the parameters # and a based on the available Type-II right-censored sample. Finally, we present a numerical example to illustrate the methods of inference developed in this paper.
2. BLUEs OF /l A N D o
Let Zi:N = (Xi:N -- #)/or, i = r + 1, r + 2. . . . . N -- s. Then, Zi:N are order statistics from the standard Laplace distribution with p.d.f.
f(z)=½e-I~l,
-~
~.
(2)
Then, Govindarajulu [3, 4] has given the following expressions for the first two single moments and the product moments of Z~:N:
-~,(N)sI(m,i)},
(3)
N. Balakrishnan et al.
372
oid: N2 ) : E ( Z ~ u ) : 2 - u [
~
and
(N){sz(m,N-i+I)
I_m=N-i+l
COV(/Z*, o-*) = - a
+ S2(m,N - i + 1)}
= cr2~,
+ ,,=~Z( N ) { s 2 ( m ' i ) + S Z ( m ' i ) } J '
(4) and
l
+ St(m, N - i + 1)Sdm, N - j + 1)} -J
(10)
F r o m eqns (6) and (7), we computed the coefficients
a i and b i for Type-II right-censored samples ( r - - 0 and s = 0(1)N - 2) for sample size N = 3(1)20. Some of these values for N = 20 are presented in Table 1. F r o m eqns (8)-(10), we computed the values of V~, V2 and V3 and these are presented in Table 2. Similar tables have been prepared by Govindarajulu [4] for the case when the available sample is symmetrically Type-II censored.
~i,~:~ = E(Zi:~Z2:u) I._ra=N-i+
<((~TB_ l~)(lTB±il ) _ ( ~TB- tl)Z
S a ( m , i ) S I ( N - j + 1, N - m ) m=l
+ ~= =j
S2(m,j) + Sl(m, i)St(m,j)
, (5)
m
Based on the B L U E s #* and a* in eqns (6) and (7), let us define
where i2
Sp(il, i2) = E n-P, p = 1, 2
p*-p a*-a P1 - a~V-1 , P2 - ax/ ~V2 and
n-il
and for i 2 >~ in, Sp(i2, it) = Sp(il, i2). F r o m the above formulae, one can then compute the variance fl~,~:u = Var(Zi:N) and the covariance fli, j:N = Cov(Z/:N, Zj:u). Let us now denote
X = (X,+I:N, X,+2:~ . . . . . X~-~:N) T, = (0~r+ t:N, ~r+2:N, • • ' , ~N s:N) T,
r+ 1
B=((fl,,j:N)), 1
.
.( 1., 1.,
.
T r ~)×t. 1)(N
Then, the B L U E s of/~ and a are given by (gT B - I~IT B - 1
_ _
gT B - l l a T B - 1 )
. . . . X ~(0[TB_ I~)(1TB_ 1 | ) -- (0cTB_ 1i)2 ~
/A*
n-s
= ~
i=r+t
3. PIVOTAL QUANTITIES AND INFERENCE FOR /t AND a
aiXi:N
(6)
p~-p ¢* • (11)
P3 . . . . . .
It is easily verified that Pt, P2 and P3 are all pivotal quantities, that is, their distributions do not involve the parameters # and a. P1 can be used to draw inference for/~ when a is known while/'3 can be used to draw inference for # when a is unknown. Similarly, P2 can be used to draw inference for a when p is unknown. The exact sampling distributions of the pivotal quantities Pt, P~ and P3 in eqn (11) are intractable. Hence, we resorted to Monte Carlo simulations and determined the 1, 2.5, 5, 10, 90, 95, 97.5 and 99% points of the distributions o f / 1 , P2 and P3 based on Type-II right-censored samples (r = 0 and s = 0(1)N - 2) for sample size N = 3(1)10, 12, 15 and 20. These values were determined through 5000 Monte Carlo runs. Results for N = 20 are presented in Table 3.
and 'ITB - t IgTB- t
=
_ _
1TB - t0~lT B - t )
4. ILLUSTRATIVE EXAMPLE
11) -- (0~TB- 11)2~ n-s
= ~
biXi:~,;
(7)
i=r+l
see, for example, David ([I0], pp. 129-130) and Balakrishnan and Cohen ([2], pp. 80-81). Furthermore, the variances and covariance of these B L U E s are given by
~ ) ( 1 T ~ ri~ ~_ (~TB= ~2 Vt,
f
(8)
32.00692, 37.75687, 43.84736, 46.26761, 46.90651, 47.26220, 47.28952, 47.59391, 48.06508, 49.25429, 50.27790, 50.48675, 50.66167, 53.33585, 53.49258, 53.56681, 53.98112, 54.94154. Then, from Table 1 with N = 20 and s = 2, we obtain the B L U E s of p and a to be #* = (0.0000 x 32.00692) + (0.000 x 37.75687)
I T B - 11
Var(a*) = o2 ~ (~TB- t ~ ) ( l ~ : q ) -- W B - q ~ J = a 2 V2,
We present below a simulated data set (with # = 50 and a = 5) in which, out of N = 20 observations, the largest two have been censored. The Type-II rightcensored sample thus obtained is as follows:
+ . . . + (0.0004 x 53.98112) (9)
+ (0.0001 x 54.94154) = 49.56095
373
BLUEs of location and scale parameters of Laplace distribution based on Type-II censored samples Table 1. Coefficients a~ and b~ for N = 20 and r = 0 s
a~
bi
s
0
0.0000 0.0000 0.0001 0.0004 0.0025 0.0105 0.0336 0.0825 O.1550 0.2153 0.2153 0.1550 0.0825 0.0336 0.0105 0.0025 0.0004 0.0001 0.0000 0.0000
- 0.0514 - 0.0514 - 0.0515 -- 0.0515 - 0.0519 -0.0530 - 0.0543 - 0.0530 - 0.0423 -0.0167 0.0167 0.0423 0.0530 0.0543 0.0530 0.0519 0.0515 0.0515 0.0514 0.0514
4
0.0000 0.0000 0.0001 0.0004 0.0025 0.0105 0.0336 0.0825 0.1550 0.2153 0.2153 0.1550 0.0825 0.0336 0.0105 0.0025 0.0004 0.0001
- 0.0573 - 0.0573 -0.0574 -0.0575 -0.0579 -0.0590 -0.0605 -0.0591 -0.0471 -0.0187 0.0187 0.0471 0.0591 0.0605 0.0590 0.0579 0.0575 0.1720
6
ai
bl
s
0.0000 0.0000 0.0000 0.0004 0.0025 0.0105 0.0335 0.0825 0.1550 0.2153 0.2153 0.1550 0.0825 0.0336 0.0106 0.0031
- 0.0648 - 0.0648 - 0.0648 -- 0.0649 - 0.0654 -0.0667 - 0.0684 - 0.0667 - 0.0532 -0.0211 0.0211 0.0533 0.0668 0.0684 0.0667 0,3246
- 0.0002 - 0.0002 - 0.0002 0.0002 0.0023 0.0103 0.0333 0.0823 0.1548 0.2153 0.2154 0.1552 0.0828 0.0487
- 0.0745 - 0.0745 - 0.0746 - 0.0747 - 0.0753 - 0.0767 -0.0786 0.0767 0.0611 -0.0240 0.0246 0.0615 0.0770 0.5277
-0.0024 -0.0024 -0.0023 -0.0020 0.0001 0.0081 0.0311 0.0802
-0.0884 -0.0884 -0.0884 -0.0886 -0.0892 -0.0909 -0.0929 -0.0901
and
a~
bi
0.1534 0.2151 0.2168 0.3043
- 0.0706 - 0.0255 0.0326 0.7806
10
-0.0133 - 0.0133 - 0.0132 - 0.0129 -0.0108 -0.0028 0.0206 0.0719 O.1505 0.8233
-0.1095 - 0.1095 - O.1095 - O.1097 -0.1104 -0.1121 -0.1133 - O.1066 - 0.0768 0.9573
12
-
-
14
--0.1244 --0.1244 --0.1243 --0.1239 --0.1210 1.6179
--0.1999 --0.1999 --0.2000 --0.2001 --0.2002 1.0001
16
-0.3571 -0.3571 -0.3570 2.0712
-0.3333 -0.3333 -0.3333 1.0000
1.9046 2.9046
- 1.0000 1.0000
18
-
0.0456 0.0456 0.0455 0.0451 0.0430 0.0341 0.0067 1.2657
O.1423 O.1423 O.1424 O.1426 0.1432 O.1440 O.1412 0.9980
which yields
a* = ( - 0 . 0 5 7 3 x 32.00692) + ( - 0 . 0 5 7 3 × 37.75687) +...
+ (0.0575 × 53.98112)
+ (0.1720 × 54.94154) = 4.81270.
(p*-0.41a*,It*
+ 0 . 4 2 4 * ) = ( 4 7 . 5 8 7 7 , 51.5823)
to be the 90)~, confidence interval for/~. Similarly, s u p p o s e we seek a 90~o c o n f i d e n c e interval for 6. T h e n , f r o m T a b l e 3 with N = 20 a n d s = 2, we find
F r o m T a b l e 2, we t h e n o b t a i n the s t a n d a r d e r r o r s of the a b o v e estimates to be Pr
o'* - o" t - - 1 . 5 1 ~ < P 2 = t r y 2 2 ~< 1.72 = 0 . 9 0
S E ( p * ) = a * x f V ~ = 4.81270 × x/0.0637 = 1.21467 w h i c h readily gives
and S E ( o * ) = c r * x ~ 2 = 4.81270 × [email protected] = 1.15204. S u p p o s e we seek a 90~o confidence interval for/1. Then, f r o m T a b l e 3 with N = 2 0 and s=2, we find Pr
{
- 0 . 4 2 ~< P3 = - -P* a ~ - P ~< 0.41} = 0.90
(
o* 1 + 1 . 7 2 ~ V 2' 1 -
al_5~~ )
= (3.4121, 7.5370)
to be the 9090 confidence interval for a.
A c k n o w l e d g e m e n t s - - T h e first and the last authors would like to thank the Natural Sciences and Engineering Research Council of Canda for funding this research.
374
N. B a l a k r i s h n a n et al. T a b l e 2. Values of I/1, Vz a n d V3 for N = 20 a n d r = 0
s
v,
v2
v3
s
v,
v2
v3
0 2 4 6 8
0.0637 0.0637 0.0637 0.0637 0.0641
0.0514 0.0573 0.0648 0.0745 0.0884
0.0000 0.0000 0.0000 0.0002 0.0024
10 12 14 16 18
0.0700 0.1021 0.2103 0.6176 4.2236
0.1095 0.1423 0.1999 0.3333 1.0000
0.0133 0.0456 0.1244 0.3571 0.9046
T a b l e 3. P e r c e n t a g e points o f / ' 1 , / ' 2 a n d P3 for N = 20 a n d r = 0 s
1%
2.5~
5~o
10~o
90%
95%
97.5%
99%
P~
0 2 4 6 8 10 12 14 16 18
-2.31 2.31 -2.31 -2.3l 2.31 -2.47 -2.59 -2.42 -2.25 - 1.62
-
1.90 1.90 1.90 1.90 1.92 -2.00 -2.12 -2.00 - 1.82 - 1.41
- 1.59 - 1.59 1.59 - 1.60 - 1.57 - 1.64 - 1.74 - 1.65 - 1.56 - 1.21
--
1.22 1.22 1.22 1.22 1.21 1.26 1.30 1.24 1.21 1.00
1.22 1.22 1.23 1.23 1.20 1.17 1.18 1.23 1.28 1.30
1.63 1.63 1,63 1.62 1.6l 1.54 1.52 1.57 1.70 1.86
1,98 1.98 1.98 1.99 1.99 1.98 1.82 1.87 2.13 2.48
2.43 2.43 2.43 2.44 2.46 2.42 2.17 2.28 2.68 3.23
P2
0 2 4 6 8 10 12 14 16 18
--2.00 1.94 -- 1.96 -- 1.88 -- 1.84 -- 1.77 -- 1.70 -- 1.62 -- 1.46 -- 0.99
-- 1.74 -- 1.75 -- 1.73 -- 1.66 -- 1.64 -- 1.60 -- 1.54 -- 1.45 -- 1.36 -- 0.97
-- 1.52 -- 1.51 -- 1.49 1.46 -- 1.46 -- 1.40 -- 1.36 -- 1.30 -- 1.23 -- 0.95
-- 1.22 -- 1.21 -- 1.22 -- 1.21 -- 1.19 -- 1.18 -- 1.16 -- 1.13 -- 1.08 -- 0.89
1.31 1.32 1.30 1.29 1.30 1.32 1.29 1.30 1.34 1.33
1.77 1.72 1.78 1.76 1.78 1.82 1.85 1.81 1.89 1.95
2.15 2.18 2.17 2.24 2.21 2.22 2.32 2.38 2.42 2.55
2.60 2.66 2.74 2.74 2.81 2.83 2.92 2.95 3.10 3.37
I°3
0 2 4 6 8 10 12 14 16 18
--0.60 0.61 --0.61 -- 0.61 -- 0.66 -- 0,90 -- 1,37 --2.53 -- 6.72 --163,43
--0.51 --0.50 --0.50 -- 0.50 -- 0.53 -- 0.70 -- 1.01 -- 1.84 -- 4.76 --74.87
--0.42 --0.42 --0.42 -- 0.42 -- 0.43 -- 0.52 --0.78 -- 1.34 -- 3.21 --35,82
--0.31 --0.32 --0.32 -- 0.32 -- 0.32 -- 0.38 --0.54 --0,88 -- 2,08 --16.31
0.32 0.32 0,32 0.33 0.33 0.33 0.34 0.44 0.65 1.23
0.41 0.41 0.41 0.42 0.42 0.42 0.44 0.53 0.77 1.41
0.51 0.5l 0.51 0.51 0.53 0.52 0.52 0.61 0,84 1.52
0,64 0.66 0.66 0.66 0.65 0.64 0.66 0.70 0.92 1.61
REFERENCES 1. A. C. C o h e n a n d B. J. Whitten, Parameter Estimation in Reliability and Life Span Models. M a r c e l D e k k e r , N e w Y o r k (1988). 2. N. B a l a k r i s h n a n a n d A. C. C o h e n , Order Statistics and InJerence: Estimation Methods. A c a d e m i c Press, San Diego (1991). 3. Z. G o v i n d a r a j u l u , Relationships a m o n g m o m e n t s of o r d e r statistics in samples from two related populations, Technometrics 5, 5 1 4 - 5 1 8 (1963). 4. Z. G o v i n d a r a j u l u , Best linear estimates u n d e r s y m m e t r i c censoring of the p a r a m e t e r s of a d o u b l e e x p o n e n t i a l p o p u l a t i o n , J. Amer. Statist. Assoc. 61, 2 4 8 - 2 5 8 (1966). 5. K. R a g h u n a n d a n a n a n d R. Srinivasan, Simplified estim a t i o n of p a r a m e t e r s in a d o u b l e e x p o n e n t i a l distribution, Technometrics 13, 689-691 (1971).
6. M. M. Ali, D. U m b a c h a n d K. M. H a s s a n e i n , E s t i m a t i o n o f q u a n t i l e s of e x p o n e n t i a l a n d d o u b l e e x p o n e n t i a l distrib u t i o n s b a s e d on two o r d e r statistics, Commun. Statist. Theor. Meth. 10, 1921-1932 (1981). 7. J . C . Shyu a n d D. B. O w e n , O n e - s i d e d tolerance intervals for the t w o - p a r a m e t e r d o u b l e e x p o n e n t i a l distribution, Commun. Statist. Simul. Comput. 15, 101-119 (1986). 8. J. C. Shyu a n d D. B. O w e n , T w o - s i d e d tolerance intervals for the t w o - p a r a m e t e r d o u b l e e x p o n e n t i a l distribution, Commun. Statist. Simul. Comput. 15, 4 7 9 - 4 9 5 (1986). 9. N. B a l a k r i s h n a n a n d R. S. A m b a g a s p i t i y a , Relationships a m o n g m o m e n t s of o r d e r statistics in s a m p l e s f r o m two related outlier m o d e l s a n d s o m e applications, Commun. Statist. Theor. Meth. 17, 2327-2341 (1988). 10. H . A . D a v i d , Order Statistics, 2nd Edn. Wiley, N e w Y o r k (1981).
Pergamon
Printed in Great Britain
0026-2714/96 $15.00+.00 0026-2714(95)00072-0
BLUEs OF LOCATION AND SCALE PARAMETERS OF LAPLACE DISTRIBUTION BASED ON TYPE-II CENSORED SAMPLES AND ASSOCIATED INFERENCE N. B A L A K R I S H N A N and M. P. C H A N D R A M O U L E E S W A R A N McMaster University, Hamilton, Ontario, Canada L8S 4K1 R. S. AMBAGASPITIYA University of Calgary, Calgary, Alberta, Canada T2N 1N4
(Received for publication 24 March 1995) Abstract--In this paper, we first present the Best Linear Unbiased Estimators (BLUEs) of the location and scale parameters, # and a, of the Laplace distribution based on Type-II censored samples. We next present some simulated percentage points for three pivotal quantities which will enable one to construct confidence intervals or carry out tests of hypotheses for the parameters # and or. Finally, we present an example to illustrate the methods of inference developed in this paper.
1. I N T R O D U C T I O N Let X,+l: N ~< X,+2:N ~ < ' " ~< X~-~:N be a doubly Type-II censored sample available from the Laplace (or double exponential) population with p.d.f.
f(x;/~, -~
1
~r) = _ e-I(x-~/~l, 2a <#<
~,a>0.
(1)
Here, out of N items placed on a life-testing experiment, the smallest r and the largest s items have been censored. The most common situation in a life-testing problem is, of course, a Type-II rightcensored sample (with r = 0 and s > 0) as the experimenter will often terminate the experiment as soon as a certain number of items have failed instead of waiting for all the items to fail; for example, see Cohen and Whitten [1] and Balakrishnan and Cohen [2]. It should be mentioned here that Govindarajulu [3] derived explicit expressions for the single and product moments of order statistics from the standard double exponential distribution, in terms of the corresponding quantities from the standard exponential distribution. By making use of these expressions, Govindarajulu [4] derived the BLUEs of the parameters/~ and a based on symmetrically Type-II censored samples (with r = s ) for N up to 20. R a g h u n a n d a n a n and Srinivasan [5] derived simplified linear estimators of /~ and a in this situation. Ali et al. [6] discussed the estimation of quantiles based on optimally selected order statistics. Shyu and Owen I-7, 8] constructed the one-sided and twosided tolerance intervals based on complete samples. Balakrishnan and Ambagaspitiya [9] discussed the 371
robustness features of various linear estimators of # and a when a single scale-outlier is present in the sample. In this paper, we first derive the BLUEs of p and a, and present the tables of coefficients and variances and covariances of these estimators for N = 20 for the case of right-censored samples (with r = 0 and s = 0 ( 2 ) N - 2). Next, we present some percentage points of three pivotal quantities determined by Monte Carlo simulations. These pivotal quantities, based on the BLUEs of/1 and a, will enable one to construct confidence intervals and carry out tests of hypotheses for the parameters # and a based on the available Type-II right-censored sample. Finally, we present a numerical example to illustrate the methods of inference developed in this paper.
2. BLUEs OF /l A N D o
Let Zi:N = (Xi:N -- #)/or, i = r + 1, r + 2. . . . . N -- s. Then, Zi:N are order statistics from the standard Laplace distribution with p.d.f.
f(z)=½e-I~l,
-~
~.
(2)
Then, Govindarajulu [3, 4] has given the following expressions for the first two single moments and the product moments of Z~:N:
-~,(N)sI(m,i)},
(3)
N. Balakrishnan et al.
372
oid: N2 ) : E ( Z ~ u ) : 2 - u [
~
and
(N){sz(m,N-i+I)
I_m=N-i+l
COV(/Z*, o-*) = - a
+ S2(m,N - i + 1)}
= cr2~,
+ ,,=~Z( N ) { s 2 ( m ' i ) + S Z ( m ' i ) } J '
(4) and
l
+ St(m, N - i + 1)Sdm, N - j + 1)} -J
(10)
F r o m eqns (6) and (7), we computed the coefficients
a i and b i for Type-II right-censored samples ( r - - 0 and s = 0(1)N - 2) for sample size N = 3(1)20. Some of these values for N = 20 are presented in Table 1. F r o m eqns (8)-(10), we computed the values of V~, V2 and V3 and these are presented in Table 2. Similar tables have been prepared by Govindarajulu [4] for the case when the available sample is symmetrically Type-II censored.
~i,~:~ = E(Zi:~Z2:u) I._ra=N-i+
<((~TB_ l~)(lTB±il ) _ ( ~TB- tl)Z
S a ( m , i ) S I ( N - j + 1, N - m ) m=l
+ ~= =j
S2(m,j) + Sl(m, i)St(m,j)
, (5)
m
Based on the B L U E s #* and a* in eqns (6) and (7), let us define
where i2
Sp(il, i2) = E n-P, p = 1, 2
p*-p a*-a P1 - a~V-1 , P2 - ax/ ~V2 and
n-il
and for i 2 >~ in, Sp(i2, it) = Sp(il, i2). F r o m the above formulae, one can then compute the variance fl~,~:u = Var(Zi:N) and the covariance fli, j:N = Cov(Z/:N, Zj:u). Let us now denote
X = (X,+I:N, X,+2:~ . . . . . X~-~:N) T, = (0~r+ t:N, ~r+2:N, • • ' , ~N s:N) T,
r+ 1
B=((fl,,j:N)), 1
.
.( 1., 1.,
.
T r ~)×t. 1)(N
Then, the B L U E s of/~ and a are given by (gT B - I~IT B - 1
_ _
gT B - l l a T B - 1 )
. . . . X ~(0[TB_ I~)(1TB_ 1 | ) -- (0cTB_ 1i)2 ~
/A*
n-s
= ~
i=r+t
3. PIVOTAL QUANTITIES AND INFERENCE FOR /t AND a
aiXi:N
(6)
p~-p ¢* • (11)
P3 . . . . . .
It is easily verified that Pt, P2 and P3 are all pivotal quantities, that is, their distributions do not involve the parameters # and a. P1 can be used to draw inference for/~ when a is known while/'3 can be used to draw inference for # when a is unknown. Similarly, P2 can be used to draw inference for a when p is unknown. The exact sampling distributions of the pivotal quantities Pt, P~ and P3 in eqn (11) are intractable. Hence, we resorted to Monte Carlo simulations and determined the 1, 2.5, 5, 10, 90, 95, 97.5 and 99% points of the distributions o f / 1 , P2 and P3 based on Type-II right-censored samples (r = 0 and s = 0(1)N - 2) for sample size N = 3(1)10, 12, 15 and 20. These values were determined through 5000 Monte Carlo runs. Results for N = 20 are presented in Table 3.
and 'ITB - t IgTB- t
=
_ _
1TB - t0~lT B - t )
4. ILLUSTRATIVE EXAMPLE
11) -- (0~TB- 11)2~ n-s
= ~
biXi:~,;
(7)
i=r+l
see, for example, David ([I0], pp. 129-130) and Balakrishnan and Cohen ([2], pp. 80-81). Furthermore, the variances and covariance of these B L U E s are given by
~ ) ( 1 T ~ ri~ ~_ (~TB= ~2 Vt,
f
(8)
32.00692, 37.75687, 43.84736, 46.26761, 46.90651, 47.26220, 47.28952, 47.59391, 48.06508, 49.25429, 50.27790, 50.48675, 50.66167, 53.33585, 53.49258, 53.56681, 53.98112, 54.94154. Then, from Table 1 with N = 20 and s = 2, we obtain the B L U E s of p and a to be #* = (0.0000 x 32.00692) + (0.000 x 37.75687)
I T B - 11
Var(a*) = o2 ~ (~TB- t ~ ) ( l ~ : q ) -- W B - q ~ J = a 2 V2,
We present below a simulated data set (with # = 50 and a = 5) in which, out of N = 20 observations, the largest two have been censored. The Type-II rightcensored sample thus obtained is as follows:
+ . . . + (0.0004 x 53.98112) (9)
+ (0.0001 x 54.94154) = 49.56095
373
BLUEs of location and scale parameters of Laplace distribution based on Type-II censored samples Table 1. Coefficients a~ and b~ for N = 20 and r = 0 s
a~
bi
s
0
0.0000 0.0000 0.0001 0.0004 0.0025 0.0105 0.0336 0.0825 O.1550 0.2153 0.2153 0.1550 0.0825 0.0336 0.0105 0.0025 0.0004 0.0001 0.0000 0.0000
- 0.0514 - 0.0514 - 0.0515 -- 0.0515 - 0.0519 -0.0530 - 0.0543 - 0.0530 - 0.0423 -0.0167 0.0167 0.0423 0.0530 0.0543 0.0530 0.0519 0.0515 0.0515 0.0514 0.0514
4
0.0000 0.0000 0.0001 0.0004 0.0025 0.0105 0.0336 0.0825 0.1550 0.2153 0.2153 0.1550 0.0825 0.0336 0.0105 0.0025 0.0004 0.0001
- 0.0573 - 0.0573 -0.0574 -0.0575 -0.0579 -0.0590 -0.0605 -0.0591 -0.0471 -0.0187 0.0187 0.0471 0.0591 0.0605 0.0590 0.0579 0.0575 0.1720
6
ai
bl
s
0.0000 0.0000 0.0000 0.0004 0.0025 0.0105 0.0335 0.0825 0.1550 0.2153 0.2153 0.1550 0.0825 0.0336 0.0106 0.0031
- 0.0648 - 0.0648 - 0.0648 -- 0.0649 - 0.0654 -0.0667 - 0.0684 - 0.0667 - 0.0532 -0.0211 0.0211 0.0533 0.0668 0.0684 0.0667 0,3246
- 0.0002 - 0.0002 - 0.0002 0.0002 0.0023 0.0103 0.0333 0.0823 0.1548 0.2153 0.2154 0.1552 0.0828 0.0487
- 0.0745 - 0.0745 - 0.0746 - 0.0747 - 0.0753 - 0.0767 -0.0786 0.0767 0.0611 -0.0240 0.0246 0.0615 0.0770 0.5277
-0.0024 -0.0024 -0.0023 -0.0020 0.0001 0.0081 0.0311 0.0802
-0.0884 -0.0884 -0.0884 -0.0886 -0.0892 -0.0909 -0.0929 -0.0901
and
a~
bi
0.1534 0.2151 0.2168 0.3043
- 0.0706 - 0.0255 0.0326 0.7806
10
-0.0133 - 0.0133 - 0.0132 - 0.0129 -0.0108 -0.0028 0.0206 0.0719 O.1505 0.8233
-0.1095 - 0.1095 - O.1095 - O.1097 -0.1104 -0.1121 -0.1133 - O.1066 - 0.0768 0.9573
12
-
-
14
--0.1244 --0.1244 --0.1243 --0.1239 --0.1210 1.6179
--0.1999 --0.1999 --0.2000 --0.2001 --0.2002 1.0001
16
-0.3571 -0.3571 -0.3570 2.0712
-0.3333 -0.3333 -0.3333 1.0000
1.9046 2.9046
- 1.0000 1.0000
18
-
0.0456 0.0456 0.0455 0.0451 0.0430 0.0341 0.0067 1.2657
O.1423 O.1423 O.1424 O.1426 0.1432 O.1440 O.1412 0.9980
which yields
a* = ( - 0 . 0 5 7 3 x 32.00692) + ( - 0 . 0 5 7 3 × 37.75687) +...
+ (0.0575 × 53.98112)
+ (0.1720 × 54.94154) = 4.81270.
(p*-0.41a*,It*
+ 0 . 4 2 4 * ) = ( 4 7 . 5 8 7 7 , 51.5823)
to be the 90)~, confidence interval for/~. Similarly, s u p p o s e we seek a 90~o c o n f i d e n c e interval for 6. T h e n , f r o m T a b l e 3 with N = 20 a n d s = 2, we find
F r o m T a b l e 2, we t h e n o b t a i n the s t a n d a r d e r r o r s of the a b o v e estimates to be Pr
o'* - o" t - - 1 . 5 1 ~ < P 2 = t r y 2 2 ~< 1.72 = 0 . 9 0
S E ( p * ) = a * x f V ~ = 4.81270 × x/0.0637 = 1.21467 w h i c h readily gives
and S E ( o * ) = c r * x ~ 2 = 4.81270 × [email protected] = 1.15204. S u p p o s e we seek a 90~o confidence interval for/1. Then, f r o m T a b l e 3 with N = 2 0 and s=2, we find Pr
{
- 0 . 4 2 ~< P3 = - -P* a ~ - P ~< 0.41} = 0.90
(
o* 1 + 1 . 7 2 ~ V 2' 1 -
al_5~~ )
= (3.4121, 7.5370)
to be the 9090 confidence interval for a.
A c k n o w l e d g e m e n t s - - T h e first and the last authors would like to thank the Natural Sciences and Engineering Research Council of Canda for funding this research.
374
N. B a l a k r i s h n a n et al. T a b l e 2. Values of I/1, Vz a n d V3 for N = 20 a n d r = 0
s
v,
v2
v3
s
v,
v2
v3
0 2 4 6 8
0.0637 0.0637 0.0637 0.0637 0.0641
0.0514 0.0573 0.0648 0.0745 0.0884
0.0000 0.0000 0.0000 0.0002 0.0024
10 12 14 16 18
0.0700 0.1021 0.2103 0.6176 4.2236
0.1095 0.1423 0.1999 0.3333 1.0000
0.0133 0.0456 0.1244 0.3571 0.9046
T a b l e 3. P e r c e n t a g e points o f / ' 1 , / ' 2 a n d P3 for N = 20 a n d r = 0 s
1%
2.5~
5~o
10~o
90%
95%
97.5%
99%
P~
0 2 4 6 8 10 12 14 16 18
-2.31 2.31 -2.31 -2.3l 2.31 -2.47 -2.59 -2.42 -2.25 - 1.62
-
1.90 1.90 1.90 1.90 1.92 -2.00 -2.12 -2.00 - 1.82 - 1.41
- 1.59 - 1.59 1.59 - 1.60 - 1.57 - 1.64 - 1.74 - 1.65 - 1.56 - 1.21
--
1.22 1.22 1.22 1.22 1.21 1.26 1.30 1.24 1.21 1.00
1.22 1.22 1.23 1.23 1.20 1.17 1.18 1.23 1.28 1.30
1.63 1.63 1,63 1.62 1.6l 1.54 1.52 1.57 1.70 1.86
1,98 1.98 1.98 1.99 1.99 1.98 1.82 1.87 2.13 2.48
2.43 2.43 2.43 2.44 2.46 2.42 2.17 2.28 2.68 3.23
P2
0 2 4 6 8 10 12 14 16 18
--2.00 1.94 -- 1.96 -- 1.88 -- 1.84 -- 1.77 -- 1.70 -- 1.62 -- 1.46 -- 0.99
-- 1.74 -- 1.75 -- 1.73 -- 1.66 -- 1.64 -- 1.60 -- 1.54 -- 1.45 -- 1.36 -- 0.97
-- 1.52 -- 1.51 -- 1.49 1.46 -- 1.46 -- 1.40 -- 1.36 -- 1.30 -- 1.23 -- 0.95
-- 1.22 -- 1.21 -- 1.22 -- 1.21 -- 1.19 -- 1.18 -- 1.16 -- 1.13 -- 1.08 -- 0.89
1.31 1.32 1.30 1.29 1.30 1.32 1.29 1.30 1.34 1.33
1.77 1.72 1.78 1.76 1.78 1.82 1.85 1.81 1.89 1.95
2.15 2.18 2.17 2.24 2.21 2.22 2.32 2.38 2.42 2.55
2.60 2.66 2.74 2.74 2.81 2.83 2.92 2.95 3.10 3.37
I°3
0 2 4 6 8 10 12 14 16 18
--0.60 0.61 --0.61 -- 0.61 -- 0.66 -- 0,90 -- 1,37 --2.53 -- 6.72 --163,43
--0.51 --0.50 --0.50 -- 0.50 -- 0.53 -- 0.70 -- 1.01 -- 1.84 -- 4.76 --74.87
--0.42 --0.42 --0.42 -- 0.42 -- 0.43 -- 0.52 --0.78 -- 1.34 -- 3.21 --35,82
--0.31 --0.32 --0.32 -- 0.32 -- 0.32 -- 0.38 --0.54 --0,88 -- 2,08 --16.31
0.32 0.32 0,32 0.33 0.33 0.33 0.34 0.44 0.65 1.23
0.41 0.41 0.41 0.42 0.42 0.42 0.44 0.53 0.77 1.41
0.51 0.5l 0.51 0.51 0.53 0.52 0.52 0.61 0,84 1.52
0,64 0.66 0.66 0.66 0.65 0.64 0.66 0.70 0.92 1.61
REFERENCES 1. A. C. C o h e n a n d B. J. Whitten, Parameter Estimation in Reliability and Life Span Models. M a r c e l D e k k e r , N e w Y o r k (1988). 2. N. B a l a k r i s h n a n a n d A. C. C o h e n , Order Statistics and InJerence: Estimation Methods. A c a d e m i c Press, San Diego (1991). 3. Z. G o v i n d a r a j u l u , Relationships a m o n g m o m e n t s of o r d e r statistics in samples from two related populations, Technometrics 5, 5 1 4 - 5 1 8 (1963). 4. Z. G o v i n d a r a j u l u , Best linear estimates u n d e r s y m m e t r i c censoring of the p a r a m e t e r s of a d o u b l e e x p o n e n t i a l p o p u l a t i o n , J. Amer. Statist. Assoc. 61, 2 4 8 - 2 5 8 (1966). 5. K. R a g h u n a n d a n a n a n d R. Srinivasan, Simplified estim a t i o n of p a r a m e t e r s in a d o u b l e e x p o n e n t i a l distribution, Technometrics 13, 689-691 (1971).
6. M. M. Ali, D. U m b a c h a n d K. M. H a s s a n e i n , E s t i m a t i o n o f q u a n t i l e s of e x p o n e n t i a l a n d d o u b l e e x p o n e n t i a l distrib u t i o n s b a s e d on two o r d e r statistics, Commun. Statist. Theor. Meth. 10, 1921-1932 (1981). 7. J . C . Shyu a n d D. B. O w e n , O n e - s i d e d tolerance intervals for the t w o - p a r a m e t e r d o u b l e e x p o n e n t i a l distribution, Commun. Statist. Simul. Comput. 15, 101-119 (1986). 8. J. C. Shyu a n d D. B. O w e n , T w o - s i d e d tolerance intervals for the t w o - p a r a m e t e r d o u b l e e x p o n e n t i a l distribution, Commun. Statist. Simul. Comput. 15, 4 7 9 - 4 9 5 (1986). 9. N. B a l a k r i s h n a n a n d R. S. A m b a g a s p i t i y a , Relationships a m o n g m o m e n t s of o r d e r statistics in s a m p l e s f r o m two related outlier m o d e l s a n d s o m e applications, Commun. Statist. Theor. Meth. 17, 2327-2341 (1988). 10. H . A . D a v i d , Order Statistics, 2nd Edn. Wiley, N e w Y o r k (1981).