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Reliability estimation and tolerance limits for Laplace distribution based on censored samples
Microelectron. Reliab., Vol. 336, No. 3, pp. 375-378, 1996 Elsevier ScienceLtd Printed in Great Britain 0026-2714/96 $15.00+ .00
Pergamon
0026-2714(95)00073-9
RELIABILITY E S T I M A T I O N AND T O L E R A N C E LIMITS F O R LAPLACE D I S T R I B U T I O N BASED O N C E N S O R E D SAMPLES 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 Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1 (Received for publication 24 March 1995)
Abstract--ln this paper, we first present an estimator for the reliability function based on the best linear unbiased estimators (BLUEs) of the location and scale parameters,/z and or, for the Laplace distribution based on Type-II censored samples. We show that this estimator is almost unbiased at varying levels of reliability. Next, we determine through Monte Carlo simulations the values of the tolerance factor tr that are necessary for the construction of lower and upper tolerance limits for the distribution. We also illustrate how these tables for tolerance limits could be used to determine lower confidence limits for the reliability. Finally, we present an example to illustrate the methods of inference developed in this paper.
explained. Finally, we consider a numerical example and illustrate all the methods of inference developed in this paper.
1. INTRODUCTION Let Xl:u ~< X2:N ~< '-" ~< XN-s:N be a Type-II rightcensored sample available from the Laplace (or double exponential) population with p.d.f.
2. RELIABILITY ESTIMATION
1
.f(x; ~, a) = - - e-I(x-~)/~l, 2a
-~
o o , - - ~ 0 .
First of all, we may observe from eqn (1) that the reliability of X at t is
(1)
Here, out of N items placed on a life-testing experiment, the largest s items have been censored. It should be mentioned here that Govindarajulu [1] 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 [2] derived the BLUEs of the parameters # and a based on symmetrically Type-II censored samples (with s largest and s smallest observations censored) for N up to 20. Extending the work of Govindarajulu I-2], Balakrishnan et al. I-3] recently tabulated the BLUEs of /~ and a based on right-censored samples, and also presented some percentage points of three pivotal quantities (determined by Monte Carlo simulations) which will enable one to construct confidence intervals and carry out tests of hypotheses for the parameters/~ and a. In this paper, we first present an estimator (based on the BLUEs of/~ and a) for the reliability function. Through Monte Carlo simulations, we show that this estimator is almost unbiased at varying levels of reliability. Next, we present the tolerance factors that are necessary for the construction of lower and upper tolerance limits, and these were determined through Monte Carlo simulations. The determination of lower confidence limits for the reliability through the use of these tables is also
~1 - ½e"-")/~, t~<#
Rx(t; #, a) = 1 - Fx(t; #, a) = [½ e-"-"~/',
t>~#.
(2) Let us denote the BLUEs of p and a determined from the given Type-II right-censored sample by #* and a*, respectively. The computational formulae for these estimators and their variances and covariances are given in Balakrishnan and Cohen ([4], pp. 80-81). As mentioned in the last section, Govindarajulu [2] presented tables of coefficients of the BLUEs p* and a* and their variances and covariance based on symmetrically censored samples for sample sizes up to 20. Similar tables were presented recently by Balakrishnan et al. [3] for the case of right-censored samples. Based on #* and a*, we propose a natural estimator for the reliability of X at t in eqn (2) to be
)'1 - ½e(*-"')/"*, t <~ #*
R~(t; U, a) = Rx(t; #*, a*) = (½e_,_u.)/~. '
t >~ l~*.
(3) In order to examine the performance of this estimator, we simulated (based on 5000 Monte Carlo runs) the values of mean and standard deviation of the estimator in eqn (3). In Table 1, we present these values when Rx(t; t~, a) values were taken to be 0.5(0.1)0.8(0.05)0.95, N = 20, and s = 0(1)10. It is apparent from this table that the estimator in eqn (3)
375
N. Balakrishnan and M. P. Chandramouleeswaran
376
Table 1. Simulated mean and standard deviation of R* for N = 20
Mean s
R
0.5
0.6
0.7
0.8
0.85
0.9
0.95
0 1 2 3 4 5 6 7 8 9 10
0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.499 0.498 0.496
0.594 0.594 0.594 0.594 0.595 0.595 0.595 0.596 0.596 0.595 0.593
0.696 0.696 0.696 0.697 0.697 0.697 0.698 0.699 0.699 0.699 0.698
0.798 0.798 0.798 0.799 0.799 0.799 0.800 0.801 0.801 0.801 0.801
0.848 0.848 0.848 0.849 0.849 0.849 0.850 0.850 0.850 0.851 0.851
0.898 0.898 0.898 0.898 0.898 0.898 0.899 0.899 0.899 0.899 0.899
0.947 0.947 0.947 0.947 0.947 0.947 0.947 0.947 0.947 0.947 0.947
0 1 2 3 4 5 6 7 8 9 10
0.1064 0.1065 0.1066 0.1067 0.1070 0.1072 0.1078 0.1081 0.1093 0.1122 0.1172
0.1013 0.1015 0.1017 0.1018 0.1022 0.1025 0.1031 0.1034 0.1044 0.1072 0.1122
0.0564 0.0571 0.0578 0.0584 0.0595 0.0606 0.0621 0.0630 0.0638 0.0646 0.0655
0.0443 0.0450 0.0457 0.0463 0.0474 0.0485 0.0498 0.0508 0.0516 0.0523 0.0529
0.0290 0.0296 0.0301 0.0306 0.0315 0.0324 0.0335 0.0344 0.0351 0.0358 0.0362
Standard deviation 0.0863 0.0867 0.0870 0.0873 0.0880 0.0886 0.0896 0.0900 0.0908 0.0929 0.0966
is almost unbiased at all the levels of reliability considered. It m a y also be noted from this table that the s t a n d a r d deviation decreases as R increases with N a n d s fixed while it increases as s increases with N and R fixed, as one would expect.
0.0670 0.0677 0.0682 0.0688 0.0698 0.0708 0.0722 0.0730 0.0737 0.0748 0.0765
The lower 7 tolerance limit for p r o p o r t i o n / 3 of the p o p u l a t i o n is given by L(X) = #* - t~.~r*,
(8)
where t~ is such t h a t
Pr{T(X) <~ ty} = 7.
(9)
3. TOLERANCE LIMITS The tolerance limit L(X) is said to be a lower 7 tolerance limit for p r o p o r t i o n fl if
Pr{ 1 - F(L(X); p, g) ~> [3} = 7,
(4)
Pr{F(L(X);/~, a) ~< 1 - fl} = 7.
(5)
T h a t is, t~ is simply the upper 1 - 7 percentage point of T(X) in eqn (7) when the sample comes from a Laplace (/t, a z) population, or equivalently, the upper 1 - 7 percentage point of the variable
or F o r a general location-scale family of distributions, D u m o n c e a u x I-5] has s h o w n t h a t tolerance limits can be constructed based on the pivotal quantity p* -- ~t
a
0"*
0-*
T(X) -
F x 1(1 _ fl),
(6)
where Fx(') denotes the s t a n d a r d cumulative distrib u t i o n function of X (with # = 0 and a = 1) and F;I(.) denotes its inverse. F o r the Laplace distribution considered here, in particular, we have F xl(1-fl)=
ln{2(1-fl)
= --ln(2fl)
if
~
1
if
o ~ fl ~<
if
½~
p*_ 1 ln{2(1-fl)} 0"*
if
½~
if
0~
O'*
# * + 1 ln(2fl) (7* O'*
when the sample is from the s t a n d a r d Laplace (0, 1) population. Also, due to the symmetry of the Laplace distribution, we have the upper y tolerance limit for the p r o p o r t i o n fl of the p o p u l a t i o n to be
u ( x ) = ~* + t,~a*,
(10)
where t~ is as given in eqn (9). We simulated (based on 5000 runs) the values of t~ for N = 20, s -- 0, 2, a n d some choices of /3 a n d ?. These values are presented in Table 2.
so that T(X) in eqn (6) becomes
/2*-T(X)=~P O'*
_P*-P+ O'*
a ln{2(1-fl)} 0"*
a ln(2fl) 0"*
if 0~
4. LOWER CONFIDENCE LIMIT FOR RELIABILITY (7)
It is well k n o w n that a relationship exists between the tolerance limits a n d confidence limits for the
Laplace distribution based on censored samples Table 2. Simulated values of tolerance factors t~.for N = 20 s
3
377
N o w ifRx(t; #, a) = fl, then F x t(1 - / 3 ) = (t - / ~ ) / a so that eqn (12) becomes
0.75
0.85
0.90
0.95
0.98
0.99
0.995
0 0.500 0.525 0.550 0.575 0.600 0.625 0.650 0.675 0.700 0.725 0.750 0.775 0.800 0.825 0.850 0.875 0.900 0.925 0.950 0.975
0.162 0.215 0.270 0.330 0.396 0.466 0.543 0.621 0.709 0.809 0.916 1.034 1.172 1.325 1.504 1.714 1.965 2.301 2.782 3.604
0.256 0.310 0.369 0.432 0.499 0.573 0.650 0.739 0.832 0.931 1.044 1.173 1.315 1.480 1.671 1.898 2.186 2.549 3.058 3.938
0.325 0.383 0.446 0.511 0.575 0.650 0.731 0.817 0.918 1.025 1.144 1.270 1.418 1.591 1.792 2.027 2.326 2.704 3.254 4.194
0.436 0.495 0.559 0.627 0.702 0.776 0.858 0.950 1.055 1.165 1.288 1.427 1.587 1.769 1.981 2.242 2.558 2.986 3.579 4.591
0.558 0.612 0.674 0.743 0.822 0.899 0.988 1.085 1.200 1.318 1.444 1.614 1.791 2.008 2.233 2.527 2.870 3.323 3.962 5.093
0.649 0.709 0.780 0.846 0.918 1.004 1.101 1.209 1.338 1.457 1.590 1.743 1.916 2.135 2.371 2.68l 3.070 3.542 4.230 5.452
0.726 0.802 0.862 0.938 1.009 1.096 1.185 1.280 1.401 1.542 1.683 1.847 2.076 2.291 2.569 2.867 3.286 3.807 4.556 5.794
pr~#*-= I~
2 0.500 0.525 0.550 0.575 0.600 0.625 0.650 0.675 0.700 0.725 0.750 0.775 0.800 0.825 0.850 0.875 0.900 0.925 0.950 0.975
0.162 0.215 0.271 0.333 0.400 0.470 0.548 0.628 0.717 0.816 0.925 1.043 1.179 1.333 1.5t5 1.732 1.995 2.335 2.806 3.624
0.255 0.313 0.372 0.435 0.501 0.573 0.653 0.739 0.834 0.940 1.056 1.184 1.327 1.494 1.688 1.920 2.209 2.570 3.099 4.015
0.323 0.384 0.445 0.512 0.580 0.656 0.740 0.831 0.927 1.032 1.153 1.289 1.443 1.618 1.815 2.061 2.370 2.769 3.338 4.297
0.437 0.493 0.555 0.623 0.697 0.775 0.862 0.964 1.065 1.179 1.305 1.448 1.615 1.811 2.029 2.297 2.633 3.053 3.642 4.692
0.568 0.629 0.694 0.765 0.837 0.917 1.015 1.114 1.229 1.353 1.487 1.642 1.816 2.025 2.280 2.561 2.925 3.378 4.06l 5.241
0.643 0.711 0.779 0.861 0.940 1.019 1.115 1.236 1.356 1.497 1.650 1.804 1.995 2.187 2.451 2.777 3.159 3.669 4.405 5.688
0.729 0.792 0.858 0.935 1.025 1.113 1.229 1.339 1.459 1.595 1.761 1.945 2.166 2.385 2.676 3.034 3.458 3.994 4.747 6.060
/t_l~, \ t.. = - - ~ ] = - F x ~ ( 1 - R*)
[
~(t-#)<<-Fxl(l-R~)}=
a*
if*\
a
,
J
,~ P r { 1 - F * ( t ; # , a ) = _ R ~ ( t ; # , a ) < ~ R * } = ? . We thus have Rx(t;/t, a ) = / 3
(13)
to be the value of
Rx(t ;#,a) t h a t makes the probability statement in eqn (13) true, and hence /3 is the lower confidence limit for Rx(t; I~, a) by the general m e t h o d of setting up a confidence interval. As D u m o n c e a u x [5] has shown, such a value exists a n d is i n d e p e n d e n t of the u n k n o w n parameters # a n d a a n d the distribution of R~:(t; I~, a) depends only on the u n k n o w n p a r a m e t e r
Rx(t; I~, ~). Table 2 can therefore be used to determine lower confidence limits for reliability based on R*(t;la, a). F o r observed values of kt* a n d a*, set
'
\
a*
/t
= --ln{2(1 -- R*)}
if
½ ~< R* < 1
= ln(2R*)
if
0 ~< R* ~< ½.
(14)
Then the value of/3 such t h a t Pr{T(X) ~< t~} = ? is the lower limit for the reliability.
5. ILLUSTRATIVE EXAMPLE Let us consider the d a t a presented earlier by B a l a k r i s h n a n et al. [3] (simulated with /a = 50 a n d a = 5) in which, out of N = 20 observations, the largest two have been censored. The Type-II rightcensored sample thus o b t a i n e d is as follows: 32.00692, 37.75687, 43.84736, 46.26761, 46.90651,
reliability function in eqn (2). While discussing the case of the n o r m a l distribution, for example, Lloyd and Lipow ([6], p. 204) stated that a lower confidence limit for the reliability function Rx(t; p, a) can be obtained by setting t = L(X) a n d then determining for a specified 7 what value of fi would make the probability statement in eqn (9) true. To this end, let us denote R* for the observed value of the estimate R~(t; el, a) in eqn (3) for the given sample. Then, by setting L ( X ) = e l * - t ~ , a * = t, we obtain
t. . . . / [ t - - ~ '
\
a*
'|*\ = - F x l ( 1 /
-R~).
(11)
Thence, we choose/3 to satisfy
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. In this case, B a l a k r i s h n a n et al. 1-3] c o m p u t e d the B L U E s of # a n d a to be I~* = 49.56095 a n d a* = 4.81270. Suppose we are interested in the reliability at t = 38 (h). Then, we have (t - g*)/a* = - 2 . 4 0 so that the point estimate of the reliability at t = 38 becomes [eqn (3)] R*(38; g, a) = 1 - ½e- 2.40 = 0.956. F r o m Table 1, by looking at N = 2 0 , s = 2 a n d R = 0.95, we may also determine the a p p r o x i m a t e s t a n d a r d error of this estimate to be 0.0301. A 95% lower tolerance limit for a p r o p o r t i o n of 0.80 of the p o p u l a t i o n is o b t a i n e d from Table 2 to be L(X) = p* - 1.615~r* = 49.56095 - 1.615(4.8127)
e r
O'*
- R~*)~ ) = ;,. ) (12)
= 41.7884 h. To determine a 95% lower confidence limit for the
378
N. Balakrishnan and M. P. Chandramouleeswaran
reliability at t = 38 h, we set t°'95
= _ (38 - p*'] = _ (38 - 49.56095") = k tr* / ~ J 2.40;
then from Table 2, we get fl---0.88 to be the 95% lower confidence limit for the reliability at t = 38 h. Acknowledgements--The first author would like to thank the Natural Sciences and Engineering Research Council of Canada for funding this research. REFERENCES
1. Z. Govindarajulu, Relationships among moments of order statistics in samples from two related populations, Technometrics 5, 514-518 (1963).
2. Z. Govindarajulu, Best linear estimates under symmetric censoring of the parameters of a double exponential population, J. Amer. Statist. Assoc. 61, 248-258 (1966). 3. N. Balakrishnan, M. P. Chandramouleeswaran and R. S. Ambagaspitiya, BLUEs of location and scale parameters of Laplace distribution based on Type-ll censored samples and associated inference, Microelectron. Reliab. (1995). 4. N. Balakrishnan and A. C. Cohen, Order Statistics and Inference." Estimation Methods. Academic Press, Boston (1991). 5. R. H. Dumonceaux, Statistical inferences for location and scale parameter distributions, Doctoral thesis, University of Missouri at Rolla, Rolla, MI (1969). 6. D. K. Lloyd and M. Lipow, Reliability: Manafement, Methods and Mathematics. Prentice-Hall, Englewood Cliffs, NJ (1962).
Pergamon
0026-2714(95)00073-9
RELIABILITY E S T I M A T I O N AND T O L E R A N C E LIMITS F O R LAPLACE D I S T R I B U T I O N BASED O N C E N S O R E D SAMPLES 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 Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1 (Received for publication 24 March 1995)
Abstract--ln this paper, we first present an estimator for the reliability function based on the best linear unbiased estimators (BLUEs) of the location and scale parameters,/z and or, for the Laplace distribution based on Type-II censored samples. We show that this estimator is almost unbiased at varying levels of reliability. Next, we determine through Monte Carlo simulations the values of the tolerance factor tr that are necessary for the construction of lower and upper tolerance limits for the distribution. We also illustrate how these tables for tolerance limits could be used to determine lower confidence limits for the reliability. Finally, we present an example to illustrate the methods of inference developed in this paper.
explained. Finally, we consider a numerical example and illustrate all the methods of inference developed in this paper.
1. INTRODUCTION Let Xl:u ~< X2:N ~< '-" ~< XN-s:N be a Type-II rightcensored sample available from the Laplace (or double exponential) population with p.d.f.
2. RELIABILITY ESTIMATION
1
.f(x; ~, a) = - - e-I(x-~)/~l, 2a
-~
o o , - - ~ 0 .
First of all, we may observe from eqn (1) that the reliability of X at t is
(1)
Here, out of N items placed on a life-testing experiment, the largest s items have been censored. It should be mentioned here that Govindarajulu [1] 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 [2] derived the BLUEs of the parameters # and a based on symmetrically Type-II censored samples (with s largest and s smallest observations censored) for N up to 20. Extending the work of Govindarajulu I-2], Balakrishnan et al. I-3] recently tabulated the BLUEs of /~ and a based on right-censored samples, and also presented some percentage points of three pivotal quantities (determined by Monte Carlo simulations) which will enable one to construct confidence intervals and carry out tests of hypotheses for the parameters/~ and a. In this paper, we first present an estimator (based on the BLUEs of/~ and a) for the reliability function. Through Monte Carlo simulations, we show that this estimator is almost unbiased at varying levels of reliability. Next, we present the tolerance factors that are necessary for the construction of lower and upper tolerance limits, and these were determined through Monte Carlo simulations. The determination of lower confidence limits for the reliability through the use of these tables is also
~1 - ½e"-")/~, t~<#
Rx(t; #, a) = 1 - Fx(t; #, a) = [½ e-"-"~/',
t>~#.
(2) Let us denote the BLUEs of p and a determined from the given Type-II right-censored sample by #* and a*, respectively. The computational formulae for these estimators and their variances and covariances are given in Balakrishnan and Cohen ([4], pp. 80-81). As mentioned in the last section, Govindarajulu [2] presented tables of coefficients of the BLUEs p* and a* and their variances and covariance based on symmetrically censored samples for sample sizes up to 20. Similar tables were presented recently by Balakrishnan et al. [3] for the case of right-censored samples. Based on #* and a*, we propose a natural estimator for the reliability of X at t in eqn (2) to be
)'1 - ½e(*-"')/"*, t <~ #*
R~(t; U, a) = Rx(t; #*, a*) = (½e_,_u.)/~. '
t >~ l~*.
(3) In order to examine the performance of this estimator, we simulated (based on 5000 Monte Carlo runs) the values of mean and standard deviation of the estimator in eqn (3). In Table 1, we present these values when Rx(t; t~, a) values were taken to be 0.5(0.1)0.8(0.05)0.95, N = 20, and s = 0(1)10. It is apparent from this table that the estimator in eqn (3)
375
N. Balakrishnan and M. P. Chandramouleeswaran
376
Table 1. Simulated mean and standard deviation of R* for N = 20
Mean s
R
0.5
0.6
0.7
0.8
0.85
0.9
0.95
0 1 2 3 4 5 6 7 8 9 10
0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.499 0.498 0.496
0.594 0.594 0.594 0.594 0.595 0.595 0.595 0.596 0.596 0.595 0.593
0.696 0.696 0.696 0.697 0.697 0.697 0.698 0.699 0.699 0.699 0.698
0.798 0.798 0.798 0.799 0.799 0.799 0.800 0.801 0.801 0.801 0.801
0.848 0.848 0.848 0.849 0.849 0.849 0.850 0.850 0.850 0.851 0.851
0.898 0.898 0.898 0.898 0.898 0.898 0.899 0.899 0.899 0.899 0.899
0.947 0.947 0.947 0.947 0.947 0.947 0.947 0.947 0.947 0.947 0.947
0 1 2 3 4 5 6 7 8 9 10
0.1064 0.1065 0.1066 0.1067 0.1070 0.1072 0.1078 0.1081 0.1093 0.1122 0.1172
0.1013 0.1015 0.1017 0.1018 0.1022 0.1025 0.1031 0.1034 0.1044 0.1072 0.1122
0.0564 0.0571 0.0578 0.0584 0.0595 0.0606 0.0621 0.0630 0.0638 0.0646 0.0655
0.0443 0.0450 0.0457 0.0463 0.0474 0.0485 0.0498 0.0508 0.0516 0.0523 0.0529
0.0290 0.0296 0.0301 0.0306 0.0315 0.0324 0.0335 0.0344 0.0351 0.0358 0.0362
Standard deviation 0.0863 0.0867 0.0870 0.0873 0.0880 0.0886 0.0896 0.0900 0.0908 0.0929 0.0966
is almost unbiased at all the levels of reliability considered. It m a y also be noted from this table that the s t a n d a r d deviation decreases as R increases with N a n d s fixed while it increases as s increases with N and R fixed, as one would expect.
0.0670 0.0677 0.0682 0.0688 0.0698 0.0708 0.0722 0.0730 0.0737 0.0748 0.0765
The lower 7 tolerance limit for p r o p o r t i o n / 3 of the p o p u l a t i o n is given by L(X) = #* - t~.~r*,
(8)
where t~ is such t h a t
Pr{T(X) <~ ty} = 7.
(9)
3. TOLERANCE LIMITS The tolerance limit L(X) is said to be a lower 7 tolerance limit for p r o p o r t i o n fl if
Pr{ 1 - F(L(X); p, g) ~> [3} = 7,
(4)
Pr{F(L(X);/~, a) ~< 1 - fl} = 7.
(5)
T h a t is, t~ is simply the upper 1 - 7 percentage point of T(X) in eqn (7) when the sample comes from a Laplace (/t, a z) population, or equivalently, the upper 1 - 7 percentage point of the variable
or F o r a general location-scale family of distributions, D u m o n c e a u x I-5] has s h o w n t h a t tolerance limits can be constructed based on the pivotal quantity p* -- ~t
a
0"*
0-*
T(X) -
F x 1(1 _ fl),
(6)
where Fx(') denotes the s t a n d a r d cumulative distrib u t i o n function of X (with # = 0 and a = 1) and F;I(.) denotes its inverse. F o r the Laplace distribution considered here, in particular, we have F xl(1-fl)=
ln{2(1-fl)
= --ln(2fl)
if
~
1
if
o ~ fl ~<
if
½~
p*_ 1 ln{2(1-fl)} 0"*
if
½~
if
0~
O'*
# * + 1 ln(2fl) (7* O'*
when the sample is from the s t a n d a r d Laplace (0, 1) population. Also, due to the symmetry of the Laplace distribution, we have the upper y tolerance limit for the p r o p o r t i o n fl of the p o p u l a t i o n to be
u ( x ) = ~* + t,~a*,
(10)
where t~ is as given in eqn (9). We simulated (based on 5000 runs) the values of t~ for N = 20, s -- 0, 2, a n d some choices of /3 a n d ?. These values are presented in Table 2.
so that T(X) in eqn (6) becomes
/2*-T(X)=~P O'*
_P*-P+ O'*
a ln{2(1-fl)} 0"*
a ln(2fl) 0"*
if 0~
4. LOWER CONFIDENCE LIMIT FOR RELIABILITY (7)
It is well k n o w n that a relationship exists between the tolerance limits a n d confidence limits for the
Laplace distribution based on censored samples Table 2. Simulated values of tolerance factors t~.for N = 20 s
3
377
N o w ifRx(t; #, a) = fl, then F x t(1 - / 3 ) = (t - / ~ ) / a so that eqn (12) becomes
0.75
0.85
0.90
0.95
0.98
0.99
0.995
0 0.500 0.525 0.550 0.575 0.600 0.625 0.650 0.675 0.700 0.725 0.750 0.775 0.800 0.825 0.850 0.875 0.900 0.925 0.950 0.975
0.162 0.215 0.270 0.330 0.396 0.466 0.543 0.621 0.709 0.809 0.916 1.034 1.172 1.325 1.504 1.714 1.965 2.301 2.782 3.604
0.256 0.310 0.369 0.432 0.499 0.573 0.650 0.739 0.832 0.931 1.044 1.173 1.315 1.480 1.671 1.898 2.186 2.549 3.058 3.938
0.325 0.383 0.446 0.511 0.575 0.650 0.731 0.817 0.918 1.025 1.144 1.270 1.418 1.591 1.792 2.027 2.326 2.704 3.254 4.194
0.436 0.495 0.559 0.627 0.702 0.776 0.858 0.950 1.055 1.165 1.288 1.427 1.587 1.769 1.981 2.242 2.558 2.986 3.579 4.591
0.558 0.612 0.674 0.743 0.822 0.899 0.988 1.085 1.200 1.318 1.444 1.614 1.791 2.008 2.233 2.527 2.870 3.323 3.962 5.093
0.649 0.709 0.780 0.846 0.918 1.004 1.101 1.209 1.338 1.457 1.590 1.743 1.916 2.135 2.371 2.68l 3.070 3.542 4.230 5.452
0.726 0.802 0.862 0.938 1.009 1.096 1.185 1.280 1.401 1.542 1.683 1.847 2.076 2.291 2.569 2.867 3.286 3.807 4.556 5.794
pr~#*-= I~
2 0.500 0.525 0.550 0.575 0.600 0.625 0.650 0.675 0.700 0.725 0.750 0.775 0.800 0.825 0.850 0.875 0.900 0.925 0.950 0.975
0.162 0.215 0.271 0.333 0.400 0.470 0.548 0.628 0.717 0.816 0.925 1.043 1.179 1.333 1.5t5 1.732 1.995 2.335 2.806 3.624
0.255 0.313 0.372 0.435 0.501 0.573 0.653 0.739 0.834 0.940 1.056 1.184 1.327 1.494 1.688 1.920 2.209 2.570 3.099 4.015
0.323 0.384 0.445 0.512 0.580 0.656 0.740 0.831 0.927 1.032 1.153 1.289 1.443 1.618 1.815 2.061 2.370 2.769 3.338 4.297
0.437 0.493 0.555 0.623 0.697 0.775 0.862 0.964 1.065 1.179 1.305 1.448 1.615 1.811 2.029 2.297 2.633 3.053 3.642 4.692
0.568 0.629 0.694 0.765 0.837 0.917 1.015 1.114 1.229 1.353 1.487 1.642 1.816 2.025 2.280 2.561 2.925 3.378 4.06l 5.241
0.643 0.711 0.779 0.861 0.940 1.019 1.115 1.236 1.356 1.497 1.650 1.804 1.995 2.187 2.451 2.777 3.159 3.669 4.405 5.688
0.729 0.792 0.858 0.935 1.025 1.113 1.229 1.339 1.459 1.595 1.761 1.945 2.166 2.385 2.676 3.034 3.458 3.994 4.747 6.060
/t_l~, \ t.. = - - ~ ] = - F x ~ ( 1 - R*)
[
~(t-#)<<-Fxl(l-R~)}=
a*
if*\
a
,
J
,~ P r { 1 - F * ( t ; # , a ) = _ R ~ ( t ; # , a ) < ~ R * } = ? . We thus have Rx(t;/t, a ) = / 3
(13)
to be the value of
Rx(t ;#,a) t h a t makes the probability statement in eqn (13) true, and hence /3 is the lower confidence limit for Rx(t; I~, a) by the general m e t h o d of setting up a confidence interval. As D u m o n c e a u x [5] has shown, such a value exists a n d is i n d e p e n d e n t of the u n k n o w n parameters # a n d a a n d the distribution of R~:(t; I~, a) depends only on the u n k n o w n p a r a m e t e r
Rx(t; I~, ~). Table 2 can therefore be used to determine lower confidence limits for reliability based on R*(t;la, a). F o r observed values of kt* a n d a*, set
'
\
a*
/t
= --ln{2(1 -- R*)}
if
½ ~< R* < 1
= ln(2R*)
if
0 ~< R* ~< ½.
(14)
Then the value of/3 such t h a t Pr{T(X) ~< t~} = ? is the lower limit for the reliability.
5. ILLUSTRATIVE EXAMPLE Let us consider the d a t a presented earlier by B a l a k r i s h n a n et al. [3] (simulated with /a = 50 a n d a = 5) in which, out of N = 20 observations, the largest two have been censored. The Type-II rightcensored sample thus o b t a i n e d is as follows: 32.00692, 37.75687, 43.84736, 46.26761, 46.90651,
reliability function in eqn (2). While discussing the case of the n o r m a l distribution, for example, Lloyd and Lipow ([6], p. 204) stated that a lower confidence limit for the reliability function Rx(t; p, a) can be obtained by setting t = L(X) a n d then determining for a specified 7 what value of fi would make the probability statement in eqn (9) true. To this end, let us denote R* for the observed value of the estimate R~(t; el, a) in eqn (3) for the given sample. Then, by setting L ( X ) = e l * - t ~ , a * = t, we obtain
t. . . . / [ t - - ~ '
\
a*
'|*\ = - F x l ( 1 /
-R~).
(11)
Thence, we choose/3 to satisfy
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. In this case, B a l a k r i s h n a n et al. 1-3] c o m p u t e d the B L U E s of # a n d a to be I~* = 49.56095 a n d a* = 4.81270. Suppose we are interested in the reliability at t = 38 (h). Then, we have (t - g*)/a* = - 2 . 4 0 so that the point estimate of the reliability at t = 38 becomes [eqn (3)] R*(38; g, a) = 1 - ½e- 2.40 = 0.956. F r o m Table 1, by looking at N = 2 0 , s = 2 a n d R = 0.95, we may also determine the a p p r o x i m a t e s t a n d a r d error of this estimate to be 0.0301. A 95% lower tolerance limit for a p r o p o r t i o n of 0.80 of the p o p u l a t i o n is o b t a i n e d from Table 2 to be L(X) = p* - 1.615~r* = 49.56095 - 1.615(4.8127)
e r
O'*
- R~*)~ ) = ;,. ) (12)
= 41.7884 h. To determine a 95% lower confidence limit for the
378
N. Balakrishnan and M. P. Chandramouleeswaran
reliability at t = 38 h, we set t°'95
= _ (38 - p*'] = _ (38 - 49.56095") = k tr* / ~ J 2.40;
then from Table 2, we get fl---0.88 to be the 95% lower confidence limit for the reliability at t = 38 h. Acknowledgements--The first author would like to thank the Natural Sciences and Engineering Research Council of Canada for funding this research. REFERENCES
1. Z. Govindarajulu, Relationships among moments of order statistics in samples from two related populations, Technometrics 5, 514-518 (1963).
2. Z. Govindarajulu, Best linear estimates under symmetric censoring of the parameters of a double exponential population, J. Amer. Statist. Assoc. 61, 248-258 (1966). 3. N. Balakrishnan, M. P. Chandramouleeswaran and R. S. Ambagaspitiya, BLUEs of location and scale parameters of Laplace distribution based on Type-ll censored samples and associated inference, Microelectron. Reliab. (1995). 4. N. Balakrishnan and A. C. Cohen, Order Statistics and Inference." Estimation Methods. Academic Press, Boston (1991). 5. R. H. Dumonceaux, Statistical inferences for location and scale parameter distributions, Doctoral thesis, University of Missouri at Rolla, Rolla, MI (1969). 6. D. K. Lloyd and M. Lipow, Reliability: Manafement, Methods and Mathematics. Prentice-Hall, Englewood Cliffs, NJ (1962).