(Double Distributions
4 METHODS OF INFERENCE .1 Maximum Likelihood Estimation
4.2 Best Linear Unbiased Estimation
Let
. .
be the doubly censored sampleavailable from a sample of size n , where the smallest r and the largest observations have been censored. From Eqs. we can compute the means, variances, and covariances of order statistics from the standard distribution and we denote them by and Further let us denote
Then the best linear unbiased estimators of 8 and based on the given doubly censored sample, are given by [see David
nan and Cohen (1991, pp.
(DOUBLE EXPONENTIAL) DISTRIBUTIONS
n - s
=
The variances and covariance of these estimators are given by
For the case where the available sample is symmetrically Type-I1 censored r = in (24.41) will equal since = Further in this case the coefficient of in in (24.37) is the same as that of
the coefficient of in in (24.38) is the same as that of in magnitude but is of opposite sign. (1966) presented tables of the coefficients and and the values of V,, and for sample sizes up to 20 and all possible choices of r = s. Balakrishnan, Chandramouleeswaran, and (1994) presented similar tables for right-censored samples of sizes up to 20 with = and s = - 2.
Table 24.1 [from Govindarajulu gives the coefficients of and in samples of sizes n = and r = = - 21/21. The final column gives the values of and
Sarhan (1954) has compared the variances of the best linear estimator of 8, the median (defined as the arithmetic mean of and when n is even), the arithmetic mean and the midrange
+
These are all unbiased estimators of 8. Table 24.2 presents the efficiencies (inverse ratio of variances, expressed as a percentage) of the last three estimators relative to the first. Figure represent these values dia- grammatically. The irregular appearance of Figure is associated with the different definition of the median in samples of odd and even sizes.
We note that the estimator - of (with known) is dis- tributed as with 2n degrees of freedom). The distribution of
METHODS OF INFERENCE
Table 24.2 Efficiency of various estimators of 8, relative to best linear unbiased estimator
In Table 24.3 the coefficients for the best linear unbiased estimators of the mean 9 and the standard deviation = and the values of
and [taken from Sarhan are presented for = 3, 4, and 5 for the case of right censoring with r = 0 and
-
2. As we men- tioned earlier, Balakrishnan, Chandramouleeswaran, and Ambagaspitiya (1994) present an extended form of this table for sizes.
to 20. - -Sample Size Sample Site Sample Size
Figure 24.2 Percentage efficiencies of the sample mean, midrange. and median in different populations.
Table 24.3 Coefficients of best linear unbiased estimators of expected value and standard deviation = for samples censored by omission of largest values
"Relative efficiency (inverse ratio of variance to that of best linear unbiased estimator using the complete sample) is shown, as a percentage, in parentheses.
interval for
4
are thenand 2
I
If neither 8 nor is known it would be possible to construct confidence intervals for 8 and respectively, using the distributions of
-
e
nand - (24.43)
which are pivotal quantities for the parameters and
4,
respectively. Bain and Engelhardt have determined exact distributions for = 3 and= 5 and have provided approximate distributions for larger n . These authors have also given the asymptotic distributions of the pivotal quantities and the powers of the associated tests of hypotheses.
For the case of complete as well as censored samples,
nan, Chandramouleeswaran, and Arnbagaspitiya considered three pivotal quantities by using the best linear unbiased estimators and in Eqs. (24.37) and (24.38) and their variances in Eqs. (24.39) and
e * - e
and -
METHODS OF INFERENCE
and making inferences on 8 when is known, on when is unknown, and on when is unknown, respectively. These authors presented some percentage points of all three pivotal quantities for sample sizes up to 20 for various choices of censoring. Edgeworth approximations for the distributions of the pivotal quantities in have been discussed by Balakrishnan,
Chandramouleeswaran, and who have also examined
their accuracy.
Referring to the tables of given by
vasan and Wharton (1982) discussed the derivation of one-sided and two-sided confidence bands on the entire cumulative distribution function
These bands are constructed using the statistics.
For example, the two-sided band on is based on the statistic
while the one-sided upper confidence contour for is based on the statistic
= -
For a if 1 , the a t h quantile of L , = a ) , then a two-sided confidence band for with confidence level a is given by the planar region bounded above by the function y =
+
I,, 1) and bounded below by y = 0*, - Srinivasan and Wharton (1982) have presented tables of simulated percentage points of and for n up to 20; these are presented in Tables 24.4 and 24.5.
and Wharton have also discussed some large-sample ap- proximations for the percentage points of and For example, by using the asymptotic result that = and = - are indepen- dent standard normal variables (in the case of the standard distribu- tion), they have shown that the limiting distribution of is the same as that of the random variable where
V y ) , y
<
This expression readily gives approximate quantiles for when is large. and Wharton (1982) have mentioned that this asymptotic approximation works quite effectively for 30.
By considering just the location-Laplace model (with = Sugiura and Naing (1989) derived improved estimators of 8 in the form of a weighted linear combination of the sample median and pairs of order statistics (with symmetric distance to both sides from the sample median) and by minimizing with respect to weights and distances. The resulting estimator has been
Table 24.4 Simulated percentage points of the statistic a
Table 24.5 Simulated percentage points of the statistic a
METHODS O F INFERENCE
shown to have smaller asymptotic variance in the second order; see also Akahira (1987,1990) and Akahira and Takeuchi (1993).
4.3 Simplified Linear Estimation
By considering the ith quasi-range, =
,
- and the ith quasi- midrange, =+
Raghunandanan and Srinivasan (1971) proposed simplified linear estimators of the parameters and 8. Their estimator of is defined to be that with the smallest variance. The estimator is presented in Table 24.6 for n up to 20, and the efficiency of this estimator relative to the best linear unbiased estimator of 8 based on the complete sample of size n (Table 24.1) is also presented in this table. The estimator8
presented in the table is also applicable when the available sample is symmetrically Type-I1 censored with r i - 1. We may note that for n = 3 and 5, the estimator in Table 24.6 is simply the sample median.Raghunandanan and Srinivasan's estimator of which is based on a symmetrically Type-I1 censored sample with r smallest and r largest observa- tions censored, is defined to be the one with minimum variance among linear estimators of the form
Table 24.6 Estimator and its efficiency
Table 24.7 Estimator and its efficiency
4 0 0.289157
+
0.300624 0.9935 0 0.231325
+
0.229000 1.0006 0
+ +
0.186515 0.9966 1 0.666667 0.304009 0.985
7 0
+ +
0.1565007 1 0.390721
+
0.234731 0.9758 0 0.134254
+ + +
0.135438 0.9978 1 0.324571
+
0.188570 0.9848 2 0.967133 0.303726 0.994
9 0
+ + +
0.1 19000 19 0.282882
+
0.158812 0.9839 2 0.790855 0.233068 0.985
10 0
+ + +
0.106392 0.99810 1 0.238741
+ +
0.137784 0.98010 2 0.681084 0.190810 0.973
10 3 1.267536 0.305295 0.997
where each takes the values 0 or 1 and D is the constant that would make the estimator unbiased; as a result D is given by
In Table 24.7 the estimator is presented for = for various choices of r. The efficiency of this estimator relative to the BLUE based on the symmetrically Type-11 censored sample (see Table 24.1) is also presented in this table. More elaborate tables have been provided by Raghunandanan and Srinivasan Similar simplified linear estimators for the normal case have been discussed in Chapter 13.
Iliescu and (1973) have presented minimum mean-square-error estimator of the form
for the parameter These authors have also presented the appropriate values of for = It may be noted that this estimator is essentially of the same form as Raghunandanan and Srinivasan's simplified linear estimator (with all taken to be 1).
METHODS OF INFERENCE