9 METHODS OF INFERENCE
9.2 Simple Linear Estimation
Upon noting that the likelihood equations for and 8 do not admit explicit solutions and hence need to be solved by numerical iterative methods, (1956) suggested a simple modification to the equation for 8 (based on the equation for that makes it easier to solve the resulting equation.
The equation for 8 given by
used in conjunction with the equation for given by
can be rewritten as
1
=
+
- log (22.74)is the estimated cumulative By replacing Iog in (22.74) with the expected value of log Kimball
METHODS OF INFERENCE 29 derived a simplified linear estimator for 8 as
which may be further approximated as
The estimator in (22.75) or in (22.76) is a linear function of the order statistics, and hence its bias and mean square error can be determined easily from means, variances and covariances of order statistics in Tables 22.3 and 22.4. Since the linear estimator in is biased, presented a table of corrective multipliers to make it unbiased; from the table it appears that for n 10 the estimator
is very nearly unbiased. Further a simplified linear estimator of may then be obtained as
Estimator of = - (Estimator of (22.78)
I
Due to the linearity of the estimator of 8, it is only natural to compare it with the best linear unbiased estimator of 8 and with its approximations proposed
by Blom and (1961).
(1966) carried out a number of comparisons of this nature. He actually discussed the type 1 distribution appropriate to minima, with lative distribution function 1 - e
,
but his results also apply to the type 1 distribution in (22.1) (with some simple changes). His results are all in terms of efficiencies, that is, ratios of the values given by (22.65) to corre- sponding variances for the estimators in question. For each estimator of 8, the parameter was estimated from (22.78). Tables 22.5 and 22.6, taken from give efficiencies for various estimators of and 8.For the small values of n considered, the asymptotic formulas used in the calculations may not be accurate, yet the tables probably give a good idea of relative efficiency and the performance of different estimators considered. It can be seen from Table 22.5 that the location parameter can be estimated with quite good accuracy using simple linear functions of order statistics;
however, it may also be noted from Table 22.6 that the situation is rather unsatisfactory should one use such simple linear functions of order statistics to estimate the scale parameter 8.
30 EXTREME VALUE DISTRIBUTIONS
Table 22.5 Efficiencies of linear unbiased estimators of for the extreme value distribution
Best linear 84.05 91.73 94.45 95.82 96.65 100.00
Blom's
approximation 84.05 91.72 94.37 95.68 96.45 100.00 Weiss's
approximation 84.05 91.73 94.41 95.74 96.53 Kimball's
approximation 84.05 91.71 94.45 95.82 96.63 Note: Efficiencies are expressed in percentages.
For the case of a Type-I1 right-censored sample from the ty e 1 extreme value distribution for minima with cdf 1 - e , Bain (1972) suggested a simple unbiased linear estimator for the scale parameter 8. This estimator was subsequently modified by Engelhardt and Bain to the form
where
n - s
= - being the order statistics from the standard type 1 extreme
Table 22.6 Efficiencies of linear unbiased estimators of for the extreme value distribution
--
Best linear 42.70 58.79 67.46 72.96 76.78 100.00
Blom's
approximation 42.70 57.47 65.39 70.47 74.07 100.00 Weiss's
approximation 42.70 58.00 66.09 71.04 74.47 Kimball's
approximation 42.70 57.32 65.04 69.88 73.25 Note: are expressed in percentages.
METHODS OF INFERENCE 31 value distribution for minima, and
for n -
r = n for n - = n , n 15, for n - = n , 16 24,
r =
+
1 f o r n n 25.By making use of the tables of means of order statistics referred to in Section 5, Bain (1972) determined exact values of k for n = 5, 15, 20, 30, 60, and 100 and n infinite and - = for integer n - s.
Engelhardt and Bain (1973) gave exact values of for n =
n = 39, 49, and 59 and infinite n. Mann and Fertig (1975) also presented
exact values of for n = and - = for inte-
ger n - s. [It needs to be mentioned that the values of given by Mann and Fertig are slightly different from those given by Engelhardt and Bain (1973) for n 40 as the choice of used by the former is different.]
Since is a scale parameter and is an unbiased estimator of improvement is possible in terms of minimum mean-square-error estimator (see Section 9.3 for more details). The improvement in efficiency becomes considerable when the censoring is heavy in the sample. As Bain (1972) noted
that for - about at most 0.5, and conse-
quently
has a smaller mean square error than when - On these grounds, an estimator that has been used in general is
which has mean square error
+
here,Values of have been tabulated by Engelhardt and Bain and Mann and Fertig (1975). From the tables of Bain (1972) and Engelhardt and Bain it is clear that the estimator in (22.79) is highly efficient;
for example, when - 0.7, the asymptotic of relative to
the bound is at least 97.7%.
The estimator in (22.79) may also be used to produce a simple linear unbiased estimator for through the moment equation
EXTREME VALUE DISTRIBUTIONS
Using the estimators and in Eqs. (22.79) and respectively, a simple linear unbiased estimator for the p t h quantile can be derived as
Confidence intervals for the parameters and based on the linear unbiased estimators and have also beeq Bain suggested approximating the distribution of by a central chi-square distri- bution with degrees of freedom when - is at most 0.5 and n at least 20. But Mann and Fertig have shown that for n 20,
,
is approximately distributed as chi-square with,
de- grees of freedom. Interestingly this approximate result holds for all values of- in This approximation arose from an observation of van Montfort (1970) that the statistics
-
i = n -
- '
all have approximately an exponential distribution with mean exactly 1 , variance approximately 1 , and covariance almost zero [see also Pyke
As aptly pointed out by Mann and Fertig since for n
-
is approximately a sum of weighted independent chi-square variables, various approximations discussed in Chapter 18 for this distribution are useful in developing approximate inference for 8 .