9 METHODS OF INFERENCE
9.6 Maximum Likelihood Estimation
Based on a random sample
.
the maximum likelihood estima- tors and satisfy the equationsand
The asymptotic variances of and are given by the lower bounds in (22.65). The asymptotic correlation coefficient between and is
Equation (22.108) can be rewritten as
this, when used in Eq. yields the following equation for
42 EXTREME VALUE DISTRIBUTIONS
It is necessary to solve (22.112) by an iterative method for Eq. (22.111) then will give If is large compared to then the rhs of (22.112) is approximately
This will provide an approximate solution to (22.112) which can sometimes be used as an initial guess for the iterative method used to solve Eq. (22.112).
The asymptotic confidence interval at significance level is given by
that is,
These are ellipses in the plane. For the estimator
= - - log
of the p t h percentile of the distribution, the asymptotic variance is given by
de (1972) has shown that the best asymptotic point predictor of the maximum of (the next) m observations is
+
and its asymptotic variance is
If the scale parameter 8 is known, then the maximum likelihood estimator of is obtained from (22.108) to be
METHODS OF INFERENCE
This estimator is not unbiased for has in fact shown that (when 8 is known)
and
While is a biased estimator of is an unbiased estimator of This is so because has an exponential distribution with expected value Consequently confidence intervals for this quantity and also for (when 8 is known) can be constructed using methods discussed in Chapter 19, Section 7.
Suppose that the available sample is a doubly censored sample ,,
.
Then the log-likelihood function based on this cen- sored sample iswhere = - are the order statistics from the standard type 1 extreme value distribution with density and its corresponding cdf. From (22.117) we obtain the likelihood equations for and to be
and
EXTREME VALUE DISTRIBUTIONS
Harter and Moore and Harter have discussed the numerical solution of the above likelihood equations. The maximum likelihood estima- tion of when is known, based on right-censored data had been discussed earlier by Harter and Moore (1967). The
matrix of the maximum likelihood estimates, and 0, determined from Eqs. and is given by [Harter (1970, pp.
where is the inverse of the matrix with
= 1 - q , -
+
log - - -and
In the equations above = = a ) = dt,
a ) = and a ) = Harter
for example, has tabulated the values of V,,, and for
q, = and = -
(1991) has discussed further the maximum likelihood estimation of the parameters and based on censored samples. Escobar and Meeker (1986) have discussed the determination of the elements of the Fisher information matrix based on censored data. carried out an
METHODS OF INFERENCE
extensive simulation study and observed the following concerning the effects of Type-I censoring on the estimation of parameters and quantiles of the Gumbel distribution using the maximum likelihood method: light censor- ing on the right may be useful in reducing the bias in estimating the parameters, while left and double censoring are useful for a wider range of censoring levels; the bias in estimating the parameters and quantiles is very small; for complete samples the MLE of overestimates while the MLE of underestimates slightly; and (4) censoring introduces an increase in the variances of the estimates.
has also discussed the maximum likelihood estimation of the parameters based on doubly Type-I censored data. Specifically, for the distribution
with and as the left- and right-censoring time points and with r lowest and largest observations censored, the likelihood function is proportional to
Note in this case that r and are random variables while X, and are fixed. The log-likelihood function is
where
46 EXTREME VALUE DISTRIBUTIONS
The maximum likelihood estimators of and 8 satisfy the equations
a
L Ga
L Hand --- - - -
where
and with
n n - s
(1991) recommended solving these equations using Newton's proce- dure.
Posner when applying the extreme value theory to error-free communication, estimated the parameters and 8 for the complete sample case by the maximum likelihood theory and justified it on the basis of its asymptotic properties. By pointing out that the asymptotic theory need not be valid for Posner's sample size = Gumbel and Mustafi (1966) showed that in fact a modified method of moments gives better results for Posner's data.
An alternative approach was taken by Balakrishnan and Varadan who approximated the likelihood equations by using appropriate linear functions and derived approximate maximum likelihood estimators of and 0. They derived these estimators for the type 1 extreme value distribution for the minimum and we present their estimators in the same form for conve- nience. [The estimators for the type 1 extreme value distribution for the maximum in (22.25) can be obtained simply by interchanging and and replacing by - and by - The likelihood equations for and
in this case are
METHODS OF INFERENCE
and
n - s
where
-
= ey and = 1-
Upon ex-panding the three functions in (22.121) and (22.122) in a Taylor series around the point = log(-log (with = 1
-
q , =+
we get the approximate expressionswhere
By making use of the above approximate expressions in (22.121) and (22.122) and solving the resulting equations, Balakrishnan and Varadan (1991) de- rived the approximate maximum likelihood estimators of and to be
1 1
and 2 ( n - r - s ) (22.124)
48 EXTREME VALUE DISTRIBUTIONS
where
Through a simulation study Balakrishnan and Varadan (1991) have displayed that the above estimators are as efficient as the maximum likelihood estima- tors, best linear unbiased estimators, and best linear invariant estimators even for samples of size as small as 10. For example, values of bias and mean square error for various estimators of and are presented in Table 22.14 for n = 10 and 20, r = 0, and some choices of s. Estimators of this form have been seen earlier in Chapters 13 and 14.
Estimators of this form based on multiply censored samples have been discussed by Balakrishnan, Gupta, and Panchapakesan and Fei, Kong, and Tang (1994).
Table 22.14 Comparison of bias and mean square error of various estimators of and for = 10 and 20 and right censoring =
are the best linear unbiased are the best linear invariant estimators, are the maximum likelihood estimators, are the approximate maximum likelihood estimators.
METHODS OF INFERENCE