Optimum quantiles for the linear estimation of the parameters of the extreme value distribution in complete and censored samples

Abstract

The present study is concerned with the determination of a few observations from a sufficiently large complete or censored sample from the extreme value distribution with location and scale parameters μ and σ, respectively, such that the asymptotically best linear unbiased estimators (ABLUE) of the parameters in Ref.[24] yield high efficiencies among other choices of the same number of observations. (All efficiencies considered are relative to the Cramer-Rao lower bounds for regular unbiased estimators.) The study is on the asymptotic theory and under Type II censoring scheme.For the estimation of μ when σ is known, it has been proved that there exists a unique optimum spacing whether the sample is complete, right censored, left censored, or doubly censored. Several tables are prepared to aid in the numerical computation of the estimates as well as to furnish their efficiencies.For the estimation of σ when μ is known, it has been observed that there does not exist a unique optimum spacing. Accordingly we have obtained a spacing based on a complete sample which yields high efficiency. A similar table as above is prepared.When both μ and σ are unknown, we have considered four different spacings based on a complete sample and chosen the one yielding highest efficiency. A table of the efficiencies is also prepared.Finally we apply the above results for the estimation of the scale and/or shape param eters of the Weibull distribution. χ-μ -exp ( -J \, -χ<χ<χ, -χ<μ<&, σ> 0 QUANTILES FOR LINEAR ESTIMATION 383

Extreme‐Value Distributions

“…Using criteria that are analogous to Theorem 2, we can verify that if F is a normal distribution, then F ∈ D (L (1) ), whereas if F is an exponential, a uniform, or a Weibull, then F ∈ D (L (2) ). Here again, the distributions L (i) are self-locking.…”

“…All the other distributions considered here can be transformed to the distribution L (1) (a, b), either by a change of variable or by a change of variable with a setting of the location parameter equal to zero. It is because of this fact that some of the literature on the Weibull distribution * with a location parameter of 0(L (2) (0, b, α)) appears under the heading of ''an extremevalue distribution,'' which is a common way of referring to the distribution L (1) (·, ·).…”

“…When the location parameter a associated with the distributions H (i) and L (i) , i = 2, 3, cannot be set equal to zero, most of the relationships mentioned before do not hold, and thus we cannot be content by just considering the distribution L (1) (a, b), We will have to consider both H (2) (a, b, α) and H (3) (a, b, α) or their duals L (3) (−a, b, α) and L (2) (−a, b, α), respectively. Estimation of the parameters a (or − a), b, and α is discussed in the next section.…”

“…Because of the popularity of the Weibull distribution, the case L (2) (a, b, α) has been investigated very extensively. We give below an outline of the results for this case, and guide the reader to the relevant references.…”

“…Let X (1) X (2) · · · X (n) be the smallest ordered observations in a sample of size n from the distribution L (2) (a, b, α). Harter and Moore [12] and also Mann et al [24, p. 186] give the three likelihood equations * and suggest procedures for an iterative solution of these.…”