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OPTIMAL LINEAR COMBINATIONS OF CONSISTENT, ASYMPTOTICALLY NORMAL ESTIMATORS (SAMPLE, GROUPED, QUANTILES, DATA, MIXED)
Abstract
Let (')(theta)(,1),(')(theta)(,2),...,(')(theta)(,k) denote k different consistent estimators of the same real parameter (theta). Consider estimators of the form (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) The optimal coefficients are the c(,i)'s which minimize the variance of (')(theta) and are given by c('T) = (c(,1),...,c(,k))('T) = (1('T)(SIGMA)('-1)1)('-1)(SIGMA)('-1)1 where 1 is a k x 1 vector of ones and (SIGMA) is the k x k positive definite covariance matrix of the(')(theta)(,i)'s. In general, the optimal coefficients depend on the value of any unknown parameters (the parameter (theta) and other nuisance parameters), hence a multi-stage procedure is the basis for all estimation methods proposed. In the first stage, a consistent, initial approximation to the optimal coefficients is calculated. In the succeeding stage(s), (')(theta)(,i) is updated using the approximation to the optimal coefficients calculated in the previous stage(s). Three different estimation problems are considered, one each in Chapters 3, 4 and 5. Asymptotic mean and covariance approximations to (SIGMA) are used instead of the much more difficult to obtain exact covariance matrix (SIGMA). The (')(theta)(,i)'s are all obtained by the same technique. The estimators proposed in Chapters 3 and 4 are BAN (best asymptoti- cally normal). In Chapter 5 the AREs of the proposed estimators are shown to be higher than the AREs of some previously proposed estimators in certain parametric families of distributions, families where the traditional maximum likelihood estimator is inconsistent. As a by-product of the process of calculating (')(theta), confidence intervals for (theta) are easily obtainable with no extra computation, in contrast to previously proposed methods.
Subject Area
Statistics
Recommended Citation
FROM, STEVEN GLEN, "OPTIMAL LINEAR COMBINATIONS OF CONSISTENT, ASYMPTOTICALLY NORMAL ESTIMATORS (SAMPLE, GROUPED, QUANTILES, DATA, MIXED)" (1985). ETD collection for University of Nebraska-Lincoln. AAI8606962.
https://digitalcommons.unl.edu/dissertations/AAI8606962