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Article Excerpt
1. INTRODUCTION
In the social sciences much empirical research proceeds in the context of partial-incomplete subject matter theories and data based on experimental designs not devised by or known to the analyst. This leads to uncertainty concerning the statistical model that is appropriate for describing the data sampling process compatible with the observed sample of data. Uncertainty regarding the appropriate statistical model, in turn, leads to uncertainty regarding appropriate estimation and inference methods. In practice, test statistics, tuning parameters, and sometimes magic are invoked to identify a single statistical model on which to base estimation and inference. Selecting one particular statistical model suffers from the possibility that a wrong choice may be made, resulting in a loss of estimation and inference accuracy. Moreover, the validity of eliminating statistical model uncertainty through the specification of a particular parametric formulation depends on information that one generally does not possess.
Given the uncertainty underlying the model discovery, estimation and inference tasks, and Stein-like possibilities for coping with it, we consider the statistical implications of combining related-competing estimation problems, where the alternative estimators encompassed by alternative models exhibit distinct and dissimilar sampling properties. The objective is to develop natural adaptive semiparametric estimation and inference methods that are free of subjective choices and tuning parameters and that have superior risk performance. In the context of the multivariate linear statistical model, we demonstrate a well-defined data-based semiparametric Stein-like (SPSL) estimator that combines estimation problems by shrinking a base estimator to a plausible alternative estimator. Asymptotic and finite-sample risk results are demonstrated and the relationship of the SPSL estimator to the family of Stein rule (SR) estimators is discussed along with risk performance properties under normality. As an application of the SPSL estimator, we demonstrate the implications of combining two alternative linear statistical models whose associated estimators differ markedly in their bias and precision sampling characteristics. Sampling experiments are used to illustrate the superior finite-sample performance of the SPSL estimator for a variety of experimental designs and normal and nonnormal sampling distributions. Bootstrap procedures are used to define and illustrate confidence set performance and serve as a basis for inference.
2. STATISTICAL MODEL AND SEMIPARAMETRIC STEIN-LIKE (SPSL) ESTIMATOR
Consider the problem of estimating the k-dimensional location parameter vector [beta] when one observes an n-dimensional sample vector y such that y = X[beta] + [epsilon], where X is an (n X k) design matrix of rank k and [epsilon] is an n-dimensional random vector such that E[[epsilon]] = and cov([epsilon]) = [[sigma].sup.2][I.sub.n]. The scale parameter [[sigma].sup.2] may be either known or unknown and no error distribution assumptions are made other than the existence of second-order moments. The objective is to estimate the unknown location vector by some estimator [delta](y) when performance is evaluated by a squared error loss measure L([beta], [delta](y)) = [parallel][beta] - [delta](y)[parallel][.sup.2]. Assuming the usual regularity conditions underlying the linear model, the conventional least squares (LS) estimator is [[delta].sup.LS](y) = [^.[beta]] = (X'X)[.sup.-1]X'y [approximately] ([beta], [[sigma].sup.2](X'X)[.sup.-1]), and under quadratic loss, is a minimax estimator with constant risk [rho]([beta], [^.[beta]]) = [[sigma].sup.2] tr(X'X)[.sup.-1].
2.1 The Stein-Like Base
As one basis for identifying model-estimator uncertainty, Stein (1955) demonstrated the inadmissibility of the conventional maximum likelihood (ML) estimator [[delta].sup.ML](y) = [^.[beta]] when estimating the multivariate normal mean [beta] under quadratic loss. Following this result, James and Stein (1961), Stein (1962), and Baranchik (1964) combined the k-variate ML estimator [^.[beta]] with a k-dimensional fixed null vector and demonstrated, under the assumption of normality, risk-dominating Stein rule (SR) estimators, such as [[delta].sup.S](y) = (1 - a/[parallel][^.[beta]][parallel][.sup.2])[^.[beta]], when < a < 2(k - 2). A very general class of estimators that improves on [^.[beta]] follows from Judge and Bock (1978), Stein (1981), and Brandwein and Strawderman (1991). For the general multivariate normal case, the class of pseudo-Bayes-Stein rules having risk less than that of [^.[beta]] is very large (see, for example, Judge and Bock 1978). Making use of Stein-like estimators, Sclove, Morris, and Radhakrishnan (1972) demonstrated the nonoptimality of preliminary test estimators as a basis for dealing with model uncertainty.
In an orthonormal k-mean context, Lindley (1962) suggested shrinking [^.[beta]] toward the grand-mean estimator and demonstrated the risk dominance of the Stein estimator when [less than or equal to] a [less than or equal to] 2(k - 3). Green and Strawderman (1991) considered a parametric statistical model setting where [^.[beta]] and [~.[beta]] are independent k-dimensional normally distributed data-based estimators with known covariance matrices [[sigma].sup.2][I.sub.k] and [[tau].sup.2][I.sub.k], and demonstrated that the best linear combination of the independent random vector estimators, under quadratic loss, yields the risk-dominating estimator
[[delta].sup.GS]([^.[beta]], [~.[beta]]) = (1 - (k - 2)[[sigma].sup.2]/[parallel][^.[beta]] - [~.[beta]][parallel][.sup.2])([^.[beta]] - [~.[beta]]) + [~.[beta]].
Given this base, Kim and White (2001) provided an expression for the asymptotic risk and bias of Green-Strawderman (GS) Stein-type estimators that also applies to cases of correlated estimators and presented some asymptotic estimation results for the case where both estimators are asymptotically unbiased.
2.2 The SPSL Estimator
Assume that, in addition to the estimator [[delta].sup.LS](y) = [^.[beta]], the following alternative statistical model and corresponding possibly biased data-based competing estimator is available:
[~.[beta]] [approximately] ([beta] + [gamma], [PHI]), (2.1)
where [gamma] is a (k X 1) bias vector and [PHI] is a positive-definite covariance matrix. Also allow the estimators to be correlated and let the covariance matrix of [[^.[beta]'] [??] [~.[beta]']] be defined by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.2)
Our objective is to identify a weighted linear combination of the two estimators with smaller expected quadratic risk than the estimator [[delta].sup.LS](y) = [^.[beta]]. Toward this end, define a new estimator
[bar.[beta]]([alpha]) = [alpha][^.[beta]] + (1 - [alpha])[~.[beta]]. (2.3)
The quadratic risk or mean squared error (MSE) of the estimator [bar.[beta]]([alpha]) is given by
MSE([bar.[beta]]([alpha])) = E[[[alpha]([^.[beta]] - [beta]) + (1 - [alpha])([~.[beta]] - [beta])]' X [[alpha]([^.[beta]] - [beta]]) + (1 - [alpha])([~.[beta]] - [beta])]] = [[alpha].sup.2] tr([[sigma].sup.2](X'X)[.sup.-1]) + (1 - [alpha])[.sup.2][tr([PHI]) + [gamma]'[gamma]] + 2[alpha](1 - [alpha]) tr([SIGMA]). (2.4)
To minimize MSE([bar.[beta]]([alpha])), the first-order necessary condition for [alpha] is
[alpha]* = 1 - [[[sigma].sup.2] tr(X'X)[.sup.-1] - tr([SIGMA])]/[[gamma]'[gamma] + [[sigma].sup.2] tr(X'X)[.sup.-1] + tr([PHI]) - 2 tr([SIGMA])]]. (2.5)
Because [[partial derivative].sup.2]MSE([bar.[beta]]([alpha]))/[partial derivative][[alpha].sup.2] > whenever [^.[beta]] and [~.[beta]] are not perfectly correlated, the optimal weighted linear combination estimator, [bar.[beta]]([alpha]*) = [alpha]*[^.[beta]] + (1 - [alpha]*)[~.[beta]], will, under quadratic loss, be superior to the LS estimator, [[beta].sup.LS](y) = [^.[beta]].
2.3 Estimating the Optimal [alpha]
Relative to the theoretically optimal [alpha] defined in (2.5), note...
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