Unbiased statistical estimation functions for parameters in presence of nuisance parameters

Chandrasekar, B.; Kale, B. K.

doi:10.1016/0378-3758(84)90043-0

Cited by 28 publications

(9 citation statements)

References 5 publications

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“…In the case where an O F -optimal estimating function exists, however, we may replace the matrix comparison by simpler scalar ones. The following result is essentially due to Chandrasekar and Kale (1984).…”

Section: Scalar Equivalences and Associated Resultsmentioning

confidence: 88%

Quasi-Likelihood and Its Application: A General Approach to Optimal Parameter Estimation

Nelson¹,

Heyde²

1998

Journal of the American Statistical Association

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This book is concerned with the general theory of optimal estimation of parameters in systems subject to random effects and with the application of this theory. The focus is on choice of families of estimating functions, rather than the estimators derived therefrom, and on optimization within these families. Only assumptions about means and covariances are required for an initial discussion. Nevertheless, the theory that is developed mimics that of maximum likelihood, at least to the first order of asymptotics. The term quasi-likelihood has often had a narrow interpretation, associated with its application to generalized linear model type contexts, while that of optimal estimating functions has embraced a broader concept. There is, however, no essential distinction between the underlying ideas and the term quasi-likelihood has herein been adopted as the general label. This emphasizes its role in extension of likelihood based theory. The idea throughout involves finding quasi-scores from families of estimating functions. Then, the quasilikelihood estimator is derived from the quasi-score by equating to zero and solving, just as the maximum likelihood estimator is derived from the likelihood score. This book had its origins in a set of lectures given in September 1991 at the 7th Summer School on Probability and Mathematical Statistics held in Varna, Bulgaria, the notes of which were published as Heyde (1993). Subsets of the material were also covered in advanced graduate courses at Columbia University in the Fall Semesters of 1992 and 1996. The work originally had a quite strong emphasis on inference for stochastic processes but the focus gradually broadened over time. Discussions with V.P. Godambe and with R. Morton have been particularly influential in helping to form my views. The subject of estimating functions has evolved quite rapidly over the period during which the book was written and important developments have been emerging so fast as to preclude any attempt at exhaustive coverage. Among the topics omitted is that of quasi-likelihood in survey sampling, which has generated quite an extensive literature (see the edited volume Godambe (1991), Part 4 and references therein) and also the emergent linkage with Bayesian statistics (e.g., Godambe (1994)). It became quite evident at the Conference on Estimating Functions held at the University of Georgia in March 1996 that a book in the area was much needed as many known ideas were being rediscovered. This realization provided the impetus to round off the project rather vi PREFACE earlier than would otherwise have been the case. The emphasis in the monograph is on concepts rather than on mathematical theory. Indeed, formalities have been suppressed to avoid obscuring "typical" results with the phalanx of regularity conditions and qualifiers necessary to avoid the usual uninformative types of counterexamples which detract from most statistical paradigms. In discussing theory which holds to the first order of asymptotics the treatment is especially informal, as be...

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Section: Scalar Equivalences and Associated Resultsmentioning

confidence: 88%

Quasi-Likelihood and Its Application: A General Approach to Optimal Parameter Estimation

Nelson¹,

Heyde²

1998

Journal of the American Statistical Association

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“…Mathematically, the inequalities (2) and (11) are closely related to some R-C type lower bounds for the variance of an estimating equation (e.g., Godambe 1960;Bhapkar 1972;Chandrasekar and Kale 1984), which is a function of Y and with mean zero. The similarities arise as we also deal with the variance of a function of Y and , namely, [¯(Y ) ¡ ® (Y; )], which can be regarded as an estimating equation when¯(Y ) is an unbiased predictor of Z.…”

Section: Discussionmentioning

confidence: 99%

Rao–Cramer Type Inequalities for Mean Squared Error of Prediction

Nayak

2002

The American Statistician

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Let Y be an observable random vector and Z be an unobservable random variable with joint density f (y; z), where is an unknown parameter vector. Considering the problem of predicting Z based on Y , we derive a Rao-Cramer type lower bound for the mean squared error (MSE) of any given predictor of Z satisfying some regularity conditions. Under unbiasedness, the lower bound does not depend on the speci c predictor, and the condition for attaining that bound can be used to easily identify the best unbiased predictors in some applications. Using the lower bound we develop a method for proving admissibility of predictors under squared error loss. Bhattacharyya type bounds and extensions of the results for vector Z are also discussed, and some examples illustratingthe utilityof the results are presented.

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“…The suggested method of finding optimal estimating function is basically an extended version of the inequality approach of Chandrasekar and Kale (1984). It seems natural to call the estimating function with optimal k as MVBSEF(k).…”

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confidence: 99%

“…The problem of finding optimal estimating functions were earlier considered by Godambe (1960), Kale (1962), Ferreira (1982), Chandrasekar and Kale (1984), Bhapkar and Srinivasan (1994), among others. In this article, we propose a method of finding the optimal standardized estimating function g * 1s for 1 in the presence of nuisance parameter 2 .…”

Section: Introductionmentioning

confidence: 98%

On Optimal Estimating Functions in the Presence of Nuisance Parameters

Mukhopadhyay

2007

Communications in Statistics - Theory and Methods

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When there is only one interesting parameter 1 and one nuisance parameter 2 , Godambe and Thompson (1974) showed that the optimal estimating function for 1 essentially is a linear function of the 1 -score, the square of the 2 -score, and the derivative of 2 -score with respect to 2 . Mukhopadhyay (2000b) generalized this result to m nuisance parameters. Mukhopadhyay (2000, 2002a,b) obtained lower bounds to the variance of regular estimating functions in the presence of nuisance parameters. Taking cues from these results we propose a method of finding optimal estimating function for 1 by taking the multiple regression equation on 1 score and Bhattacharyya's (1946) scores with respect to 2 . The result is extended to the case of m nuisance parameters.

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Unbiased statistical estimation functions for parameters in presence of nuisance parameters

Cited by 28 publications

References 5 publications

Quasi-Likelihood and Its Application: A General Approach to Optimal Parameter Estimation

Quasi-Likelihood and Its Application: A General Approach to Optimal Parameter Estimation

Rao–Cramer Type Inequalities for Mean Squared Error of Prediction

On Optimal Estimating Functions in the Presence of Nuisance Parameters

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