Maximum Likelihood Estimation of Functional Relationships

ISBN-13: 9780387977218 / Angielski / Miękka / 1992 / 110 str.

Nico J. D. Nagelkerke

Opis

The theory of functional relationships concerns itself with inference from models with a more complex error structure than those existing in regression models. We are familiar with the bivariate linear relationship having measurement errors in both variables and the fact that the standard regression estimator of the slope underestimates the true slope. One complication with inference about parameters in functional relationships, is that many of the standard properties of likelihood theory do not apply, at least not in the form in which they apply to e.g. regression models. This is probably one of the reasons why these models are not adequately discussed in most general books on statistics, despite their wide applicability. In this monograph we will explore the properties of likelihood methods in the context of functional relationship models. Full and conditional likelihood methods are both considered. Possible modifications to these methods are considered when necessary. Apart from exloring the theory itself, emphasis shall be placed upon the derivation of useful estimators and their second moment properties. No attempt is made to be mathematically rigid. Proofs are usually outlined with extensive use of the Landau 0(.) and 0(.) notations. It is hoped that this shall provide more insight than the inevitably lengthy proofs meeting strict standards of mathematical rigour.

Spis treści

1:Introduction.- I.Introduction.- II.Inference.- III.Controlled variables.- IV.Outline of the following chapters.- 2:Maximum likelihood estimation of functional relationships.- I.Introduction.- II.Maximization of the likelihood under constraints.- A.Direct elimination.- B.The Lagrange multiplier method.- III.The conditional likelihood.- IV.Maximum likelihood estimation for multivariate normal distributions with known covariance matrix.- A.Derivation of the normal equations.- B.The simple linear functional relationship.- C.Estimation using Sprent’s generalized residuals.- D.Non-linear models.- E.Inconsistency of non-linear ML estimators.- F.Linearization of the normal equations.- V.Maximum likelihood estimation for multivariate normal distributions with unknown covariance matrix.- A.Estimation with replicated observations.- B.Estimation without replicated observations.- C.A saddlepoint solution to the normal equations.- VI.Covariance matrix of estimators.- A.The asymptotic method.- B.The bootstrap.- C.The jackknife.- VII.Error distributions depending on the true variables.- VIII.Proportion of explained variation.- 3:The multivariate linear functional relationship.- I.Introduction.- II.Identifiability.- III.Heteroscedastic errors.- A.Known error covariance matrix.- B.Unknown error covariance matrix.- IV.Homoscedastic errors.- A.Known error covariance matrix.- B.Misspecification.- C.The eigenvalue method.- D.Unknown error covariance matrix.- V.Factor space.- VI.The asymptotic distribution of the parameter estimators.- A.Asymptotic covariance matrix.- B.Consistency and asymptotic normality.- C.Hypothesis tests.- VII.Replicated observations.- VIII.Instrumental variables.- References.