The Asymptotic Properties of ML Estimators when Sampling from Associated Populations

Abstract

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Ill-conditioned Information Matrices and the Generalized Linear Model: an Asymptotically Biased Estimation Approach

“…Maximum likelihood estimation has the large sample properties of consistency and asymptotic normality of Ä allowing conventional tests of significance. Schaefer (1979) points out that most of the theoretical work on the asymptotic properties of maximum likelihood estimators for independent, nonidentically distributed responses has already been done (Bradley and Gart (1962)). Bradley and Gart's work essentially require that the following two assumptions hold: ' l i) Ix,,I isboundedforalliandj; _ ii) l §_rr_1_N·'(X'VX) = Q, for Q positive detinite with linite determinant.…”

On the Parametric Maximum Likelihood Estimator for Independent but Non-identically Distributed Observations with Application to Truncated Data

“…For the sake of generality, we consider these both cases. In the case of an infinite number of distinct probability distribution functions, the assumptions that are sufficient conditions for consistency and asymptotic normality of the parametric maximum likelihood estimator are slighlty different than in the paper of Bradley and Gart [1]. In the remaining of this paper are presented the assumptions, the theorems and the proofs of the asymptotic properties of the parametric maximum likelihood estimator.…”

“…Chanda [3] solved the normal equations to prove the consistency of this estimator whereas Lehmann [7] studied the sign of a function of the log-likelihood on a sphere with center the true value of the parameter vector. While Bradley and Gart [1] developed the extension of the proof of Chanda [3] for independent but non-identically distributed observations, there is no extension of the proof of Lehmann [7]. There are some other proofs of asymptotic properties like the proof using empirical processes and exposed by Van der Vaart and Wellner [8] that yields to different statements of assumptions that may not be easy to verify in specific situations.…”

“…In the present article, we develop the extension of the proof of Lehmann [7] in the case of independent but non-identically distributed observations. In their paper, Bradley and Gart [1] considered two cases: either the number of distinct probability distribution functions that can be observed in the population from which the sample comes from is finite or this number is infinite. For the sake of generality, we consider these both cases.…”

"Análise de um modelo de regressão com erros nas variáveis multivariado com intercepto nulo"

“…, p, assim como também existem parâmetros comuns a todos os grupos (µ, σ 2 x e σ 2 ). Bradley & Gart (1962) classifica tais grupos como populações associadas (populações com distribuições diferentes, porém que podem ter alguns parâmetros em comum). Sob condições de regularidade, Bradley & Gart (1962) estudam a distribuição assintótica dos estimadores de máxima verossimilhança de θ quando os dados têm estas características, mais precisamente:…”