Second-order covariance matrix of maximum likelihood estimates in generalized linear models

Covariance Matrix Formula for Exponential Family Nonlinear Models

“…The matrices C and D are easily formed from the second-and third-order derivatives of the nonlinear predictor = f x i with respect to the parameters in and the inverse of the information matrix. For generalized linear models, C and D vanish and (9) reduces to −2 PHZ d P T which coincides with the corresponding expression obtained by Cordeiro (2004).…”

“…We decompose these covariances in simple three matrix formulas: the first form is the first-order asymptotic covariance matrix ofˆ and the remaining terms are corrections obtained for generalized linear models (Cordeiro, 2004) and exponential family nonlinear models, respectively. These corrections depend on the first three derivatives of the nonlinear predictor with respect to the parameters in , the variance and inverse link functions and their first two and three derivatives, respectively, and the precision parameter .…”

“…are diagonal matrices of order n defined by Cordeiro (2004), K = X T W X is the information matrix of order p for defined here without the precision parameter , P = K −1 X T , X = is the local derivative matrix, Z = z ij = XP = X X T W X −1 X T is an n × n positive semidefinite matrix of rank p, Z d = diag z 11 z nn and C = diag c 11 c nn are diagonal matrices of order n, where c ii = tr X i K −1 with X i being a p × p matrix with elements 2 i r s = rs i . Furthermore, the n × p matrix D is obtained from:…”

“…Some others asymptotics are presented by Wei (1998). The purpose of this article is to obtain a general matrix formula for the second-order covariance of the MLEˆ in exponential family nonlinear models, thus generalizing the result of Cordeiro (2004) which holds only for generalized linear models with known precision parameter.…”

Aperfeiçoamento de métodos estatísticos em modelos de regressão da família exponencial

“…Uma expressão para o coeficiente de assimetria assintótico de ordem n −1/2 para a distribuição dos EMVs dos parâmetros β que modelam a média e para o parâmetro de precisão foi obtida por Cordeiro & Cordeiro (2001). Em Cordeiro (2004) foi apresentada uma fórmula geral para a matriz de covariância assintótica de ordem n −2 dos EMVs do parâmetro β, considerando o parâmetro de dispersão conhecido. Este resultado foi estendido por Cordeiro et al (2006) considerando o parâmetro de dispersão desconhecido, porém o mesmo para todas as observações.…”