Asymptotic Expansions for Confidence Limits in the Presence of Nuisance Parameters, with Applications

Abstract

SUMMARY Series expansions for confidence limits based on consistent estimators are obtained in the presence of nuisance parameters. The maximum likelihood estimator is treated as a special case and the necessary characteristics of its sampling distribution are obtained in terms of basic quantities derivable from the likelihood function. The theory is applied to the problem of setting separate confidence limits for the scale and shape parameters of the Weibull distribution. Comparisons are made with earlier simulation studies.

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

“…The cumulant generating function of the MLEÿ of ÿ up to order n −2 can be obtained from Peers and Iqbal (1985). Using their expression, we ÿnd immediately the n −2 asymptotic covariance ij between the estimatesÿ i andÿ j as…”

Covariance Matrix Formula for Exponential Family Nonlinear Models

“…Using general expressions from Peers and Iqbal (1985) we find the asymptotic covariances rs between estimatorsˆ r andˆ s up to order n −2 in exponential family nonlinear models (Cordeiro and Paula, 1989;Wei, 1998). 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.…”

“…For a general model for which the log likelihood depends on a p-vector of unknown parameters, the cumulant generating function of the MLEˆ of up to order O n −2 , where n is the sample size, was obtained by Peers and Iqbal (1985). Using their expression, we can find the n −2 asymptotic covariance ij between any two estimatorsˆ i andˆ j by:…”

“…In Sec. 2 we review the results of Peers and Iqbal (1985) and present the cumulants of log likelihood derivatives needed to Downloaded by [Tufts University] at 14:33 19 November 2014 find the n −2 covariance matrix ofˆ . Section 3 gives a general matrix formula for the n −2 asymptotic covariance matrix ofˆ in exponential family nonlinear models.…”

An Empirical Investigation of Methods for Interval Estimation of the Log Odds Ratio in Discriminant Analysis

“…Critchley & Ford (1985), this assumes normality of Method E2. Davis (1987) gives results on the sampling cumulants of 0, and related statistics, which can be utilized using a technique of Peers & Iqbal (1985) to construct confidence intervals for 0. Rigby (1982) considers a Bayesian approach to the problem.…”