Abstract
This paper presents an integrated framework for estimation and inference from generalized linear models using adjusted score equations that result in mean and median bias reduction. The framework unifies theoretical and methodological aspects of past research on mean bias reduction and accommodates, in a natural way, new advances on median bias reduction. General expressions for the adjusted score functions are derived in terms of quantities that are readily available in standard software for fitting generalized linear models. The resulting estimating equations are solved using a unifying quasi-Fisher scoring algorithm that is shown to be equivalent to iteratively reweighted least squares with appropriately adjusted working variates. Formal links between the iterations for mean and median bias reduction are established. Core model invariance properties are used to develop a novel mixed adjustment strategy when the estimation of a dispersion parameter is necessary. It is also shown how median bias reduction in multinomial logistic regression can be done using the equivalent Poisson log-linear model. The estimates coming out from mean and median bias reduction are found to overcome practical issues related to infinite estimates that can occur with positive probability in generalized linear models with multinomial or discrete responses, and can result in valid inferences even in the presence of a high-dimensional nuisance parameter.
Highlights
The flexibility of generalized linear models (McCullagh and Nelder 1989) in handling count, categorical, positive and real-valued responses under a common modelling framework has made them a typical choice in applications but Electronic supplementary material The online version of this article contains supplementary material, which is available to authorized users.B Ioannis Kosmidis fYi (y; θi, φ) = exp yθi − b(θi ) − c1(y) φ/mi − 1a 2 − mi φ+ c2(y) for some sufficiently smooth functions b(·), c1(·), a(·) and c2(·), and fixed observation weights m1, . . . , mn
Given that the dispersion parameter φ appears in the expression for sβ in (1) only multiplicatively, the maximum likelihood (ML) estimate of β can be computed without knowledge of the value of φ
Given that the extra term in the iteratively reweighted least squares (IWLS) update for median bias reduction in (11) depends on the parameters only through the response means, the same extra step of rescaling the Poisson means before the IWLS update of the parameters will result in an iteration that delivers the median BR estimates of the multinomial logistic regression model using the equivalent Poisson log-linear model
Summary
The flexibility of generalized linear models (McCullagh and Nelder 1989) in handling count, categorical, positive and real-valued responses under a common modelling framework has not only made them a typical choice in applications but Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11222-019-09860-6) contains supplementary material, which is available to authorized users.
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