Saleh, Asma; (2021) Reduced-bias estimation of some non-standard models. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

There is a persisting interest in methods that reduce bias in the estimation of parametric models. There is already a wide range of methods that achieve that goal, with a few of them also delivering beneficial side effects. For example, the bias-reducing adjusted scores approach of Firth (1993) has been shown to always deliver finite estimates in models like logistic regression even when the maximum likelihood (ML) estimator takes infinite values. Other proposals (e.g. reduced-bias M estimation in Kosmidis and Lunardon (2020), and indirect inference of Kuk (1995) have been shown to be able to reduce estimation bias even in cases where the model is partially specified, such as for general M-estimators. In this thesis, we examine the applicability, evaluate the performance and compare a range of bias reduction methods such as the bias-reducing adjusted score equations of Firth (1993), indirect inference and reduced-bias M estimation, in terms of their impact on estimation and inference, in well-used model classes in econometrics and statistics, which are beyond the various standard models that bias reduction methods have been used for before. In particular, we study the Heckit regression model which handles non-randomly selected samples where the observed range of the dependent variable is censored, i.e. it is only partially known whether it is above or below a fixed threshold. We also examine accelerated failure time models which are parametric survival models for censored lifetime observations. Finally, we consider two stratified models Sartori (2003) where interest lies in the estimation of a parameter in the presence of a set of nuisance parameters, whose dimension increases with the number of strata. The main challenge with these models is that even basic requirements, like consistency of the ML estimator, are not necessarily satisfied Neyman and Scott (1948). We focus on binomial matched pairs where the ML estimate of the parameter of interest may be infinite due to data separation. We propose a penalised version of the log-likelihood function based on adjusted responses which always results in a finite estimator of the log odds ratio. The probability limit of the penalised adjusted log-likelihood estimator is derived and it is shown that in certain settings the ML, conditional and modified profile log-likelihood estimators drop out as special cases of the former estimator. It is found that for the models of censored data, Firth adjustments are not available in closed form whereas indirect inference and reduced-bias M estimation are applicable and are an improvement over traditional ML estimation.

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