Essays on estimation strategies addressing label-switching in Gaussian mixtures of semi- and non-parametric regressions

Abstract

Gaussian mixtures of non-parametric regressions (GMNRs) are a flexible class of Gaussian mixtures of regressions (GMRs). These models assume that some or all of the parameters of GMRs are non-parametric functions of the covariates. This flexibility gives these models wide applicability for studying the dependence of one variable on one or more covariates when the underlying population is made up of unobserved subpopulations. The predominant approach used to estimate the GMRs model is maximum likelihood via the Expectation-Maximisation (EM) algorithm. Due to the presence of non-parametric terms in GMNRs, the model estimation poses a computational challenge. A local-likelihood estimation of the non-parametric functions via the EM algorithm may be subject to label-switching. To estimate the non-parametric functions, we have to define a local-likelihood function for each local grid point on the domain of a covariate. If we separately maximise each local-likelihood function, using the EM algorithm, the labels attached to the mixture components may switch from one local grid point to the next. The practical consequence of this label-switching is characterised by non-parametric estimates that are non-smooth, exhibiting irregular behaviour at local points where the switch took place. In this thesis, we propose effective estimation strategies to address label-switching. The common thread that underlies the proposed strategies is the replacement of the separate maximisations of the local-likelihood functions with simultaneous maximisation. The effectiveness of the proposed methods is demonstrated on finite sample data using simulations. Furthermore, the practical usefulness of the proposed methods is demonstrated through applications on real data.