A1232
Title: Regularized maximum likelihood estimation for the random coefficients model
Authors: Fabian Dunker - University of Canterbury (New Zealand) [presenting]
Emil Mendoza - University of Canterbury (New Zealand)
Marco Reale - University of Canterbury (New Zealand)
Abstract: A popular way to model unobserved heterogeneity in population is the linear random coefficient model $Y_i = \beta_{i,1}X_{i,1} + \beta_{i,2} X_{i,2} + \ldots + \beta_{i,d} X_{i,d}$. It is assumed that the observations $(\mathbf{X}_i,Y_i),$ $i=1,\ldots,n,$ are i.i.d. where $\mathbf{X}_i=(X_{i,1}, \ldots, X_{i,d})$ is a $d$-dimensional vector of regressors. The random coefficients $\boldsymbol{\beta}_i=(\beta_{i,1}, \ldots, \beta_{i,d}),$ $i=1,\ldots,n$ are unobserved i.i.d. realizations of an unknown $d$-dimensional distribution with density $f_{\boldsymbol{\beta}}$ independent of $\mathbf{X}_i$. The aim is to propose a quasi-maximum likelihood method to estimate the joint density distribution of the random coefficients. This method implicitly involves the inversion of the Radon transformation in order to reconstruct the joint distribution, and hence is an inverse problem. To add stability to the solution, Tikhonov-type regularization methods are applied. Nonparametric estimation for the joint density of $\boldsymbol{\beta}$ based on kernel methods or Fourier inversion has been proposed in recent years. Most of these methods assume a heavy-tailed design density $f_\mathbf{X}$. The convergence of the quasi maximum likelihood method is analyzed without assuming heavy tails for $f_\mathbf{X}$, and performance is illustrated by applying the method on simulated and real data.
