Regularized Maximum Likelihood Estimation for the Random Coefficients Model

TL;DR

This paper introduces a regularized quasi-maximum likelihood approach for estimating the joint distribution of random coefficients in a model, addressing stability issues in inverse Radon transform problems without heavy tail assumptions.

Contribution

It proposes a novel regularized estimation method for the joint density in random coefficient models, improving stability and convergence analysis over existing nonparametric techniques.

Findings

Method successfully estimates joint density in simulations.

Regularization enhances stability of the inverse problem.

Applicable to real data with improved robustness.

Abstract

The random coefficients model $Y_i={\beta_0}_i+{\beta_1}_i {X_1}_i+{\beta_2}_i {X_2}_i+\ldots+{\beta_d}_i {X_d}_i$, with $\mathbf{X}_i$, $Y_i$, $\mathbf{\beta}_i$ i.i.d, and $\mathbf{\beta}_i$ independent of $X_i$ is often used to capture unobserved heterogeneity in a population. We propose a quasi-maximum likelihood method to estimate the joint density distribution of the random coefficient model. This method implicitly involves the inversion of the Radon transformation in order to reconstruct the joint distribution, and hence is an inverse problem. Nonparametric estimation for the joint density of $\mathbf{\beta}_i=({\beta_0}_i,\ldots, {\beta_d}_i)$ based on kernel methods or Fourier inversion have been proposed in recent years. Most of these methods assume a heavy tailed design density $f_\mathbf{X}$. To add stability to the solution, we apply regularization methods. We analyze the convergence of the method without assuming heavy tails for $f_\mathbf{X}$ and illustrate performance by applying the method on simulated and real data. To add stability to the solution, we apply a Tikhonov-type regularization method.