Regularized maximum likelihood estimation for the random coefficients model

The random coefficients model Yi=β0i+β1iX1i+β2iX2i+…+βdiXdi, with 𝐗i, Yi, 𝜷i i.i.d, and 𝜷i independent of 𝐗i 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. To add stability to the solution, we apply Tikhonov-type regularization methods. Nonparametric estimation for the joint density of βi=(β0i,…,βdi) based on kernel methods or Fourier inversion have been proposed in recent years. Most of these methods assume a heavy tailed design density f𝐗. We analyze the convergence of the quasi maximum likelihood method without assuming heavy tails for f𝐗 and illustrate performance by applying the method on simulated and real data.