MCMC methods for wavelet representations in single index models

Abstract

Single index models are a special type of nonlinear regression model that are partially linear and play an important role in fields that employ multidimensional regression models. A wavelet series is thought of as a good approximation to any function in the space. There are two ways to represent the function: one in which all wavelet coefficients are used in the series, and another that allows for shrinkage rules. We propose posterior inference for the two wavelet representations of the function.

To implement posterior inference, we define a hierarchial (mixture) prior model on the scaling(wavelet) coefficients. Since from the two representations a non-zero coefficient has information about the features of the function at a certain scale and location, a prior model for the coefficient should depend on its resolution level. In wavelet shrinkage rules we use "pseudo-priors" for a zero coefficient.

In single index models a direction theta affects estimates of the function. Transforming theta to a spherical polar coordinate is a convenient way of imposing the constraint. The posterior distribution of the direction is unknown and we employ a Metropolis algorithm and an independence sampler, which require a proposal distribution. A normal distribution is proposed as the proposal distribution for the direction. We introduce ways to choose its mode (mean) using the independence sampler.

For Monte Carlo simulations and real data we compare performances of the Metropolis algorithm and independence samplers for the direction and two functions: the cosine function is represented only by the smooth part in the wavelet series and the Doppler function is represented by both smooth and detail parts of the series.