# MCMC methods for wavelet representations in single index models

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.

Advisor:Hart, Jeffrey D.; Vannucci, Marina; Mallick, Bani; DeBlassie, Dante

School:Texas A&M University

School Location:USA - Texas

Source Type:Master's Thesis

Keywords:single index models

ISBN:

Date of Publication:08/01/2003