# Posterior Inference in Single-Index Models

• Published : 2004.04.01

#### Abstract

A single-index model is useful in fields which employ multidimensional regression models. Many methods have been developed in parametric and nonparametric approaches. In this paper, posterior inference is considered and a wavelet series is thought of as a function approximated to a true function in the single-index model. The posterior inference needs a prior distribution for each parameter estimated. A prior distribution of each coefficient of the wavelet series is proposed as a hierarchical distribution. A direction $\beta$ is assumed with a unit vector and affects estimate of the true function. Because of the constraint of the direction, a transformation, a spherical polar coordinate $\theta$, of the direction is required. Since the posterior distribution of the direction is unknown, we apply a Metropolis-Hastings algorithm to generate random samples of the direction. Through a Monte Carlo simulation we investigate estimates of the true function and the direction.

#### References

1. Journal of the American Statistical Association v.92 Generalized Partially Linear Single-Index Models Carroll,R.J.;Fan,J.;Gilbels,I.;Wand,M.P. https://doi.org/10.2307/2965697
2. Nonparametric Smoothing and Lack-of-Fit Tests Hart,J.D.
3. Generalized Linear Models McCullagh,P.;Nelder,J.A.
4. Journal of Chemical Physics v.21 Equations of State Calculations by Fast Computing Machines Metropolis,N.;Rosenbluth,A.;Rosenbluth,M.;Teller,A.;Teller,E. https://doi.org/10.1063/1.1699114
5. Ph.D. dissertation, Dept. Statistics MCMC Methods for Wavelet Representation in Single Index Models Park,C.G.
6. Econometrica v.54 Consistent Estimation of Scaled Coefficients Stoker,T.M. https://doi.org/10.2307/1914309
7. Journal of the American Statistical Association v.97 Penalized Spline Estimation for Partially Linear Single-Index Models Yu,Y.;Ruppert,D. https://doi.org/10.1198/016214502388618861
8. Bayesian Inference in Wavelet Based Models MCMC Methods in Wavelet Shrinkage: Non-Equally Spaced Regression, Density and Spectral Density Estimation Muller,P.;Vidakovic,B.;P.Muller(ed.)B.Vidakovic(ed.)