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### Posterior Inference in Single-Index Models

Park, Chun-Gun   (Department of Statistics, Seoul National UniversityUU0000691  ); Yang, Wan-Yeon   (Department of Applied Statistics, Kyungwon UniversityUU0000122  ); Kim, Yeong-Hwa   (Department of Statistics, Chung-Ang UniversityUU0001197  );
• #### 초록

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.

• #### 주제어

Single-index model .   Wavelet series .   Daubechies wavelet .   Posterior inference .   Hierarchical distribution .   Metropolis-Hastings algorithm.

• #### 참고문헌 (8)

1. Consistent Estimation of Scaled Coefficients , Stoker,T.M. , Econometrica / v.54,pp.1461-1481,
2. MCMC Methods in Wavelet Shrinkage: Non-Equally Spaced Regression, Density and Spectral Density Estimation , Muller,P.;Vidakovic,B.;P.Muller(ed.)B.Vidakovic(ed.) , Bayesian Inference in Wavelet Based Models / v.,pp.,
3. Hart,J.D. , Nonparametric Smoothing and Lack-of-Fit Tests / v.,pp.,
4. MCMC Methods for Wavelet Representation in Single Index Models , Park,C.G. , Ph.D. dissertation, Dept. Statistics / v.,pp.,
5. Penalized Spline Estimation for Partially Linear Single-Index Models , Yu,Y.;Ruppert,D. , Journal of the American Statistical Association / v.97,pp.1042-1054,
6. Equations of State Calculations by Fast Computing Machines , Metropolis,N.;Rosenbluth,A.;Rosenbluth,M.;Teller,A.;Teller,E. , Journal of Chemical Physics / v.21,pp.1087-1091,
7. McCullagh,P.;Nelder,J.A. , Generalized Linear Models / v.,pp.,
8. Generalized Partially Linear Single-Index Models , Carroll,R.J.;Fan,J.;Gilbels,I.;Wand,M.P. , Journal of the American Statistical Association / v.92,pp.477-489,

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