Nonparametric Estimation Based Bayes Method

In the field of nonparametric estimation, Bayesian method is not only an extensively used but also a very effective means. In this paper, we applied Bayesian idea to address a kind of nonparametric estimation problem: local polynomial estimation with variable bandwidth on a bounded interval and the trend function of time series of nonparametric estimation. For the first problem, Fan and Gijbels(1996) advise an idea of local polynomial estimation with variable bandwidth: partition the interval into some subintervals according to the data size, on each subinterval applied local polynomial estimation method and determined the bandwidth according to some suitable criterions. In this paper, we regarded both the number of subinterval and the end point of subinterval as random parameters, then applied reversible-jump MCMC methods to this problem. For the second problem, we considered the multiscale Bayesian analysis of trend function of time series with AR(p) errors from the point of scale space . That is to say, we regarded the bandwidth(h) in the trend function's kernel estimation as random parameter. Through the Bayesian analysis of h, we analysed the trend function's long-term, middle-term and short-term cases. In order to sampling from posterior distrubution, we applied the Gibbs sampling algorithm to this problem and obtained the Markov-chain samples. In the end, we given some simulated examples and summarization.