Bernshteĭn–von Mises theorems for nonparametric function analysis via locally constant modelling: A unified approach

Various statistical models involve a certain function, say ff, like the mean regression as a function of a covariate, the hazard rate as a function of time, the spectral density of a time series as a function of frequency, or an intensity as a function of geographical position, etc. Such functions are often modelled parametrically, whether for frequentist or Bayesian uses, and under weak conditions there are so-called Bernshteĭn–von Mises theorems implying that these two approaches are large-sample equivalent. Results of this nature do not necessarily hold up in nonparametric and high-dimensional setups, however. The aim of the present paper is to exhibit a unified framework and methodology for both frequentist and Bayesian nonparametric analysis, involving priors that set ff constant over windows, and where the number mm of such windows grows with sample size nn. Applications include nonparametric regression, maximum likelihood with nonparametrically varying parameter functions, hazard rates being functions of covariates, and nonparametric analysis of stationary time series. We work out conditions on the number and sizes of the windows under which Bernshteĭn–von Mises type theorems can be established, with the prior changing with sample size via the growing number of windows. These conditions entail e.g. that if m∝nαm∝nα, then α∈(14,12) is required. Illustrations of the general methodology are given, including setups with nonparametric regression, hazard rate estimation, and inference about frequency spectra for stationary time series.