Nonparametric Estimation of Irregular Functions with Independent or Autocorrelated Errors
This paper shows how to estimate non-smooth univariate functions that are either discontinuous or have discontinuities in the first or higher order derivatives. The estimation is carried out by modelling the function as a linear combination of terms from one of a variety of nonorthonormal bases, such that each basis has elements that are themselves non-smooth. The advantage of such an approach over wavelet based alternatives is that the sample size need not be a power of two, nor do the observations of the independent variable have to be equally spaced. To estimate the nonparametric regression we use a hierarchical Bayesian framework, with the function estimate calculated using its posterior mean. To efficiently capture any irregularities in the functions we use bases with approximately as many elements as there are observations. The computation is made tractable by using a ‘focused’ Markov chain Monte Carlo sampling scheme. Both the hierarchical Bayesian model and the focused sampling scheme are extended to the case where the errors in a regression are potentially autocorrelated. The efficiency of the resulting estimators for both the independent error case and the autocorrelated case is demonstrated using simulation experiments based on several functions commonly examined in the literature.
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