Nonparametric Maximum Likelihood Estimation by the Method of Sieves

Consistency of Bayes Estimates for Nonparametric Regression: Normal Theory

“…There is closely related work on sieves and on model selection; see Geman and Hwang (1982), Shibata (1981; or Stone (1982). Hierarchical priors for regression in finite-dimensional settings go back to Lindley and Smith (1972).…”

The Fast Gauss Transform

“…For each target box C, we need to evaluate the total field due to sources in all boxes B, at each target in C. Because the range of the Gaussian e -tsI 2 /6 is O(v') aiad the boxes have side lengths rV/ 7 , only a fixed number of source boxes B can contribute more than QE to the field in a given target box C, where Q = EN=I jqj] and c is a specified precision. Indeed, if we cut off the sum over all B after including the (2n + 1)d nearest boxes to C, we incur an error bounded by Qe -2 2 2 .…”

“…A similar transform, with F a lower-dimensional subset of R , occurs when one solves any initial/boundary value problem for the heat equation by means of potential theory [1,6,9,13,14]. Other examples occur in vortex methods [31, tomography [11], and nonparametric statistics [7,15]. Finally, a common analytical tool is mollification; one approximates an arbitrary function f by the family of smooth rapidly decreasing functions…”

Asymptotic Theory for Sieve Estimators in Semimartingale Regression Models.

“…Grenander (1981), Ibragimov and Khasminskii (1981) and Geman and Hwang (1982) have studied the case where p= , Y1 1 1 and M is a Wiener process. Nguyen and Pham (1982) have treated the linear diffusion process with p = 1.…”