On the Estimation of a Probability Density Function by the Maximum Penalized Likelihood Method
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In this article, a class of probability density estimates can be obtained by penalizing the likelihood by a functional which depends on the roughness of the logarithm of the density.Abstract:
: A class of probability density estimates can be obtained by penalizing the likelihood by a functional which depends on the roughness of the logarithm of the density. The limiting case of the estimates as the amount of smoothing increasing has a natural form which makes the method attractive for data analysis and which provides a rationale for a particular choice of roughness penalty. The estimates are shown to be the solution of an unconstrained convex optimization problem, and mild natural conditions are given for them to exist. Rates of consistency in various norms and conditions for asymptotic normality and approximation by a Gaussian process are given, thus breaking new ground in the theory of maximum penalized likelihood density estimation. (Author)read more
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Practical Approximate Solutions to Linear Operator Equations When the Data are Noisy
TL;DR: It is shown that the weighted cross-validation estimate of $\hat \lambda $ estimates the value of $\lambda $ which minimizes $({1 / n) E\sum
olimits_{j = 1}^n {[(\mathcal{K}f_{n,\lambda } )(t_j ) - (\mathcal(K)f)(t-j )]} ^2 $ .