dslcd: Evaluation of a smoothed log-concave maximum likelihood estimator at given points

Description

This function evaluates the density function of a smoothed log-concave maximum likelihood estimator at a point or points.

Usage

dslcd(x, lcd, A=hatA(lcd))

Arguments

Point (or matrix of points) at which the smoothed log-concave maximum likelihood estimator should be evaluated

lcd

Object of class "LogConcDEAD" (typically output from mlelcd)

A positive definite matrix that determines the degree of smoothing, typically taken as the output of hatA(lcd)

Value

A vector of smoothed log-concave maximum likelihood estimate values, as evaluated at the points x.

Details

The smoothed log-concave maximum likelihood estimator is a fully automatic nonparametric density estimator, obtained as a canonical smoothing of the log-concave maximum likelihood estimator. More precisely, it equals the convolution $\hat{f} * \phi_{d,\hat{A}}$, where $\phi_{d,\hat{A}}$ is the density function of d-dimensional multivariate normal with covariance matrix $\hat{A}$. Typically, $\hat{A}$ is taken as the difference between the sample covariance and the covariance of fitted log-concave maximum likelihood density. Therefore, this estimator matches both the empirical mean and empirical covariance. The estimate is evaluated numerically either by Gaussian quadrature in two dimensions, or in higher dimensions, via a combinatorial method proposed by Grundmann and Moeller (1978). Details of the computational aspects can be found in Chen and Samworth (2011). In one dimension, explicit expression can be derived. See logcondens for more information.

For examples, see mlelcd

References

Chen, Y. and Samworth, R. J. (2012), Smoothed log-concave maximum likelihood estimation with applications Submitted. http://arxiv.org/abs/1102.1191v4

Grundmann, A. and Moeller, M. (1978) Invariant Integration Formulas for the N-Simplex by Combinatorial Methods SIAM Journal on Numerical Analysis, Volume 15, Number 2, 282-290.