This function evaluates the density function of a smoothed log-concave maximum likelihood estimator at a point or points.
dslcd(x, lcd, A=hatA(lcd))A vector of smoothed log-concave maximum likelihood estimate
values, as evaluated at the points x.
Point (or matrix of points) at which the smoothed log-concave
maximum likelihood estimator should be evaluated
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)
Yining Chen
Madeleine Cule
Robert Gramacy
Richard Samworth
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
Chen, Y. and Samworth, R. J. (2013) Smoothed log-concave maximum likelihood estimation with applications Statist. Sinica, 23, 1373-1398. https://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.
dlcd, hatA, mlelcd