dslcd {LogConcDEAD} | R Documentation |
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
x |
Point (or |
lcd |
Object of class |
A |
A positive definite |
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
Value
A vector
of smoothed log-concave maximum likelihood estimate
values, as evaluated at the points x
.
Author(s)
Yining Chen
Madeleine Cule
Robert Gramacy
Richard Samworth
References
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.