dlcd {LogConcDEAD}

Evaluation of a log-concave maximum likelihood estimator at a point

Description

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

Usage

dlcd(x,lcd, uselog=FALSE, eps=10^-10)

Arguments

x

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

lcd

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

uselog

Scalar logical: should the estimator should be calculated on the log scale?

eps

Tolerance for numerical stability

Details

A log-concave maximum likelihood estimate \hat{f}_n is satisfies \log \hat{f}_n = \bar{h}_y for some y \in R^n, where

\bar{h}_y(x) = \inf \lbrace h(x) \colon h \textrm{ concave }, h(x_i) \geq y_i \textrm{ for } i = 1, \ldots, n \rbrace.

Functions of this form may equivalently be specified by dividing C_n, the convex hull of the data into simplices C_j for j \in J (triangles in 2d, tetrahedra in 3d etc), and setting

f(x) = \exp\{b_j^T x - \beta_j\}

for x \in C_j, and f(x) = 0 for x \notin C_n. The estimated density is zero outside the convex hull of the data.

The estimate may therefore be evaluated by finding the appropriate simplex C_j, then evaluating \exp\{b_j^T x - \beta_j\} (if x \notin C_n, set f(x) = 0).

For examples, see mlelcd.

Value

A vector of maximum likelihood estimate (or log maximum likelihood estimate) values, as evaluated at the points x.

Author(s)

Madeleine Cule

Robert Gramacy

Richard Samworth

See Also

mlelcd