dlcd: Evaluation of a log-concave maximum likelihood estimator at a...

In LogConcDEAD: Log-Concave Density Estimation in Arbitrary Dimensions

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

LogConcDEAD documentation built on April 6, 2023, 1:11 a.m.