LogConcDEAD (version 1.5-5)

dlcd: 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

Value

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

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.

See Also

mlelcd