This function evaluates the density function of a log-concave maximum likelihood estimator at a point or points.
dlcd(x,lcd, uselog=FALSE, eps=10^-10)A vector of maximum likelihood estimate (or log
maximum likelihood estimate) values, as evaluated at the points x.
Point (or matrix of points) at which the maximum
likelihood estimator should be evaluated
Object of class "LogConcDEAD" (typically output from
mlelcd)
Scalar logical: should the estimator should be calculated on the
log scale?
Tolerance for numerical stability
Madeleine Cule
Robert Gramacy
Richard Samworth
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.
mlelcd