# evaluateLogConDens: Evaluates the Log-Density MLE and Smoothed Estimator at... In logcondens: Estimate a Log-Concave Probability Density from Iid Observations

## Description

Based on a "dlc" object generated by logConDens, this function computes the values of

\hat φ_m(t)

\hat f_m(t) = exp(\hat φ_m(t))

\hat F_m(t) = int_{x_1}^t exp(\hat φ_m(x)) dx

\hat f_m^*(t) = exp(\hat φ_m^*(t))

\hat F_m^*(t) = int_{x_1}^t \exp(\hat φ_m^*(x)) dx

at all real number t in xs. The exact formula for \hat F_m and t \in [x_j,x_{j+1}] is

\hat F_m(t) = \hat F_m(x_j) + (x_{j+1}-x_j) J(\hat φ_j, \hat φ_{j+1}, (t-x_j)/(x_{j+1}-x_j))

for the function J introduced in Jfunctions. Closed formulas can also be given for \hat f_m^*(t) and \hat F_m^*(t).

## Usage

 1 evaluateLogConDens(xs, res, which = 1:5, gam = NULL, print = FALSE) 

## Arguments

 xs Vector of real numbers where the functions should be evaluated at. res An object of class "dlc", usually a result of a call to logConDens. which A (sub-)vector of 1:5 specifying which of the above quantities should be computed. gam Only necessary if smoothed = TRUE. The standard deviation of the normal kernel. If equal to NULL, gam is chosen such that the variances of the original sample x_1, …, x_n and \hat f_n^* coincide. See logConDens for details. print Progress in computation of smooth estimates is shown.

## Value

Matrix with rows (x_{0, i}, \hat φ_m(x_{0, i}), \hat f_m(x_{0, i}), \hat F_m(x_{0, i}), \hat f_m^*(x_{0, i}), \hat F_m^*(x_{0, i})) where x_{0,i} is the i-th entry of xs.

## Author(s)

Kaspar Rufibach, kaspar.rufibach@gmail.com,
http://www.kasparrufibach.ch

## Examples

  1 2 3 4 5 6 7 8 9 10 11 ## estimate gamma density set.seed(1977) x <- rgamma(200, 2, 1) res <- logConDens(x, smoothed = TRUE, print = FALSE) ## compute function values at an arbitrary point xs <- (res$x[100] + res$x[101]) / 2 evaluateLogConDens(xs, res) ## only compute function values for non-smooth estimates evaluateLogConDens(xs, res, which = 1:3) 

### Example output

Warning messages:
1: In rgl.init(initValue, onlyNULL) : RGL: unable to open X11 display
2: 'rgl_init' failed, running with rgl.useNULL = TRUE