MLE: Unconstrained piecewise linear MLE In logcondens: Estimate a Log-Concave Probability Density from Iid Observations

Description

Given a vector of observations x = (x_1, …, x_m) with pairwise distinct entries and a vector of weights w =(w_1, …, w_m) s.t. ∑_{i=1}^m w_i = 1, this function computes a function \hat φ_{MLE} (represented by the vector (\hat φ_{MLE}(x_i))_{i=1}^m) supported by [x_1, x_m] such that

L(φ) = ∑_{i=1}^m w_i φ(x_i) - ∑_{j=1}^{m-1} (x_{j+1} - x_j) J(φ_j, φ_{j+1})

is maximal over all continuous, piecewise linear functions with knots in {x_1, …, x_m}.

Usage

 1 MLE(x, w = NA, phi_o = NA, prec = 1e-7, print = FALSE) 

Arguments

 x Vector of independent and identically distributed numbers, with strictly increasing entries. w Optional vector of nonnegative weights corresponding to x_m. phi_o Optional starting vector. prec Threshold for the directional derivative during the Newton-Raphson procedure. print print = TRUE outputs log-likelihood in every loop, print = FALSE does not. Make sure to tell R to output (press CTRL+W).

Value

 phi Resulting column vector (\hat φ_{MLE}(x_i))_{i=1}^m. L Value L(\hat φ_{MLE}) of the log-likelihood at \hat φ_{MLE}. Fhat Vector of the same length as x with entries \hat F_{MLE,1} = 0 and \hat F_{MLE,k} = ∑_{j=1}^{k-1} (x_{j+1} - x_j) J(φ_j, φ_{j+1}) for k >= 2.

Note

This function is not intended to be invoked by the end user.

Author(s)

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

logcondens documentation built on May 2, 2019, 6:11 a.m.