MLE: Unconstrained piecewise linear MLE

In logcondens: Estimate a Log-Concave Probability Density from Iid Observations

Unconstrained piecewise linear MLE

Description

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

L(\phi) = \sum_{i=1}^m w_i \phi(x_i) - \sum_{j=1}^{m-1} (x_{j+1} - x_j) J(\phi_j, \phi_{j+1})

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

Usage

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 {\bold{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 (\widehat \phi_{MLE}(x_i))_{i=1}^m.
L	Value L(\widehat \phi_{MLE}) of the log-likelihood at \widehat \phi_{MLE}.
Fhat	Vector of the same length as {\bold{x}} with entries \widehat F_{MLE,1} = 0 and \widehat F_{MLE,k} = \sum_{j=1}^{k-1} (x_{j+1} - x_j) J(\phi_j, \phi_{j+1}) for k \ge 2.

Note

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

Author(s)

Kaspar Rufibach, [e-mail],
http://www.kasparrufibach.ch

Lutz Duembgen, [e-mail],
https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html

logcondens documentation built on Aug. 22, 2023, 5:06 p.m.