MLE {logcondens}R Documentation

Unconstrained piecewise linear MLE

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. sum_{i=1}^m w_i = 1, this function computes a function widehat varphi_{MLE} (represented by the vector (widehat varphi_{MLE}(x_i))_{i=1}^m) supported by [x_1, x_m] such that

L(varphi) = sum_{i=1}^m w_i varphi(x_i) - sum_{i=1}^m w_i sum_{j=1}^{m-1} (x_{j+1} - x_j) J(varphi_j, varphi_{j+1})

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

Usage

MLE(x, w = NA, phi_o = NA, prec = 10^(-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}, where w_1 > 0 and w_m > 0. These raw weights are normalized in order to sum to one. Default: w_i = 1/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 varphi_{MLE}(x_i))_{i=1}^m.
L Value L(widehat varphi_{MLE}) of the log-likelihood at widehat varphi_{MLE}.
Fhat Vector of the same length as {x} with entries widehat F_{MLE,1} = 0 and

widehat F_{MLE,k} = sum_{j=1}^{k-1} (x_{j+1} - x_j) J(varphi_j, varphi_{j+1})


for k >= 2.

Author(s)

Kaspar Rufibach, kaspar.rufibach@gmail.com

Lutz Duembgen, duembgen@stat.unibe.ch,
http://www.staff.unibe.ch/duembgen


[Package logcondens version 1.3.3 Index]