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logcondens (version 2.1.8)

MLE: 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)

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 \(k \ge 2.\)

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).