reparametrizations

phieta

etaphi

Vector of independent and identically distributed numbers, with strictly increasing entries.

Vector with entries $\eta_i = \eta(x_i).$

Vector with entries $\phi_i = \phi(x_i).$

Given a vector $(\phi_1, \ldots, \phi_m)$ representing the values of a piecewise linear concave function at
 $x_1, \ldots, x_m,$ <code><a rd-options="" href="/link/etaphi?package=logcondens&version=2.1.5" data-mini-rdoc="logcondens::etaphi">etaphi</a></code> returns a column vector with the entries 
$${\bold{\eta}} = \Bigl(\phi_1, \Bigl(\eta_1 + \sum_{j=2}^m (x_i-x_{i-1})\eta_i\Bigr)_{i=2}^m\Bigr).$$ 
The function <code><a rd-options="" href="/link/phieta?package=logcondens&version=2.1.5" data-mini-rdoc="logcondens::phieta">phieta</a></code> returns a vector with the entries 
$${\bold{\phi}} = \Bigl(\eta_1, \Bigl(\frac{\phi_i-\phi_{i-1}}{x_i-x_{i-1}}\Bigr)_{i=2}^m\Bigr).$$

htest

nonparametric

Given independent and identically distributed observations X(1), ..., X(n), compute the maximum likelihood estimator (MLE) of a density as well as a smoothed version of it under the assumption that the density is log-concave, see Rufibach (2007) and Duembgen and Rufibach (2009). The main function of the package is 'logConDens' that allows computation of the log-concave MLE and its smoothed version. In addition, we provide functions to compute (1) the value of the density and distribution function estimates (MLE and smoothed) at a given point (2) the characterizing functions of the estimator, (3) to sample from the estimated distribution, (5) to compute a two-sample permutation test based on log-concave densities, (6) the ROC curve based on log-concave estimates within cases and controls, including confidence intervals for given values of false positive fractions (7) computation of a confidence interval for the value of the true density at a fixed point. Finally, three datasets that have been used to illustrate log-concave density estimation are made available.

Kaspar Rufibach

logcondens

Estimate a Log-Concave Probability Density from Iid Observations

reparametrizations function

Given a vector $(\phi_1, \ldots, \phi_m)$ representing the values of a piecewise linear concave function at
 $x_1, \ldots, x_m,$ <code><a rd-options='' href='etaphi'>etaphi</a></code> returns a column vector with the entries 
$${\bold{\eta}} = \Bigl(\phi_1, \Bigl(\eta_1 + \sum_{j=2}^m (x_i-x_{i-1})\eta_i\Bigr)_{i=2}^m\Bigr).$$ 
The function <code><a rd-options='' href='phieta'>phieta</a></code> returns a vector with the entries 
$${\bold{\phi}} = \Bigl(\eta_1, \Bigl(\frac{\phi_i-\phi_{i-1}}{x_i-x_{i-1}}\Bigr)_{i=2}^m\Bigr).$$

reparametrizations: Changes Between Parametrizations

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