internal: Functions for estimation of a log-concave probability mass function via maximum likelihood
Description
Internal functions for the estimation of a log-concave probability mass function.
These functions are not intended to be called by the user directly.
Direction Compute vector that points in direction of $\max L(\psi)$ via Newton step.
dMLE Compute the vector $\psi$ s.t. the log-likelihood function $L$, as implemented in LikFunk, is maximized
over all PMFs (under no additional restrictions, though).
GradientL Gradient of LikFunk.
HesseL Hesse matrix of LikFunk.
J00 Function introduced in Section 2.3 in Weyermann (2007), defined as
$$J^{\delta_k}(\psi_k, \psi_{k+1}) := \sum_{j=0}^{\delta_k} \exp \Bigl((1-j/\delta_k)\psi_k + (j/\delta_k) \psi_{k+1} \Bigr).$$
This function is used to compute the value of the log-likelihood in LikFunk.
J10 Derivative of $J^{\delta_k}(\psi_k, \psi_{k+1})$ w.r.t to the first argument.
J11 Derivative of $J^{\delta_k}(\psi_k, \psi_{k+1})$ w.r.t to both arguments.
J20 Second derivative of $J^{\delta_k}(\psi_k, \psi_{k+1})$ w.r.t to the first argument.
LikFunk The log-likelihood function for the discrete log-concave MLE.
LocalCoarsen Auxiliary function.
LocalConcavity Auxiliary function.
LocalExtend Auxiliary function.
LocalMLE Auxiliary function.
LocalNormalize Auxiliary function.
StepSize Auxiliary function.References
Balabdaoui, F., Jankowski, H., Rufibach, K., and Pavlides, M. (2013).
Maximum likelihood estimation and confidence bands for a discrete log-concave distribution.
J. R. Stat. Soc. Ser. B Stat. Methodol., 75(4), 769--790.
Weyermann, K. (2007).
An Active Set Algorithm for Log-Concave Discrete Distributions.
MSc thesis, University of Bern (Supervisor: Lutz Duembgen).