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).