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(ψ) via Newton step.

dMLE Compute the vector ψ 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^{δ_k}(ψ_k, ψ_{k+1}) := ∑_{j=0}^{δ_k} \exp \Bigl((1-j/δ_k)ψ_k + (j/δ_k) ψ_{k+1} \Bigr).

This function is used to compute the value of the log-likelihood in LikFunk.

J10 Derivative of J^{δ_k}(ψ_k, ψ_{k+1}) w.r.t to the first argument.

J11 Derivative of J^{δ_k}(ψ_k, ψ_{k+1}) w.r.t to both arguments.

J20 Second derivative of J^{δ_k}(ψ_k, ψ_{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.

Author(s)

Kaspar Rufibach (maintainer) kaspar.rufibach@gmail.com
http://www.kasparrufibach.ch
Fadoua Balabdaoui fadoua@ceremade.dauphine.fr
http://www.ceremade.dauphine.fr/~fadoua
Hanna Jankowski hkj@mathstat.yorku.ca
http://www.math.yorku.ca/~hkj
Kathrin Weyermann

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).

See Also

All these functions are used by the function logConDiscrMLE.