# internal: Functions for estimation of a log-concave probability mass... In logcondiscr: Estimate a Log-Concave Probability Mass Function from Discrete i.i.d. Observations

## 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) [email protected]
http://www.kasparrufibach.ch
Hanna Jankowski [email protected]
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).

All these functions are used by the function `logConDiscrMLE`.