# new.beta.val: Calculating the new parameter beta In pendensity: Density Estimation with a Penalized Mixture Approach

## Description

Calculating the direction of the Newton-Raphson step for the known beta and iterate a step size bisection to control the maximizing of the penalized likelihood.

## Usage

 `1` ```new.beta.val(llold, penden.env) ```

## Arguments

 `llold` log likelihood of the algorithm one step before `penden.env` Containing all information, environment of pendensity()

## Details

We terminate the search for the new beta, when the new log likelihood is smaller than the old likelihood and the step size is smaller or equal 1e-3. We calculate the direction of the Newton Raphson step for the known beta_t and iterate a step size bisection to control the maximizing of the penalized likelihood

\eqn{l(beta,lambda0)}

. This means we set

\eqn{beta[t+1]=beta[t]-(2/v)*sp(beta,lambda0)*(-Jp(beta[t],lambda0))^-1}

with s_p as penalized first order derivative and J_p as penalized second order derivative. We begin with v=0. Not yielding a new maximum for a current v, we increase v step by step respectively bisect the step size. We terminate the iteration, if the step size is smaller than some reference value epsilon (eps=1e-3) without yielding a new maximum. We iterate for new parameter beta until the new log likelihood depending on the new estimated parameter beta differ less than 0.1 log-likelihood points from the log likelihood estimated before.

## Value

 `Likelie` corresponding log likelihood `step` used step size

## Author(s)

Christian Schellhase <[email protected]>

## References

Density Estimation with a Penalized Mixture Approach, Schellhase C. and Kauermann G. (2012), Computational Statistics 27 (4), p. 757-777.

pendensity documentation built on May 29, 2017, 9:04 a.m.