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.

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

`llold` |
log likelihood of the algorithm one step before |

`penden.env` |
Containing all information, environment of pendensity() |

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.

`Likelie` |
corresponding log likelihood |

`step` |
used step size |

Christian Schellhase <cschellhase@wiwi.uni-bielefeld.de>

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

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