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.
new.beta.val(llold, penden.env)log likelihood of the algorithm one step before
Containing all information, environment of pendensity()
corresponding log likelihood
used step size
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 $$l_p(\beta_t,\lambda_0)$$. This means we set $$\beta_{t+1}=\beta_t - 2^{-v} \{s_p(\beta_t,\lambda_0) \cdot (-J_p(\beta_t,\lambda_0))^{-1}\}$$ 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.
Density Estimation with a Penalized Mixture Approach, Schellhase C. and Kauermann G. (2012), Computational Statistics 27 (4), p. 757-777.