Calculating the second order derivative of the likelihood function of the pendensity approach w.r.t. the parameter beta. Thereby, for later use, the program returns the second order derivative with and without the penalty.
Derv2(penden.env, lambda0)Containing all information, environment of pendensity()
smoothing parameter lambda
second order derivative w.r.t. beta with penalty
second order derivative w.r.t. beta without penalty. Needed for calculating of e.g. AIC.
We approximate the second order derivative in this approach with the negative fisher information. $$J(\beta)= - \frac{\partial^2 l(\beta)}{\partial \beta \ \partial \beta^T} \approx \sum_{i=1}^n s_i(\beta) s_i^T(\beta) .$$ Therefore we construct the second order derivative of the i-th observation w.r.t. beta with the outer product of the matrix Derv1.cal and the i-th row of the matrix Derv1.cal. The penalty is computed as $$\lambda D_m$$.
Density Estimation with a Penalized Mixture Approach, Schellhase C. and Kauermann G. (2012), Computational Statistics 27 (4), p. 757-777.