Derv2: Calculating the second order derivative with and without penalty
Description
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.
Usage
Derv2(penden.env, lambda0)
Arguments
penden.env
Containing all information, environment of pendensity()
lambda0
smoothing parameter lambda
Value
Derv2.pensecond order derivative w.r.t. beta with penalty
Derv2.calsecond order derivative w.r.t. beta without penalty. Needed for calculating of e.g. AIC.
Details
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 $$\lambda D_m$$.
References
Density Estimation with a Penalized Mixture Approach, Schellhase C. and Kauermann G. (2012), Computational Statistics 27 (4), p. 757-777.