K-nearest neighbour kernel density estimation, the choice of optimal k

The k-nearest neighbour kernel density estimationmethod is a special type of the kernel density estimation method with the local choice of the bandwidth. An advantage of this estimator is that smoothing varies according to the number of observations in a particular region. The crucial problem is how to estimate the value of the parameter k. In the paper we discuss the problem of choosing the parameter k in a way that minimizes the value of the asymptotic mean integrated square error (AMISE). We define the class of the modified cosine densities that meet the requirements given by the AMISE. The results are compared in a simulation study.