Kernel Density Smoothing Using Probability Density Functions and Orthogonal Polynomials

This article is the first of a series devoted to providing a way to correctly explore stock market data through kernel smoothing methods. Here, we are mainly interested in kernel density smoothing, our approach revolves around introducing and testing the goodness of fit of some non-classical kernels based on probability density functions and orthogonal polynomials, the latter ones are of interest to us when they are of order two and above. For each kernel, we use a modified version of the "rules of thumb" principle in order to compute a smoothing parameter that would offer optimal smoothing for a reasonable computational cost. Compared to the Gaussian kernel, some of the tested kernels have provided a better Chi-square statistic, especially the kernels of order 2 based on Hermite and Laguerre polynomials. These results are illustrated using data from the Moroccan stock market.