Beta Kernel Estimation For Multivariate Probability Density Functions

We propose a nonparametric kernel estimator, extended from one-dimension to multi-dimensional case, for the density function of multivariate bounded data. As frequently observed in financial field, the variables maybe non-negative and completely bounded (e.g., in the unit interval), we consider kernel estimators using non-negative kernels to estimate density functions with compact supports. The kernels are chosen from a family of beta densities. By using the beta kernel, in a product kernel, the suggested estimator becomes simple in implementation, non-negative and free of boundary bias. On the aspect of theoretical properties, firstly we investigate the mean integrated squared error properties, including optimal bandwidth. Secondly, we establish the uniform strong consistency and asymptotic normality. On the aspect of simulation and application, several examples are provided to illustrate the theoretical properties. In bandwidth selection, we provide the least squares cross-validation method. Then a detailed simulation study investigates the performance of the estimators. Employing the kernel estimator for survival function by beta kernels, applications are also provided with finance data, such as the gold and silver price, and stock market in China.