Rate of complete second-order moment convergence and theoretical applications

Abstract

The purpose of this work is to present a novel mode of convergence, complete second-order moment convergence with rate, which implies almost complete convergence and gives a smaller rate of convergence. Indeed, this mode is easier to obtain and gives better performances than those of the almost complete convergence in the case of the nonparametric estimators with kernels of the density function, of the distribution function and of the quantile function. A great advantage of the proposed approach is that less conditions are imposed to the kernel function thanks to the use of the mean squared error expression.

Keywords

• Kernel density estimator
• kernel distribution estimator
• kernel quantile estimator
• complete moment convergence
• nonparametric estimation
• rate of convergence