Estimation of a Multivariate Density Function Using Delta Sequences

This paper studies some asymptotic properties of density estimates $\hat{f}$ of $f$ based on $d$-variate delta sequences. The mean-square consistency, almost sure consistency, and asymptotic normality of $\hat{f}$ have been obtained as corollaries to the $L_1$ convergence properties of these delta sequences. Estimators based on kernel functions, orthogonal series, and some histogram methods can be obtained as special cases of $\hat{f}$.