© 2007 Bull. Georg. Natl. Acad. Sci. saqarTvelos mecnierebaTa erovnuli akademiis moambe , 17 17 17 17 175, , , , , ½1, , , , , 200 200 200 200 2007 BULLETIN OF THE GEORGIAN NATIONAL ACADEMY OF SCIENCES, 175, ½1, 200 , 200 , 200 , 200 , 2007 Mathematics On the Wolverton-Wagner Estimate of a Distribution Density Elizbar Nadaraya * , Petre Babilua ** * Academy Member, I. Javakhishvili Tbilisi State University ** I. Javakhishvili Tbilisi State University ABSTRACT. The result of the work consists mainly in obtaining the limit distribution of an integral quadratic deviation of the Wolverton-Wagner nonparametric estimate of a multidimensional distribution density. © 2007 Bull. Georg. Natl. Acad. Sci. Key words: nonparametric estimate, recursive estimate, convergence in distribution. 1. Let n X X X , , , 2 1 K be a sequence of independent, equally distributed random variables with values in a Euclidean p-dimensional space p R , 1 ≥ p , whose distribution density is () x f , ( ) p x x x , , 1 K = . As is known, Rosenblatt [1] and Parzen [2] gave the following definition of an empirical kernel density ( ) () x f RP n , which is based on the sampling n X X X , , , 2 1 K : ( ) () ( ) ( ) ∑ = − = n i i n p n RP n X x a K n a x f 1 , where () x K is a given kernel and { } n a is a sequence of positive integers converging monotonically to infinity. Wolverton and Wagner [3] introduce the following definition of an empirical density ( ) () x f W n that differs little from ( ) () x f RP n but is recurrent: ( ) () ( ) ( ) ( ) () ( ) ( ) n n p n W n n j i i p j W n X x a K n a x f n n X x a K a n x f − + − = − = − = ∑ 1 1 1 1 . The recurrent definition of probability density estimates ( ) () x f W n has two obvious advantages: 1) there is no need to memorize data, i.e. if the estimate ( ) () x f W n 1 − is known, then ( ) () x f W n can be calculated by means of the last observation n X only, without using the sampling 1 2 1 , , , − n X X X K ; 2) the asymptotic dispersion of the estimate ( ) () x f W n does not exceed the dispersion of the estimate ( ) () x f RP n . The aim of this work is to study the asymptotics of a mean value of the integral of the squared error