On an Integral Square Deviation Measure with the Weight of “Delta-Functions ” of the Rosenblatt-Parzen Probability Density Estimator

ABSTRACT. The limit distribution of an integral square deviation with the weight of “delta-functions ” of the Rosenblatt-Parzen probability density estimator is defined. Also, the limit power of the goodness-of-fit test constructed by means of this deviation is investigated. © 2008 Bull. Georg. Natl. Acad. Sci. Key words: distribution density, goodness-of-fit test, power, consistency, limit distribution. 1. Limit distributions of some global measures of distributions of estimates ()xfn of the density ()xf such as, for example, an integral square deviation constructed by means of the so-called weight function ()xW not depending on n were studied in P. Bickel and M. Rosenblatt [1], E. Nadaraya ([2, 3]), P. Hall [4] and other works.. The theory of the asymptotic behavior of an integral mean-square error ( ) ( ) ( ) () () ∫ − = dxxfxfxWEWffR nnnn 2;, , (1) is developed in the work of T. T. Cai and M. G. Low [5], where ( ) () ()0l− = xaWaxW nnn, {}na is a sequence of positive numbers, ( ) 0≥xW is a Borel-measurable function and 0l is some fixed point. If in (1) we put ( ) ()11 2 1 ≤≤− = xIxW and pass to the limit as ∞→na for fixed n, then, roughly speaking, ( ) ( ) () ()200~;, ll ffEWffR nnn − −. If, however, we put 1≡na in (1) for all n, 00 =l and assume that ( ) 0≥xW i