THE ANALYSIS OF NONPARAMETRIC MIXTURE PROPERTIES WITH A PROBABLITY DENSITY OF A MULTIDIMENSIONAL RANDOM VARIABLE

Abstract

The asymptotic properties of a mixture with nonparametric estimations of probability density with a multidimensional random variable are researched in this article. They are compared with the properties of the traditional Rosenblatt–Parzen type nonparametric probability density estimation, depending on the quantity of the composed mixture and dimension of the random variable.

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The application of nonparametric statistics methods based on the estimations of Rosenblatt–Parzen type probability density [1; 2] is a rapidly developing modelling method of priori uncertainty systems. However, when the research conditions of the system are complicated, there appear methodical and computing difficulties in traditional nonparametric algorithms and models; this can be clearly observed during the processing of statistical data in great amounts. The perspective “detour” direction of the arisen problems consists in the application of decomposition principles of training samples according to their size, and the application of the parallel calculation technology. The purpose of this work is to prove the effective usage of decomposition principles when processing largescale arrays of statistical data, on the basis of the asymptotic properties’ analysis for a nonparametric estimation of probability density mixture. Let sample V = (xi , i = 1, n) from n independent observations of k – dimensional random variable ( v , 1, ) x = x v = k be with a probability density p(x) . The type p(x) is a priori unknown. Let’s divide sample V into T observation groups ( i , ) j j V = x i∈ I , j = 1, T . Multiple observation numbers x in the group with number j shall be identified as j I . While: ( ) 1 1, T j j I I i n U = = = . The quantity j j n = I of units in samples ( i , ) j j V = x i∈ I is equal and equals n n T = . At each sample j V let us construct a nonparametric estimation of probability density with a multidimensional random variable x [1]: ( ) 1 1 1 j k i v v j k i I v v v v x x p x n c ∈ = c ⎛ − ⎞ = Φ⎜ ⎟ ⎝ ⎠ Σ Π Π , j = 1, T . (1) In statistics (1), the nuclear function ( ) v Φ u is satisfied to conditions of normalization, positivity, and symmetry. The parameters of nuclear ( ) v v c = c n functions decrease with the increase of n . Let the intervals of component v x value change for vector x be identical. In these conditions it is reasonable to assume that the values of coefficients v c in nonparametric estimations of probability densities ( ) j p x , j = 1, T are identical and equal to с .