«Актуальные вопросы в сфере социально-экономических, технических и естественных наук и информационных технологий» (3-4 апреля 2014г.)

Kuzmenko K. V., Lysenko N. A.,Timoshenko Zh. I.

Oles Honchar Dnipropetrovsk National University

ESTIMATING OF THE LAWS OF PROBABILITY DISTRIBUTIONS ON THE BASIS OF ORTHONORMAL POLYNOMIALS OF LAGUERRE

Most complete characteristic of statistics is the probability density function. The problem of estimating the probability density function has been studied for a long time, it was devoted a lot of work, but it remains relevant today [1]. This report examines one of the non-parametric methods, namely projection estimation [2; 3]. Projection methods are based on the decomposition of the desired density in a series of orthonormal system of functions.

Suppose that in the control sample are obtained measurements of random variables  формула . Rating distribution law on the basis of series of orthonormal polynomials will be:

  формула , (1)

where  формула  – the functions that make up an orthonormal system;  формула  – weight function, and the coefficients  формула :

формула ;

 normalizing factor  формула :

формулаформула ;

формула  – the number of terms of the series, which is determined by the rule of «two zeros», described in [4]. From the structure of (1) follows that if the type of functions  формула  is similar to the distribution law  формула , we can expect that the number of terms формула  sufficient to effect recovery, will be small. Thus, the Laguerre polynomials  формула  with weight function  формулаформула  are recommended to restore asymmetric laws of probability distribution, similar to the exponential. Are Laguerre polynomials  формула  with weight function  формулаформула  are used in the restoration of distribution laws with weak asymmetry.

Difficulties in the practical application of this method of estimating the probability distribution of the laws are in a choose of type of weighting function with limited input data.

In this paper we investigated the error recovery laws of probability distribution series of orthonormal Laguerre polynomials of two types. In the computational experiment generated sampling distribution laws Rayleigh Weibull, Gamma and exponential distribution. Experimental samples of random variables with strong and weak asymmetry were restored by recovered two types of orthonormal Laguerre polynomials  формула and формула , error values ​​obtained are summarized in the table. Practical recommendations about estimation of laws of distribution of probabilities by ranks of orthonormalized polynoms of Laguerre in the conditions of limitation of experimental data are offered.

The list of references:

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