Optimal selection of the number of sampling intervals in domain of variation of a one-dimensional random variable in estimation of the probability density

Testing of hypothesis o f random variables independence on the basis of nonparametric algorithm of pattern recognition

“…At the proof of the theorem the technique offered in work [3] and developed in [4][5][6][7][8] is used.…”

Testing of hypothesis of two-dimensional random variables independence on the basis of algorithm of pattern recognition

“…At the proof of the theorem the technique offered in work [3] and developed in [4][5][6][7][8] is used.…”

Comparison of the Effectiveness of Methods for Sampling the Range of Variation of Random Quantities in Synthesis of Nonparametric Estimates of Probability Density

“…The problem of large samples can be avoided by using a nonparametric estimate of the probability density, the synthesis of which is based on compressing the initial statistical data [9][10][11]. The condition of a minimum asymptotic expression for its mean square deviation (MSD) is used to determine a procedure for optimal choice of the number of sampling intervals in the range of variation of the random variable [12].Here an analysis of the approximation properties of a nonparametric estimate of the probability density is used to compare the most widespread procedures for sampling the range of variation of normally distributed random quantities.Synthesis of a Nonparametric Estimate of the Probability Density. Consider a sample V = (x i , i =⎯⎯⎯ 1, n) of n independent values of a univariate random quantity x with an unknown probability density p(x).…”

“…The problem of large samples can be avoided by using a nonparametric estimate of the probability density, the synthesis of which is based on compressing the initial statistical data [9][10][11]. The condition of a minimum asymptotic expression for its mean square deviation (MSD) is used to determine a procedure for optimal choice of the number of sampling intervals in the range of variation of the random variable [12].…”

“…From the condition of a minimum asymptotic expression for the MSD of p(x) from p(x), we obtain a procedure for optimal choice of the number of sampling intervals [12]: (2) which is determined by the form of the probability density to be recovered, its domain of definition Δ, and the volume n of initial statistical data. Here, This relationship is objective, since it does not depend on the form of the kernel functions of the nonparametric estimate (1) of the probability density.…”