Analytical Description of Empirical Probability Distribution Functions
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ytical Description of Empirical Probability Distribution Functions P. A. Iosifova and A. V. Kirillina, * a
Moscow Aviation Institute, Moscow, Russia *e-mail: [email protected]
Received October 6, 2019; revised October 7, 2019; accepted October 7, 2019
Abstract—The selection of an analytical expression approximating an empirical probability distribution function is considered. For specific examples, the problems that arise in the analysis of data from simulations and tests of aerospace products are identified. These problems cannot be solved by classical statistical methods. A universal approach based on estimation of the distance between the empirical and hypothetical distribution functions permits the selection of the best solution from those available. Keywords: empirical probability distributions, theoretical probability distributions, distribution moments, agreement of ordinal statistics, verification, sample size DOI: 10.3103/S1068798X20080122
The methods used in practice to analyze measurement data may be divided into two groups: (1) parametric methods; (2) nonparametric or distributionfree methods. In statistics, the most complete results are obtained for parametric methods, which permit the derivation of the most accurate and reliable results from the smallest samples. Numerous tables and programs implement these methods and assess the accuracy or reliability of the results. However, these methods are sensitive to violations of the initial assumptions: deviation of the actual probability distributions from those assumed in statistical analysis. Methods in the second group are robust. In other words, they are relatively insensitive to deviations from the initial assumptions. However, although these methods are promising, the results obtained are more modest. That hinders their practical use. In addition, to match the accuracy or reliability of parametric methods, a much larger sample size is required. Therefore, in constructing algorithms for the analysis of measurement results, parametric methods are generally preferred.
tity of interest and calculation of its empirical moments n
n
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3 4 m3 = 1 ( xi − x ) ; m4 = 1 ( xi − x ) , n i =1 n i =1 we make a hypothesis regarding the possibility of describing this quantity by a particular distribution. Then, by means of a set of criteria, we verify the agreement of this null hypothesis with the experimental data. Where necessary, this procedure is repeated for a different null hypothesis. The adoption of a particular null hypothesis is based primarily on information accumulated in a number of applied fields [1]. However, the formulation of a null hypothesis often causes problems. For example, the vertical speed in airplane landing may be described by a gamma distribution, a log-normal distribution, or a Weibull distribution. All these distributions meet the basic requirements. Which should we choose? In classical statistics, numerous methods permit formalization of the solution to this problem [1, 2].
SELECTION OF APPROXIMATING DISTRIBUTIONS The
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