Jacobi probability distribution for approximation of emperic statistic distributions

Gladkiy, E.G.; Office, Yuzhnoye State Design

doi:10.15407/itm2017.01.107

Teh. Meh.

2017

DOI: 10.15407/itm2017.01.107

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Jacobi probability distribution for approximation of emperic statistic distributions

E.G. Gladkiy¹,

Yuzhnoye State Design Office²

Abstract: The paper purpose is to demonstrate the opportunities of the Jacobi probability distributions for fitting the statistical populations. The universal Pearson and Johnson systems of the distributions, the generalized lambda distribution and the Gram-Charlier distribution, which are widely used for fitting statistical populations, are analyzed. It is pointed out that the main disadvantage of these distributions is that they do not take into account real limited ranges of variations in the random variables. The pa… Show more

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Cited by 2 publications

References 7 publications

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Construction and analysis of universal 2D distributions with a bounded rectangular variation domain

Hladkyi¹,

Perlyk²

2022

Teh. Meh.

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When solving parametric reliability problems, one often has to construct distributions of statistical data to find the probability of containment in the operability region. This paper considers the problem of 2D statistical ensemble fitting. The use of a 2D normal distribution in statistical data description is not always justified because statistical ensembles rather frequently (at the level of marginal components and a stochastic relationship between them) have properties different from the normal case. From a practical standpoint, it is desirable for researchers to describe 2D statistical ensembles with the use of universal distributions, which allow one to cover a wide range of source data using a single analytical form. In the process of fitting, account should be made of bounded ranges of random variables. The paper considers who universal distribution construction methods, which are based on 1D orthogonal Jacobi polynomial expansions. In these distributions, the random variable range is a rectangle. In the first method, a 2D distribution is constructed using a direct expansion in the 1D Jacobi polynomials. A 2D Jacobi distribution function and regression lines are obtained, and methods to fit it are considered. In theory, a distribution obtained in this way can be used, up to the fourth order inclusive, for marginal and even reduced moments different from the normal case. However, its real capabilities are limited to values of reduced moments (1D and even) that differ from the normal case only very slightly. Otherwise, the probability surface may enter negative ranges with the occurrence of multiple modes. The second way to construct a 2D distribution is to use a normal copula and 1D Jacobi distributions as components. The resulting 2D distribution allows one to deal with 1D distributions different from the normal case and linear correlation. This approach is justified because, according to research data, it is a linear stochastic relationship that relates a significant part of 2D statistical ensembles, and marginal distributions deviate from the normal case. Regression lines of a distribution of this kind are obtained, and it is shown that they are curved because marginal distributions differ from the normal one. The paper considers the practical example of fitting a 2D ensemble of characteristics of a liquid-propellant rocket engine some components of which are related via a linear stochastic relationship (the parameters that characterize a nonlinear stochastic relationship proved to be insignificant) and have 1D distributions different from the normal one. The fitted and observed frequencies are in rather good agreement. It is shown that a distribution based on a normal copula is more universal, and it is recommended for practical calculations.

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Construction and analysis of universal 2D distributions with a bounded rectangular variation domain

Hladkyi¹,

Perlyk²

2022

Teh. Meh.

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Universal spline-perturbed distribution

Hladkyi,

Perlyk

2023

Teh. Meh.

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This paper considers the problem of probability distribution construction for a random variable from known numerical characteristics. The problem is of importance in determining the parametric reliability of engineering systems when the numerical characteristics (in particular, the bias and the kurtosis) of an output parameter (state variable) are determined by analytical methods and its distribution must be recovered. This may be done using a four-parameter universal distribution, which allows one to cover certain ranges (preferably, as wide as possible) of the bias and kurtosis coefficients using a single analytical form. The most familiar universal distribution is Gram-Charlier’s, which is a deformation of the normal distribution obtained using a Chebyshev-Hermite orthogonal polynomial expansion. However, in the general case, Gram-Charlier’s distribution function is not a steadily increasing one. For some combinations of the bias and kurtosis coefficients, the density curve may exhibit negative values and multiple modes. Because of this, a search for other universal distributions to cover wider ranges of the bias and kurtosis coefficients is of current importance. The paper analyzes a method of universal probability distribution construction by multiplying the normal density by a perturbing polynomial in the form of a spline (referred to as the spline-perturbed distribution). The idea of a distribution of this type was proposed earlier to account for a nonzero bias coefficient. The spline is constructed based on Hermite’s interpolating polynomials of the third degree with two knots, which have a minimum of parameters and possess a locality property The basic distribution is constructed for a four-knot spline. The paper further develops and generalizes the spline-perturbed distribution to nonzero bias and kurtosis coefficients. Two cases are considered. The first case is a composition of two splines that have four and five knots, respectively. The former and the latter allow one to account for the bias and the kurtosis, respectively. Integral equations are obtained to find the values at the knots of both splines and construct the distribution. The second case is more general and uses one five-knot Hermite spline. The paper shows a way to construct a generalized spline-perturbed distribution without any negative density values or any multiple modes. The knot points are chosen using an enumerative technique. Conditions for the absence of negative density values and multiple nodes are identified.

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Jacobi probability distribution for approximation of emperic statistic distributions

Cited by 2 publications

References 7 publications

Construction and analysis of universal 2D distributions with a bounded rectangular variation domain

Construction and analysis of universal 2D distributions with a bounded rectangular variation domain

Universal spline-perturbed distribution

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