scholarly journals Methodical aspects of statistical modeling of two-dimensional systems of random variables

2020 ◽  
Vol 1 (1) ◽  
pp. 43-57
Author(s):  
P. Kosobutskyy
Keyword(s):  
Statistical Modeling ◽  
Random Variables ◽  
Practical Interest ◽  
Statistical Processing ◽  
Random Variable ◽  
Point Of View ◽  
Original Research ◽  
Two Dimensional ◽  
Random Data ◽  
Statistical Regularities

According to the analysis of literature sources, the statistical processing of measurement results is not always given due attention. Unfortunately, appropriate algorithms are often limited to simplified statistical procedures, without the proper justification of the objective function, including to check the quality of processing of random data. Therefore, the author plans to publish a series of articles on statistical modeling, which will include the results of original research by the author and others. In this article are considered the methodological aspects of statistical modeling of two-dimensional systems with random data, physical substantiation of correlation regularities of statistical relations between random variables is given, since or the problem of establishing the law of distribution of random variable has practical interest from the point of view of modeling statistical regularities of model "signal + noise".

Random packing of an interval. II

1979 ◽  
Vol 11 (03) ◽  
pp. 591-602
Author(s):  
David Mannion
Keyword(s):  
Fine Particles ◽  
Random Variables ◽  
Random Variable ◽  
Difficult Problem ◽  
Random Packing ◽  
Point Of View ◽  
Initial Size ◽  
Moderate Size ◽  
The Individual

We showed in [2] that if an object of initial size x (x large) is subjected to a succession of random partitions, then the object is decomposed into a large number of terminal cells, each of relatively small size, where if Z(x, B) denotes the number of such cells whose sizes are points in the set B, then there exists c, (0 < ≦ 1), such that Z(x, B)x −c converges in probability, as x → ∞, to a random variable W. We show here that if a parent object of size x produces k offspring of sizes y 1, y 2, ···, y k and if for each k x - y 1 - y 2 - ··· - y k (the ‘waste’ or the ‘cover’, depending on the point of view) is relatively small, then for each n the nth cumulant, Ψ n (x, B), of Z(x, B) satisfies Ψ n (x, B)x -c → κ n (B), as x → ∞, for some κ n (B). Thus, writing N = x c , Z(x, B) has approximately the same distribution as the sum of N independent and identically distributed random variables (The determination of the distribution of the individual appears to be a difficult problem.) The theory also applies when an object of moderate size is broken down into very fine particles or granules.

Estimation of a nonlinear functional from the probability density when optimizing nonparametric decision functions

2021 ◽  
pp. 14-20
Author(s):  
Aleksandr V. Lapko ◽  
Vasiliy A. Lapko
Keyword(s):  
Probability Density ◽  
Decision Rules ◽  
Random Variables ◽  
Random Variable ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Two Dimensional ◽  
Probability Density Estimation ◽  
Nonlinear Functional ◽  
Approximation Errors

A method for estimating the nonlinear functional of the probability density of a two-dimensional random variable is proposed. It is relevant when implementing procedures for fast bandwidths selection in the problem of optimization of kernel probability density estimates. The solution of this problem allows to significantly improve the computational efficiency of nonparametric decision rules. The basis of the proposed approach is the analysis of the formula for the optimal bandwidth of the kernel probability density estimation. In this case, the bandwidth of kernel functions is represented as the product of an indeterminate parameter and the average square deviations of random variables. The main component of an undefined parameter is a nonlinear functional of the probability density. The considered functional is determined by the type of probability density and does not depend on the density parameters. For a family of two-dimensional lognormal laws of distribution of independent random variables, the approximation errors of the considered nonlinear functional from the probability density are determined. The possibility of applying the proposed methodology when evaluating nonlinear functionals of probability densities that differ from the lognormal distribution laws is investigated. An analysis is made of the effect of the resulting approximation errors on the root-mean-square criteria for restoring a non-parametric estimate of the probability density of a two-dimensional random variable.

scholarly journals Analysis of the folded normal distribution of a random variable

2021 ◽  
pp. 111-122
Author(s):  
Степан Алексеевич Рогонов ◽  
Илья Сергеевич Солдатенко
Keyword(s):  
Probability Distribution ◽  
Analytical Solution ◽  
Normal Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Point Of View ◽  
Practical Solution ◽  
Absolute Value ◽  
Actual Error ◽  
Folded Normal Distribution

Анализ поведения случайных величин после различных преобразований можно применять при решении многих нетривиальных задач. В частности, решения, которые невозможно выразить аналитически, с точки зрения практической применимости способны давать результаты с точностью, достаточной для вычислений, вынося невыразимую невязку аналитического решения далеко за рамки требуемой погрешности. В настоящей работе исследовано поведение модуля нормально распределенной случайной величины и выяснено, при каких условиях можно пренебречь операцией взятия абсолютного значения и аппроксимировать модуль случайной величины {\it похожим} распределением вероятностей. The analysis of the behavior of random variables after various transformations can be used in the practical solution of many non-trivial problems. In particular, solutions that cannot be expressed purely analytically, from the point of view of practical applicability, are able to give results with accuracy sufficient for real calculations, taking the inexpressible discrepancy of the analytical solution far beyond the actual error. In this paper, the behavior of the modulus of a normally distributed random variable is investigated and it is found out under what conditions it is possible to neglect the operation of taking an absolute value and approximate the modulus of a random variable with a {\it similar} probability distribution.

Random packing of an interval. II

1979 ◽  
Vol 11 (3) ◽  
pp. 591-602
Author(s):  
David Mannion
Keyword(s):  
Fine Particles ◽  
Random Variables ◽  
Random Variable ◽  
Difficult Problem ◽  
Random Packing ◽  
Point Of View ◽  
Initial Size ◽  
Moderate Size ◽  
The Individual

We showed in [2] that if an object of initial size x (x large) is subjected to a succession of random partitions, then the object is decomposed into a large number of terminal cells, each of relatively small size, where if Z(x, B) denotes the number of such cells whose sizes are points in the set B, then there exists c, (0 < ≦ 1), such that Z(x, B)x−c converges in probability, as x → ∞, to a random variable W. We show here that if a parent object of size x produces k offspring of sizes y1, y2, ···, yk and if for each k x - y1 - y2 - ··· - yk (the ‘waste’ or the ‘cover’, depending on the point of view) is relatively small, then for each n the nth cumulant, Ψn (x, B), of Z(x, B) satisfies Ψn (x, B)x-c → κn (B), as x → ∞, for some κn(B). Thus, writing N = xc, Z(x, B) has approximately the same distribution as the sum of N independent and identically distributed random variables (The determination of the distribution of the individual appears to be a difficult problem.) The theory also applies when an object of moderate size is broken down into very fine particles or granules.

A Different Way to Look at Random Variables

2016 ◽  
pp. 179-199
Author(s):  
Erio Castagnoli ◽  
Marzia De Donno ◽  
Gino Favero ◽  
Paola Modesti
Keyword(s):  
Decision Theory ◽  
Decision Maker ◽  
Classical Problem ◽  
Random Variables ◽  
Random Variable ◽  
Point Of View ◽  
Alternative Approach ◽  
The World ◽  
Sure Thing Principle ◽  
Points Of View

A classical problem in Decision Theory is to represent a preference preorder among random variables. The fundamental Debreu's Theorem states that, in the discrete case, a preference satisfies the so-called Sure Thing Principle if and only if it can be represented by means of a function that can be additively decomposed along the states of the world where the random variables are defined. Such a representation suggests that every discrete random variable may be seen as a “histogram” (union of rectangles), i.e., a set. This approach leads to several fruitful consequences, both from a theoretical and an interpretative point of view. Moreover, an immediate link can be found with another alternative approach, according to which a decision maker sorts random variables depending on their probability of outperforming a given benchmark. This way, a unified approach for different points of view may be achieved.

Selective statistical estimates of multi-year monthly concentrations of biogenous pollution of small rivers of the Northwest zone of the Russian Federation

2020 ◽  
pp. 9-13
Author(s):  
Sergey Kovalenko
Keyword(s):  
Statistical Processing ◽  
Point Of View ◽  
Small Rivers ◽  
Statistical Parameters ◽  
Monthly Data ◽  
Statistical Estimates ◽  
Scientific Task ◽  
The Russian Federation ◽  
Agricultural Areas

The management of surface watercourses is an urgent scientific task. The article presents the results of statistical processing of long-term monthly data of field observations of hydrological and hydrochemical parameters along the Upper Yerga small river in the Vologda region. Sampling estimates of statistical parameters are obtained, autocorrelation and correlation analyzes are performed. The limiting periods from the point of view of pollution for water receivers receiving wastewater from drained agricultural areas are identified.

scholarly journals Stochastic Comparisons of Some Distances between Random Variables

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 981
Author(s):  
Patricia Ortega-Jiménez ◽  
Miguel A. Sordo ◽  
Alfonso Suárez-Llorens
Keyword(s):  
Financial Risk ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Random Variable ◽  
Distribution Functions ◽  
Stochastic Comparisons ◽  
Stochastic Orderings ◽  
Absolute Value ◽  
The Difference

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

scholarly journals Fully degenerate Bell polynomials associated with degenerate Poisson random variables

2021 ◽  
Vol 19 (1) ◽  
pp. 284-296
Author(s):  
Hye Kyung Kim
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Combinatorial Identities ◽  
Bell Polynomials ◽  
Poisson Random Variable ◽  
Central Moments ◽  
Special Polynomials ◽  
Special Numbers And Polynomials ◽  
New Type ◽  
Two Parameters

Abstract Many mathematicians have studied degenerate versions of quite a few special polynomials and numbers since Carlitz’s work (Utilitas Math. 15 (1979), 51–88). Recently, Kim et al. studied the degenerate gamma random variables, discrete degenerate random variables and two-variable degenerate Bell polynomials associated with Poisson degenerate central moments, etc. This paper is divided into two parts. In the first part, we introduce a new type of degenerate Bell polynomials associated with degenerate Poisson random variables with parameter α > 0 \alpha \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the n n th moment of the degenerate Poisson random variable, special numbers and polynomials. In the second part, we consider the fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α > 0 \alpha \gt 0 and β > 0 \beta \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the two-variable fully degenerate Bell polynomials. We show their connection with the degenerate Poisson central moments, special numbers and polynomials.

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

A STUDY OF TWO‐DIMENSIONAL HEAD WAVES IN FLUID AND SOLID SYSTEMS

Geophysics ◽  
1963 ◽  
Vol 28 (4) ◽  
pp. 563-581 ◽  
Author(s):  
John W. Dunkin
Keyword(s):  
Half Space ◽  
Vertical Displacement ◽  
Point Of View ◽  
Head Wave ◽  
Apparent Velocity ◽  
Two Dimensional ◽  
Multiple Reflections ◽  
Transient Wave ◽  
Line Load ◽  
Head Waves

The problem of transient wave propagation in a three‐layered, fluid or solid half‐plane is investigated with the point of view of determining the effect of refracting bed thickness on the character of the two‐dimensional head wave. The “ray‐theory” technique is used to obtain exact expressions for the vertical displacement at the surface caused by an impulsive line load. The impulsive solutions are convolved with a time function having the shape of one cycle of a sinusoid. The multiple reflections in the refracting bed are found to affect the head wave significantly. For thin refracting beds in the fluid half‐space the character of the head wave can be completely altered by the strong multiple reflections. In the solid half‐space the weaker multiple reflections affect both the rate of decay of the amplitude of the head wave with distance and the apparent velocity of the head wave by changing its shape. A comparison is made of the results for the solid half‐space with previously published results of model experiments.

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