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Normality assumption in statistical data analysis
P. 763–768.
Shatskikh S. Y., Мелкумова Л. Э.
The article is devoted to normality assumption in statistical data analysis. It gives a short historical review of the development of scientific views on the normal law and its applications. It also briefly covers normality tests and analyzes possible consequences of using the normality assumption incorrectly.
In book
Vol. 1638: ITNT 2016, Information Technology and Nanotechnology. , CEUR-WS.org, 2016.
Gnetov F., Konakov V., / Series arXiv "math". 2025. No. 2512.04667.
We establish a central limit theorem, a local limit theorem, and a law of large numbers for a natural
random walk on a symmetric space M of non-compact type and rank one. This class of spaces, which
includes the complex and quaternionic hyperbolic spaces and the Cayley hyperbolic plane, generalizes
the real hyperbolic space Hn. Our approach introduces ...
Added: December 5, 2025
Kratz M., Prokopenko E., Extremes 2023 Vol. 26 No. 3 P. 509–544
We build a sharp approximation of the whole distribution of the sum of iid heavy-tailed random vectors, combining mean and extreme behaviors. It extends the so-called ’normex’ approach from a univariate to a multivariate framework. We propose two possible multi-normex distributions, named d-Normex and MRV-Normex. Both rely on the Gaussian distribution for describing the mean behavior, ...
Added: February 20, 2025
Glinskiy V., Artem Logachov, Logachova O. et al., Mathematics 2024 Vol. 12 No. 21 Article 3319
We investigate the asymptotic properties of the plug-in estimator for the Jeffreys divergence, the symmetric variant of the Kullback–Leibler (KL) divergence. This study focuses specifically on the divergence between discrete distributions. Traditionally, estimators rely on two independent samples corresponding to two distinct conditions. However, we propose a one-sample estimator where the condition results from a ...
Added: February 19, 2025
Logachov A.V., Mogulskii A. A., Yambartsev A. A., Siberian Electronic Mathematical Reports 2024 Vol. 21 No. 2 P. 914–926
We obtain a bound for the convergence rate in the central limit theorem for the number of triangles in a heterogeneous Erdos-Renyi graphs. Our approach is reminiscent of Hoeffding decomposition (a common technique in the theory of U-statistics). We show that the centered and normalized number of triangles asymptotically behaves as the normalized sum of ...
Added: February 19, 2025
Glinskiy V., Ismayilova Y., Khrushchev S. et al., Mathematics 2024 Vol. 12 No. 16 Article 2523
The Jarque–Bera test is commonly used in statistics and econometrics to test the hypothesis that sample elements adhere to a normal distribution with an unknown mean and variance. This paper proposes several modifications to this test, allowing for testing hypotheses that the considered sample comes from: a normal distribution with a known mean (variance unknown); ...
Added: December 7, 2024
Ulyanov V. V., Mathematics 2024 Vol. 12 No. 14 Article 2276
In 1733, de Moivre, investigating the limit distribution of the binomial distribution, was the first to discover the existence of the normal distribution and the central limit theorem (CLT). In this review article, we briefly recall the history of classical CLT and martingale CLT, and introduce new directions of CLT, namely Peng’s nonlinear CLT and ...
Added: October 31, 2024
Lebedeva S., Сервис в России и за рубежом 2024 Т. 18 № 1(110) С. 15–26
The article below presents a hypothesis about the normal distribution of Russian tourists in the context of their psychographic states within the framework of motivational-target dynamics, i.e. for various types of tourism. The hypothesis of a normal distribution is not rejected for cultural, gastronomic, sports, health and event tourism. The author revealed that tourists can ...
Added: May 28, 2024
Lebedeva S., Вестник Национальной академии туризма 2020 № 2 (54) С. 13–17
The purpose of this work is testing the hypothesis about the normal distribution of Russian tourists along the psychographic spectrum "allocentrism - psychocentrism". Today, the majority of published works are descriptive. This publication has a practical focus. Moreover, this is an example of statistical methods usage on practice to study the tourism industry. ...
Added: May 7, 2024
Nikita Puchkin, Vladimir Ulyanov, Annales de l'institut Henri Poincare (B) Probability and Statistics 2023 Vol. 59 No. 3 P. 1508–1529
We show that external randomization may enforce the convergence of test statistics to their limiting distributions in particular cases. This results in a sharper inference. Our approach is based on a central limit theorem for weighted sums. We apply our method to a family of rank-based test statistics and a family of phi-divergence test statistics ...
Added: September 3, 2023
Shchegolev A., Управление большими системами: сборник трудов 2023 № 102 С. 5–14
The class of nonlinear Markov processes is characterized by the dependence of the current state of the process on its current distribution in addition to the dependence on the previous state. Due to this feature, these processes are characterized by complex limit behavior and ergodic properties, for which the usual criteria for Markov processes are ...
Added: June 12, 2023
Bobkov S., Ulyanov V. V., Theory of Probability and Its Applications 2022 Vol. 66 No. 4 P. 537–549
We give a short overview of the results related to the refined forms of the central limit theorem, with a focus on independent integer-valued random variables (r.v.'s). In the independent and non-identically distributed (non-i.i.d.) case, an approximation is then developed for the distribution of the sum by means of the Chebyshev--Edgeworth correction containing the moments ...
Added: February 22, 2022
Sawada T., Frontiers in Psychology 2021 Vol. 12 Article 762418
This study describes how the conditions in the Central Limit theorem (CLT) are usually not satisfied in empirical Psychological studies by comparing the formulation of the CLT with a common experimental procedure used in empirical Psychological studies. This explains why the CLT cannot assure that the population follows a normal distribution no matter how large ...
Added: November 9, 2021
Carlier G., Eichinger K., Kroshnin A., SIAM Journal on Mathematical Analysis 2021 Vol. 53 No. 5 P. 5880–5914
In this paper, we investigate properties of entropy-penalized Wasserstein barycenters introduced in [J. Bigot, E. Cazelles, and N. Papadakis, SIAM J. Math. Anal., 51 (2019), pp. 2261--2285] as a regularization of Wasserstein barycenters [M. Agueh and G. Carlier, SIAM J. Math. Anal., 43 (2011), pp. 904--924]. After characterizing these barycenters in terms of a system of Monge--Ampère ...
Added: October 27, 2021
Ayvazyan S. A., Ulyanov V. V., , in: Сборник материалов V-й Международной конференции по стохастическим методам: The 5th International Conference on Stochastic Methods (ICSM5). 23-27 November 2020, Russia, Moscow.: M.: RUDN, 2020. P. 21–23.
We consider the typical behavior of the weighted sums of independent identically distributed random vectors in k-dimensional space. It is shown that in this case the rate of convergence in the multivariate central limit theorem is of order O(1/n) up to logarithmic factor. This extends the one-dimensional Klartag and Sodin result. ...
Added: March 23, 2021
Kroshnin A., Spokoiny V., Suvorikova A., Annals of Applied Probability 2021 Vol. 31 No. 3 P. 1264–1298
n this work we introduce the concept of Bures-Wasserstein barycenter $Q_*$, that is essentially a Fr\'echet mean of some distribution $P$ supported on a subspace of positive semi-definite Hermitian operators $\mathbb{H}_{+}(d)$.
We allow a barycenter to be constrained to some affine subspace of $\mathbb{H}_{+}(d)$ and provide conditions ensuring its existence and uniqueness.
We also investigate convergence and concentration properties ...
Added: October 30, 2020
Bobkov S., , in: Geometric Aspects of Functional AnalysisVol. 1: Israel Seminar (GAFA) 2017-2019.: Springer, 2020. Ch. 5 P. 71–97.
We consider rates of approximation of distributions of weighted sums of independent, identically distributed random variables by the Edgeworth correction of the 4-th order. ...
Added: August 2, 2020
Gribkova N., Probability and Mathematical Statistics 2017 Vol. 37 No. 1 P. 101–118
In this paper, we propose a new approach to the investigation of asymptotic properties of trimmed L-statistics and we apply it to the Cramér type large deviation problem. Our results can be compared with those in Callaert et al. (1982) – the first and, as far as we know, the single article where some results ...
Added: February 28, 2020
Smirnov S. N., Aequationes Mathematicae 2019 Vol. 93 No. 2 P. 445–449
In this short note we show that a weak version of Bernstein’s characterization of the normal distribution implies the local integrability of a measurable solution of the Cauchy functional equation; the linearity of a solution of the Cauchy functional equation is an easy consequence of its local integrability. In its turn, this weak version of ...
Added: December 26, 2019
Sherstinova T., Martynenko G., , in: Digital Transformation and Global Society. Fourth International Conference, DTGS 2019, St. Petersburg, Russia, June 19–21, 2019, Revised Selected Papers.: Springer, 2019. P. 719–731.
Digital technologies provide new possibilities for studying cultural heritage. Thus, literature research involving large text corpora allows to set and solve theoretical problems which previously had no prospects for their decision. For example, it has become possible to model the literary system for some defi-nite literary period (i.e., for the Silver Age of Russian literature) ...
Added: October 31, 2019
Nikitin Y. Y., Petrov V. V., Zaitsev A. Y. et al., Vestnik of the St. Petersburg University: Mathematics 2018 Vol. 51 No. 2 P. 201–232
This is the first in a series of reviews devoted to the scientific achievements of the Leningrad–St. Petersburg school of probability and statistics in the period from 1947 to 2017. It is devoted to limit theorems for sums of independent random variables—a traditional subject for St. Petersburg. It refers to the classical limit theorems: the ...
Added: October 1, 2019
Tsitovich I. I., Информационные процессы 2018 Т. 18 № 4 С. 335–365
In the paper, we consider the score value of some functional of conditionally normally distributed random variables, which is linked to the problem of the sequential hypothesis testing. It is ascertained that the evaluation was limited in growth, which is responsible for setting the average length of observations. ...
Added: September 4, 2019
Molchanov S., Panov V., Stochastics-An International Journal of Probability and Stochastic Processes 2019 Vol. 91 No. 5 P. 754–772
In this paper, we consider limit laws for the model, which is a generalisation of the random energy model (REM) to the case when the energy levels have the mixture distribution. More precisely, the distribution of the energy levels is assumed to be a mixture of two normal distributions, one of which is standard normal, ...
Added: November 19, 2018