Statistical Theorem Research Articles

Developing Statistical Thinking in Schoolchildren: From Theory to Practice

Since 2022, the school course «Probability and Statistics» has been taught as a compulsory subject in all schools in Russia, and questions on it are included in the Unified State Exam. However, international experience in teaching the basics of statistics shows that this is not an easy task. It includes not only teaching students how to prove the theorems of mathematical statistics and explain the calculation of the median and variance, but also contributes to the development of statistical thinking in them – the ability to make decisions based on the analysis of empirical data. According to the authors, the statistical community cannot and should not replace teachers who promote the teaching of statistics at school, but it has the opportunity to make a significant contribution to this process.The article presents the experience of the All-Russian School competition on statistics «Trend», the All-Russian teacher competition for the best practices in teaching statistics «We train the best!», with the support of the territorial bodies of state statistics in St. Petersburg, Kirov and Rostov regions; master classes for hundreds of teachers teaching statistics in grades 7–11. In addition, the article presents plans for organizing a competition on civil statistics aimed at developing skills in working with statistical data, as well as teaching the basics of statistical analysis, the use of visualization tools, and the formation of citizenship through familiarity with objective statistical information

Self-correlated spatial random variables: From an auto- to a sui- model respecification

This paper marks the 50-year publication anniversary of Besag's seminal spatial auto- models paper. His classic article synthesizes generic autoregressive specifications (i.e., a response variable appears on both sides of a regression equation and/or probability function equal sign) for the following six popular random variables: normal, logistic (i.e., Bernoulli), binomial, Poisson, exponential, and gamma. Besag dismisses these last two while recognizing failures of both as well as the more scientifically critical counts-oriented auto-Poisson. His initially unsuccessful subsequent work first attempted to repair them (e.g., pseudo-likelihood estimation), and then successfully revise them within the context of mixed models, formulating a spatially structured random effects term that effectively and efficiently absorbs and accounts for spatial autocorrelation in geospatial data. One remaining weakness of all but the auto-normal is a need to resort to Markov chain Monte Carlo (MCMC) techniques for legitimate estimation purposes. Recently, Griffith succeeded in devising an innovative uniform distribution genre—sui-uniform random variables—that accommodates spatial autocorrelation, too. Its most appealing feature is that, by applying two powerful mathematical statistical theorems (i.e., the probability integral transform, and the quantile function), it redeems Besag's auto- model failures. This paper details conversion of Besag's initial six modified variates, exemplifying them with both simulation experiments and publicly accessible real-world georeferenced data. The principal outcome is valuable spatial statistical advancements, with special reference to Moran eigenvector spatial filtering.

Improving Likert scale big data analysis in psychometric health economics: reliability of the new compositional data approach

Bipolar psychometric scales data are widely used in psychologic healthcare. Adequate psychological profiling benefits patients and saves time and costs. Grant funding depends on the quality of psychotherapeutic measures. Bipolar Likert scales yield compositional data because any order of magnitude of agreement towards an item assertion implies a complementary order of magnitude of disagreement. Using an isometric log-ratio (ilr) transformation the bivariate information can be transformed towards the real valued interval scale yielding unbiased statistical results increasing the statistical power of the Pearson correlation significance test if the Central Limit Theorem (CLT) of statistics is satisfied. In practice, however, the applicability of the CLT depends on the number of summands (i.e., the number of items) and the variance of the data generating process (DGP) of the ilr transformed data. Via simulation we provide evidence that the ilr approach also works satisfactory if the CLT is violated. That is, the ilr approach is robust towards extremely large or infinite variances of the underlying DGP increasing the statistical power of the correlation test. The study generalizes former results pointing out the universality and reliability of the ilr approach in psychometric big data analysis affecting psychometric health economics, patient welfare, grant funding, economic decision making and profits.

Short cycles of random permutations with cycle weights: Point processes approach

We study the asymptotic behavior of short cycles of random permutations with cycle weights. More specifically, on a specially constructed metric space whose elements encode all possible cycles, we consider a point process containing all information on cycles of a given random permutation on {1,…,n}. The main result of the paper is the distributional convergence with respect to the vague topology of the above processes towards a Poisson point process as n→∞ for a wide range of cycle weights. As an application, we give several limit theorems for various statistics of cycles.

Separable sample covariance matrices under elliptical populations with applications

This paper is to investigate the spectral properties of separable covariance matrices under elliptical populations. The separable covariance matrix model can handle both cross-row and cross-column correlations thus gain more popularity recently. Under the high-dimensional setting where the dimension p p and the sample size n n tend to infinity proportionally, we find the limit of the empirical spectral distribution and establish the central limit theorems (CLT) for linear spectral statistics of such kinds of sample covariance matrices. Some applications of our established CLT are also given.

Stancu variant of Jakimovski-Leviatan-Durrmeyer operators involving Brenke type polynomials

<p style='text-indent:20px;'>Karaisa [<xref ref-type="bibr" rid="b29">29</xref>] presented Jakimovski- Leviatan- Durrmeyer type operators by means of Appell polynomials. In a similar manner, Wani et al. [<xref ref-type="bibr" rid="b43">43</xref>] proposed a sequence of Jakimovski-Leviatan-Durrmeyer type operators involving Brenke type polynomials which include Appell polynomials and Hermite polynomials. We note that the definitions of the operators given in both these papers are not correct. In the present article, we introduce a Stancu variant of the operators considered in [<xref ref-type="bibr" rid="b43">43</xref>] after correcting their definition. The definition of the operator proposed in [<xref ref-type="bibr" rid="b29">29</xref>] may be similarly corrected. We establish the Korovkin type approximation theorem and the rate of convergence by means of the usual modulus of continuity, Peetre's K-functional and the class of Lipschitz type functions for our operators. Next, we discuss the Voronovskaja and Gr<inline-formula><tex-math id="M1">\begin{document}$ \ddot{u} $\end{document}</tex-math></inline-formula>ss Voronovskaja type asymptotic theorems. Finally, we study the convergence of these operators in a weighted space and the Korovkin type weighted statistical approximation theorem.</p>

Statistical Ergodic Theorem in Symmetric Spaces for Infinite Measures

Statistical Ergodic Theorem in Symmetric Spaces for Infinite Measures

Central limit theorems for local network statistics

Summary Subgraph counts, in particular the number of occurrences of small shapes such as triangles, characterize properties of random networks. As a result, they have seen wide use as network summary statistics. Subgraphs are typically counted globally, making existing approaches unable to describe vertex-specific characteristics. In contrast, rooted subgraphs focus on vertex neighbourhoods, and are fundamental descriptors of local network properties. We derive the asymptotic joint distribution of rooted subgraph counts in inhomogeneous random graphs, a model that generalizes most statistical network models. This result enables a shift in the statistical analysis of graphs, from estimating network summaries to estimating models linking local network structure and vertex-specific covariates. As an example, we consider a school friendship network and show that gender and race are significant predictors of local friendship patterns.

Analysis of Several Basic Formulae of Statistical Methods in Cosmology

As a matter of fact, statistic methods are widely applied in astrophysics in various aspects including particle motions, Big Bang theory, observations, and so on. With this in mind, this article focuses on statistical methods in cosmology. On this basis, the study aims to throw new light on understanding of these theorems. To be specific, it first discusses the history of statistical methods applied in physics especially in cosmology. In the main part, the paper analyzes some basic statistical theorems by modifying a current version of proof, or refining the proof through detailed analysis. For each analyzed formula, the paper introduces some recent applications in order to offer a better understanding of its applications in cosmology. In addition, the findings also suggest that even for proven theorems, the proof process still has room for optimization and improvements. Overall, this study comprehends the understanding of statistical methods in cosmology as well as sheds light on guiding further simplification and derivation of theoretical astrophysics.

How to think clearly about the central limit theorem.

The central limit theorem (CLT) is one of the most important theorems in statistics, and it is often introduced to social sciences researchers in an introductory statistics course. However, the recent replication crisis in the social sciences prompts us to investigate just how common certain misconceptions of statistical concepts are. The main purposes of this article are to investigate the misconceptions of the CLT among social sciences researchers and to address these misconceptions by clarifying the definition and properties of the CLT in a manner that is approachable to social science researchers. As part of our article, we conducted a survey to examine the misconceptions of the CLT among graduate students and researchers in the social sciences. We found that the most common misconception of the CLT is that researchers think the CLT is about the convergence of sample data to the normal distribution. We also found that most researchers did not realize that the CLT applies to both sample means and sample sums, and that the CLT has implications for many common statistical concepts and techniques. Our article addresses these misconceptions of the CLT by explaining the preliminaries needed to understand the CLT, introducing the formal definition of the CLT, and elaborating on the implications of the CLT. We hope that through this article, researchers can obtain a more accurate and nuanced understanding of how the CLT operates as well as its role in a variety of statistical concepts and techniques. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Detecting communities in attributed networks through bi-direction penalized clustering and its application

Exploiting heterogeneous information in attributed networks to improve the performance of community detection has attracted considerable research attention. Although variational graph autoencoder (VGAE)-based methods have been proven to be effective strategies, they perform community detection based on assumptions regarding the dimension of embedding and the number of communities, limiting their effectiveness and applicability. In this study, we combined VGAE-based methods and a bi-direction penalized clustering algorithm (BiPClust) for community detection. Our approach addresses the issues of dimension selection and community number determination by automatically optimizing penalized clustering. Both the computational algorithm and statistical theorems confirm that BiPClust effectively mitigates the impacts of redundant embedding and determines the unknown number of communities. Furthermore, applying the proposed methods to community detection on benchmark datasets and syndicated investment networks in China reveals that BiPClust surpasses other methods in performance.

Functional central limit theorems for local statistics of spatial birth–death processes in the thermodynamic regime

Functional central limit theorems for local statistics of spatial birth–death processes in the thermodynamic regime

Nonlinear Approximation by $q$-Favard-Sz{\'a}sz-Mirakjan Operators of Max-Product Kind

In this study, nonlinear $q$-Favard-Sz{\'a}sz-Mirakjan operators of max-product kind are defined and approximation properties of these operators are investigated. Classical approximation and $A$-statistical approximation theorems are given.

Analytic and Asymptotic Properties of the Generalized Student and Generalized Lomax Distributions

Analytic and asymptotic properties of the generalized Student and generalized Lomax distributions are discussed, with the main focus on the representation of these distributions as scale mixtures of the laws that appear as limit distributions in classical limit theorems of probability theory, such as the normal, folded normal, exponential, Weibull, and Fréchet distributions. These representations result in the possibility of proving some limit theorems for statistics constructed from samples with random sizes in which the generalized Student and generalized Lomax distributions are limit laws. An overview of known properties of the generalized Student distribution is given, and some simple bounds for its tail probabilities are presented. An analog of the ‘multiplication theorem’ is proved, and the identifiability of scale mixtures of generalized Student distributions is considered. The normal scale mixture representation for the generalized Student distribution is discussed, and the properties of the mixing distribution in this representation are studied. Some simple general inequalities are proved that relate the tails of the scale mixture with that of the mixing distribution. It is proved that for some values of the parameters, the generalized Student distribution is infinitely divisible and admits a representation as a scale mixture of Laplace distributions. Necessary and sufficient conditions are presented that provide the convergence of the distributions of sums of a random number of independent random variables with finite variances and other statistics constructed from samples with random sizes to the generalized Student distribution. As an example, the convergence of the distributions of sample quantiles in samples with random sizes is considered. The generalized Lomax distribution is defined as the distribution of the absolute value of the random variable with the generalized Student distribution. It is shown that the generalized Lomax distribution can be represented as a scale mixture of folded normal distributions. The convergence of the distributions of maximum and minimum random sums to the generalized Lomax distribution is considered. It is demonstrated that the generalized Lomax distribution can be represented as a scale mixture of Weibull distributions or that of Fréchet distributions. As a consequence, it is demonstrated that the generalized Lomax distribution can be limiting for extreme statistics in samples with random size. The convergence of the distributions of mixed geometric random sums to the generalized Lomax distribution is considered, and the corresponding extension of the famous Rényi theorem is proved. The law of large numbers for mixed Poisson random sums is presented, in which the limit random variable has a generalized Lomax distribution.

Evidence for a short-lived resonance state in enzyme catalysis via rate-equationconvolution.

At the cellular level, all biological function relies on enzymes to provide catalytic acceleration of essential biochemical processes driving cellular metabolism. The enzyme is presumed to lower the activation energy barrier separating reactants from products, but the precise mechanism remains unresolved. Here we examine the temperature dependence of the enzyme-catalyzed dissociation of p-nitrophenyl-α-D-glucopyranoside (pNPG), a chromogenic analog for maltose, isomaltose, and sucrose disaccharide sugars, into p-nitrophenol (pNP) and glucose (monosaccharide). The enzymes of interest are the wild type and mutant forms of glucosidase MalL produced by the probiotic bacterium Bacillus subtilis. The per-enzyme production rates k(T) for the pNPG→ glucose reaction all show a characteristic temperature profile with an Arrhenius-like (approximately exponential) slow acceleration at low temperatures, rising through a point of inflexion to reach a maximum, then turning over to decline steeply towards zero production at high temperatures. This asymmetric profile is found to be well fitted by convolving an exponential growth function f(T) with a Gaussian temperature distribution g(T) to produce an exponentially modified Gaussian function h(T). To give a physical interpretation of the convolution components, we make the temperature mapping Θ≡T_{ref}-T where T_{ref} marks the temperature at which a given mutant becomes fully denatured (unfolded) and therefore inactive, then convert the convolution components to probability density functions which obey the convolution theorem of statistics. Working in Θ space, we identify f(Θ) as the density function for an Arrhenius-like transition from ground-state A to metastable-state B, and g(Θ) as the Gaussian distribution of offset-temperature fluctuations for the metastable state. By mapping the standard thermodynamic relations for temperature and energy fluctuations to the enzyme frame of reference, we are able to derive an expression for the lifetime for the metastable B state. For the 15 enzyme experiments, we obtain a mean value 〈Δt〉≳(29.0±1.3)×10^{-15}s, in remarkably good agreement with the ∼30-fs estimate for the period of glycosidic bond oscillations extracted from published infrared spectroscopy. We suggest that the metastable B state provides a low-energy target that has the effect of lowering the activation energy barrier by presenting an alternative axis for the reaction coordinate.

Statistical Korovkin-type theorem for monotone and sublinear operators

"In this paper we generalize the result on statistical uniform convergence in the Korovkin theorem for positive and linear operators in C([a; b]), to the more general case of monotone and sublinear operators. Our result is illustrated by concrete examples."

Application of Bayes Theorem to Accident Analysis and Classification

In today's social context, "data" is a very common term. Data Science, Data analysis and other applications on different levels of Data have become very popular topics and even jobs. The main topic to be analyzed in this paper is Bayesian theorem. Bayesian theorem is a basic theorem in statistics, but with sufficient data support, Bayesian theorem can be applied in many fields. In this paper, this paper will analyze the application of Bayes' theorem in these two aspects: Bayes Theorem to Accident Analysis and Bayes Theorem to Classification. In terms of Bayes Theorem to Accident Analysis, this paper will analyze three papers that use Bayesian theorem to analyze accidents. In these papers, the authors have used Bayesian theorem to analyze different accidents. These accidents are: The causes and analysis of engineering accidents in mining areas, the analysis of causes of roof accidents in coal mines, and the analysis of the causes of highway traffic accidents. way of accident. In terms of Bayes Theorem to Classification, most of the authors use naive bayes to classify different data sets. The benefits of this classification are that whenever new data enters, researchers can easily classify new data sets. Incoming data is categorized.

Central limit theorem of linear spectral statistics of high-dimensional sample correlation matrices

A high-dimensional sample correlation matrix is an important random matrix in multivariate statistical analysis. Its central limit theory is one of the main theoretical bases for making statistical inferences on high-dimensional correlation matrices. Under the high-dimensional framework in which the data dimension tends to infinity proportionally with the sample size, we establish the central limit theorems (CLT) for the linear spectral statistics (LSS) of sample correlation matrices in two settings: (1) the population follows an independent component structure; (2) the population follows an elliptical structure, including some heavy-tailed distributions. The results show that the CLTs of the LSS of the sample correlation matrices are very different in the two settings. In particular, even if the population correlation matrix is an identity matrix, the CLTs are different in the two settings. An application of our two established CLTs is provided.

Spherical Sherrington–Kirkpatrick Model for Deformed Wigner Matrix with Fast Decaying Edges

We consider the 2-spin spherical Sherrington–Kirkpatrick model whose disorder is given by a deformed Wigner matrix of the form \(W+\lambda V\), where W is a Wigner matrix and V is a random diagonal matrix with i.i.d. entries. Assuming that the density function of the entries of V decays faster than a certain rate near the edges of its spectrum, we prove the sharp phase transition of the limiting free energy and its fluctuation. In the high temperature regime, the fluctuation of the free energy converges in distribution to a Gaussian distribution, whereas it converges to a Weibull distribution in the low temperature regime. We also prove several results for deformed Wigner matrices, including a local law for the resolvent entries, a central limit theorem of the linear spectral statistics, and a theorem on the rigidity of eigenvalues.

Analyzing the Financial Statements of Companies listed on the National Stock Exchange using the Benford’s Law

Accounting numbers normally follow Benford’s mathematical rule. These findings are so unpredictable that knowledge manipulators typically refuse to comply with the law. Based on that information, accounting data that are submitted fraudulently can be identified. The law provides, however, for instances where data containing errors are presented to be detected. This paper examines the Indian companies ‘ financial statements of the last 10 years listed on the “National stock exchange of India” and testing whether they adhere to a statistical theorem called The Benford Law. The rule of Benford notes that the numbers are not uniformly distributed in a sequence of numerical data but instead obey a certain logarithmic rule such that the numbers beginning with a smaller digit have a greater frequency to occur than those beginning with larger digits. They use the first and the second digit test of Benford law to check whether businesses are following the logarithmic distribution.