Classical Probability Research Articles

NEUTROSOPHIC EXTENSION OF THE NEGATIVE BINOMIAL DISTRIBUTION: PROPERTIES AND APPLICATIONS TO INTERVAL-VALUED DATA

In real world while addressing the challenges of ambiguity and vagueness, the problems call for a scientific approach to quantify the uncertainties. The negative binomial distribution, a cornerstone in classical probability theory, traditionally examines the probability of achieving the kth success in n trials, following failures, with precise observations. However, the available literature does not address the occurrence of observations in the intervals and hence the handling of interval-valued data remained unattended. This study introduces the neutrosophic negative binomial distribution which is extended from the classical version by incorporating both indeterminacy and crisp intervals. We derive various properties of this generalized distribution which includes; characteristic function, probability function and moment generating function. This study also presents the reliability analysis metrics like cumulative hazard rate, Mills ratio, survival hazard rate, odds ratio and reversed hazard rate. This study explores the order statistics for neutrosophic negative binomial distribution, encompassing with joint, maximum, minimum and median order statistics. Special case studies are discussed which demonstrate the application of the proposed distribution, paving the way for addressing complex problems that combine conventional approaches with non-precisely determined information.

Probability Updating in the Classical and the Quantum Model

Imagine a Bayesian decision agent who is keen to invest only in technologies or businesses that are conductive to achieving a sustainable economic future. Being initially keen to invest in a certain technology, subsequently two pieces of new information are received that both individually reduce the agent’s inclination to make that investment. In such a situation, it would be natural to assume that the simultaneous consideration of both pieces of information should further reduce the agent’s initial enthusiasm; after all the Bayesian method has been called “ nothing but common sense reduced to calculation.” Somewhat surprisingly, making general statements like this about the double conditional probability requires substantial additional assumptions in classical Bayesian probability theory. We investigate four schemes of assumptions that allow ‘reasonable’ conclusions about the double conditional probability. We compare this with two schemes for the quantum probability model where the double conditional results from sequential updating through projections.

The convergence properties for randomly weighted sums of widely negative dependent random variables under sub-linear expectations with related statistical applications

In this paper, we study the complete convergence and complete moment convergence for randomly weighted sums of arrays of rowwise widely negative dependent random variables in sub-linear expectation space under some appropriate conditions, which extend and improve the corresponding ones in classical probability space to the case of sub-linear expectation space. And we obtain a strong law of large numbers for the randomly weighted sums of arrays of rowwise widely negative dependent random variables. As applications of our main results, we not only present a result on the complete consistency for the weighted estimator in a nonparametric regression model, but also obtain the complete consistency for the least squares estimators in errors-in-variables regression models based on widely negative dependent errors under sub-linear expectations. We perform some numerical simulations to verify the validity of the theoretical results and a real example is analysed for illustration.

Weighted sums and Berry-Esseen type estimates in free probability theory

We study weighted sums of free identically distributed self-adjoint random variables with weights chosen randomly from the unit sphere and show that the Kolmogorov distance between the distribution of such a weighted sum and Wigner’s semicircle law is of order n-12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{-\\frac{1}{2}}$$\\end{document} with high probability. Replacing the Kolmogorov distance by a weaker pseudometric, we obtain a rate of convergence of order n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{-1}$$\\end{document}, thus providing a free analog of the Klartag-Sodin result in classical probability theory. Moreover, we show that our ideas generalize to the setting of sums of free non-identically distributed bounded self-adjoint random variables leading to a new rate of convergence in the free central limit theorem.

Correlation analysis and joint probability density function model of wind pressures: Focusing on multivariate wind loads field on low-rise building under typhoon climate

The characteristics of the multivariate wind loads field on the roof are crucial to the wind-resistant design of low-rise buildings, which contain the correlation characteristics in space and probability characteristics in the time domain. This paper proposes a framework for constructing a Joint Probability Density Function (Joint PDF) model for a multivariate wind loads field. It provides a detailed correlation analysis for the first time. This paper employs wind pressure data collected from the roof of a low-rise building during Typhoon Muifa. It was found that the correlation becomes more robust with increasing roof pitch and the wind pressures are strongly correlated with a correlation coefficient exceeding 0.50 when the roof pitch is above 15°. The mixture distribution model is applied to the probability density function fitting procedure of wind pressure time series under typhoon climate, and the fitting effect is significantly better than other classical probability density functions. The optimal copula function is determined according to the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) for estimating the Joint PDF. The results reveal that Gumbel-copula and Student-copula have the highest proportion in optimal copula functions, accounting for over 90% of the total copula functions. Then, a bivariate Joint PDF for wind pressures are established with the optimal copula function. Additionally, the comparison between measured bivariate Joint PDF and that constructed using copula functions verifies the accuracy of the proposed framework for constructing Joint PDFs. The Joint PDF of wind pressures can enhance the understanding of the stochastic characteristics of local wind load fields on roofs, and the correlation characteristics in space provide crucial references for improving the accuracy of wind load random field simulation and saving the cost of wind resistance design.

On complete convergence for weighted sums of m-widely acceptable random variables under sub-linear expectations and its statistical applications

. In this article, by means of Rosenthal-type inequality, we study the complete convergence and strong law of large numbers for weighted sums of m-widely accpetable random variables under sub-linear expectations, which extend some existing results in classical probability space. As applications, we investigate the complete consistency and strong consistency of weighted estimators in nonparametric regression models under sub-linear expectations. In addition, we provide some simulation studies to verify the finite sample performance of the theoretical results.

Quantum state over time is unique

The conventional framework of quantum theory treats space and time in vastly different ways by representing temporal correlations via quantum channels and spatial correlations via multipartite quantum states—an imbalance absent in classical probability theory. Since Leifer and Spekkens [] called for a causally neutral formulation of quantum theory in their seminal work, numerous attempts have been made to rectify this asymmetry by proposing a dynamical description of a quantum system encapsulated by a static , without a definite consensus on which one is most appropriate. In this paper, we propose sets of operationally motivated axioms for quantum states over time alternative to the ones proposed by Fullwood and Parzygnat [], which we show is unable to induce a unique quantum state over time. Our proposed axioms are better suited to describe quantum states over any spacetime regions beyond two points. Through this reformulation, we prove that the Fullwood-Parzygnat state over time uniquely satisfies all these operational axioms, unifying the bipartite spacetime correlations of quantum systems. Published by the American Physical Society 2024

A novel extended Kumaraswamy distribution and its application to COVID‐19 data

AbstractIn the field of biomedical research, data characteristics often exhibit significant variability, challenging the applicability of classical probability distributions for biomedical data modeling. In this research, we introduce a novel four‐parameter distribution called the odd beta prime Kumaraswamy (OBPK) distribution, derived from the odd beta prime generalized family of distributions and the Kumaraswamy distribution. This distribution exhibits greater flexibility compared with the traditional Kumaraswamy distribution. Importantly, the OBPK distribution offers symmetrical, right‐skewed, left‐skewed, bathtub, N‐shaped, J‐shaped, and reversed J‐shaped densities, as well as varied hazard functions, including increasing, decreasing, bathtub, increasing‐decreasing, upside‐down bathtub, and N‐shaped forms. These curvature patterns make it particularly useful for modeling biomedical data. We derive some basic properties of the OBPK distribution. Parameters are estimated using the maximum likelihood approach, and the performance of various estimators is assessed via Monte Carlo simulations. Finally, four applications to real data on COVID‐19 mortality rates from different countries are evaluated to demonstrate the importance of the OBPK model over other extended versions of the Kumaraswamy models. The empirical findings suggest that the OBPK model outperforms other extended versions of the Kumaraswamy models in these applications.

Uncertainty measures: A critical survey

Classical probability is not the only mathematical theory of uncertainty, or the most general. Many authors have argued that probability theory is ill-equipped to model the ‘epistemic’, reducible uncertainty about the process generating the data. To address this, many alternative theories of uncertainty have been formulated. In this paper, we highlight how uncertainty theories can be seen as forming clusters characterised by a shared rationale, are connected to each other in an intricate but interesting way, and can be ranked according to their degree of generality. Our objective is to propose a structured, critical summary of the research landscape in uncertainty theory, and discuss its potential for wider adoption in artificial intelligence.

Resource theory of Kirkwood-Dirac imaginarity

As an extension of classical probability distribution, the Kirkwood-Dirac distribution (KDD) was discussed by Kirkwood in 1933 and Dirac 1945, independently. Recently, it has been proved that nonclassical values (negative and non-real values) of the KDD have the ability of outperforming their classical counterparts in quantum computation, quantum measurement and so on. In this work, by dividing quantum states into KD-real (KD-free) and KD-imaginary (KD-resource) ones based on the KDD of a state, we establish a resource theory for KD-imaginarity with respect to a pair of bases (A, B), called the resource theory of Kirkwood-Dirac imaginarity. This theory is different from the resource theory of imaginarity of quantum states with respect to one basis A, where the free states are those that have real density matrices under the basis A.

A class of sequential multi-hypothesis tests

In this article, we deal with sequential testing of multiple hypotheses. In the general scheme of construction of optimal tests based on the backward induction, we propose a modification which provides a simplified (generally speaking, and suboptimal) version of the optimal test, for any particular criterion of optimization. We call this DBC version (the one with Dropped Backward Control) of the optimal test. In particular, for the case of two simple hypotheses, dropping backward control in the Bayesian test produces the classical sequential probability ratio test (SPRT). Similarly, dropping backward control in the modified Kiefer-Weiss solutions produces Lorden’s 2-SPRTs. In the case of more than two hypotheses, we obtain in this way new classes of sequential multi-hypothesis tests, and investigate their properties. The efficiency of the DBC-tests is evaluated with respect to the optimal Bayesian multi-hypothesis test and with respect to the matrix sequential probability ratio test (MSPRT) by Armitage. In a multihypothesis variant of the Kiefer-Weiss problem for binomial proportions the performance of the DBC-test is numerically compared with that of the exact solution. In a model of normal observations with a linear trend, the performance of the DBC-test is numerically compared with that of the MSPRT. Some other numerical examples are presented. In all the cases the proposed tests exhibit a very high efficiency with respect to the optimal tests (more than 99.3% when sampling from Bernoulli populations) and/or with respect to the MSPRT (even outperforming the latter in some scenarios).

Lazy Multi-Label Classification algorithms based on Non-Parametric Predictive Inference

Multi-Label Classification (MLC) extends standard classification in the sense that an instance might belong to multiple labels simultaneously. Many lazy approaches to MLC have been proposed so far. The majority of them, to classify an instance, use statistical estimators from the neighboring instances based on classical probability theory. In this work, we propose lazy algorithms for MLC that employ the Non-Parametric Predictive Inference Model (NPI-M) for the statistical estimators based on the neighboring instances. It is shown that our proposed lazy MLC algorithms are more suitable to tackle the class-imbalance problem that usually arises in MLC, especially when data contain label noise. This issue is corroborated via an exhaustive experimental analysis.

Studies on the Marchenko–Pastur Law

In free probability, the theory of Cauchy–Stieltjes Kernel (CSK) families has recently been introduced. This theory is about a set of probability measures defined using the Cauchy kernel similarly to natural exponential families in classical probability that are defined by means of the exponential kernel. Within the context of CSK families, this article presents certain features of the Marchenko–Pastur law based on the Fermi convolution and the t-deformed free convolution. The Marchenko–Pastur law holds significant theoretical and practical implications in various fields, particularly in the analysis of random matrices and their applications in statistics, signal processing, and machine learning. In the specific context of CSK families, our study of the Marchenko–Pastur law is summarized as follows: Let K+(μ)={Qmμ(dx);m∈(m0μ,m+μ)} be the CSK family generated by a non-degenerate probability measure μ with support bounded from above. Denote by Qmμ•s the Fermi convolution power of order s>0 of the measure Qmμ. We prove that if Qmμ•s∈K+(μ), then μ is of the Marchenko–Pastur type law. The same result is obtained if we replace the Fermi convolution • with the t-deformed free convolution t.

Characterizing the geometry of the Kirkwood–Dirac-positive states

The Kirkwood–Dirac (KD) quasiprobability distribution can describe any quantum state with respect to the eigenbases of two observables A and B. KD distributions behave similarly to classical joint probability distributions but can assume negative and nonreal values. In recent years, KD distributions have proven instrumental in mapping out nonclassical phenomena and quantum advantages. These quantum features have been connected to nonpositive entries of KD distributions. Consequently, it is important to understand the geometry of the KD-positive and -nonpositive states. Until now, there has been no thorough analysis of the KD positivity of mixed states. Here, we investigate the dependence of the full convex set of states with positive KD distributions on the eigenbases of A and B and on the dimension d of the Hilbert space. In particular, we identify three regimes where convex combinations of the eigenprojectors of A and B constitute the only KD-positive states: (i) any system in dimension 2; (ii) an open and dense probability one set of bases in dimension d = 3; and (iii) the discrete-Fourier-transform bases in prime dimension. Finally, we show that, if for example d = 2m, there exist, for suitable choices of A and B, mixed KD-positive states that cannot be written as convex combinations of pure KD-positive states. We further explicitly construct such states for a spin-1 system.

Vývoj a interpretace pojmu pravděpodobnost

Considerations related to randomness appear relatively late in European history, at the turn of the Middle Ages and the modern era. They initially concern to the chances of winning in various games or situations and later move on to introduce classical and geometric probabilities. From a mathematical point of view, the probability calculus is completed by Kolmogorov’s axiomatic theory. However, many open questions, problems and paradoxes remain in the way probability is perceived and interpreted. The four main directions in the concept of probability (logical, frequentist, subjective and propensity) are closely related to the way of perceiving randomness (epistemological or ontological). Reflections on the evolution of the perception of randomness and the interpretation of probability bring the ability to navigate the basic principles and findings of science and contributes to a deeper understanding of the entire issue.

Complete convergence theorems for weighted sums of extended negatively dependent random variables under sub-linear expectations

. This article aims to study and establish the Lai law for the weighted sum of extended negatively dependent random variables under sub-linear expectations. Using inequalities under sub-linear expectation spaces, we extend the complete convergence of weighted sums from classical probability space to sub-linear expectation space.

On ZLindley distribution: statistical properties and applications

The modeling and analysis of lifetime data is critical in many areas, including actuarial science, management, engineering, medicin, physics, biology, hydrology, and computer science. Classical probability distributions have been used to manage data sets of varied sizes. However, considerable issues occur when real-world data does not fit into any of the classical or conventional probability models. Consequently, there arises a crucial requirement to enhance the flexibility of existing probability models by including the blending of two distributions or introducing additional parameters. This work introduces a ZLindley distribution with a single parameter. Both symmetric and left-skewed data can utilize the proposed model. We generate statistical features such as the mode, moments, quantile function, and moment-generating function to accurately represent the usefulness of the suggested distribution. We compute the parameter using the maximum likelihood estimation approach. A thorough simulation exercise evaluates the goodness-of-fit performance. We demonstrate the applicability and flexibility of the new recommended distribution by using a real dataset.

Effective Action in Free Probability

Recent works have explored relations between classical and quantum statistical physics on the one hand and Voiculescu's theory of free probability on the other. Motivated by these results, the present work focuses on the notion of effective action, which is closely related to the large deviation rate function in classical probability and one-particle irreducible correlation functions in quantum field theories. The central aim is to understand how it can be defined and studied in free probability. In this respect, we introduce a suitable diagrammatic formalism.

ON THE MOMENTS OF THE 3-PARAMETER GOMPERTZ DISTRIBUTION

Gompertz distribution is a classical probability distribution extensively used in actuarial science, reliability, and survival analysis. Gompertz distribution also plays a role in various fields, such as biology, economics, and marketing analysis. Some extensions of this distribution have been studied and applied to various problems. In this article, we are concerned with some statistical properties of a 3-parameter Gompertz distribution. This extension of the Gompertz distribution introduced has been used in studying competing risk survival analysis. Our main results are the derivation of moments of this distribution and other statistical properties related to moments, such as moment generating function, mean residual life function, mean inactivity time and Lorenz curve. These results will serve as a complement to the theoretical aspect of the analysis of the distribution.

Adulteration detection of edible oil by one‐class classification and outlier detection

AbstractEdible oil adulteration is a mostly practiced phenomenon. However, the traditional discriminant methods fail to detect oil adulteration involving more than one adulterant. Recently, one‐class classifiers were built for food or oil authentication. Unfortunately, as it is hard to determine the application domain of the one‐class classifier, high prediction error was obtained for real samples in market surveillance. In this study, a new method was developed based on one‐class classification and outlier detection for edible oil adulteration detection in market surveillance. The model population was constructed using Monte Carlo sampling of unidentified inspected samples to select the plateau region exhibiting the highest accumulated absolute centered residual (ACR) values. Subsequently, the number of models in the plateau region was validated by the theoretical ones calculated by the classical probability model. The models in the plateau region with the highest cumulative accumulated ACR values were used to identify adulterated oils. Furthermore, the cross‐validation was conducted by comparing identification results from two different Monte Carlo sampling ratios to ensure the accuracy of our method. Both single adulteration and multiple adulteration of peanut oils were prepared to validate our method. Moreover, this method was used to detect adulteration of sesame oils, which have already been identified by the markers in our previous study. The validation results of three datasets indicated that this method could effectively identify adulterated samples and therefore provide a novel solution for inspecting potential adulteration in practice.