Fractional Hermite-Hadamard-type inequalities represent a significant area of study in convex analysis due to their extensive applications in mathematical and applied sciences. These inequalities provide powerful tools for estimating the integral mean of a convex function in terms of its values at the endpoints of a given interval. In this paper, we focus on the development and refinement of fractional Hermite-Hadamardtype inequalities for the class of twice differentiable m-convex functions. Utilizing advanced analytical techniques, such as Ho¨lder’s inequality and the power mean integral inequality, we derive new bounds that generalize existing results in the literature. These findings not only extend classical inequalities to a broader class of convex functions but also provide sharper and more versatile estimations. The presented results are expected to have significant implications in various mathematical domains, including fractional calculus, optimization, and mathematical modeling. This work contributes to the ongoing efforts to generalize and refine integral inequalities by incorporating fractional operators and broader convexity assumptions, offering a deeper understanding of the behavior of m-convex functions under fractional integration.
Higher-order $U$-statistics abound in fields such as statistics, machine learning, and computer science, but are known to be highly time-consuming to compute in practice. Despite their widespread appearance, a comprehensive study of their computational complexity is surprisingly lacking. This paper aims to fill this gap by presenting several results related to the computational aspect of $U$-statistics. First, we derive a useful decomposition from a $m$-th order $U$-statistic to a linear combination of $V$-statistics with orders not exceeding $m$, which are generally more feasible to compute. Second, we explore the connection between exactly computing $V$-statistics and Einstein summation, a tool often used in computational mathematics and quantum computing to accelerate tensor computations. Third, we provide an optimistic estimate of the time complexity for exactly computing $U$-statistics, based on the treewidth of a particular graph associated with the $U$-statistic kernel. The above ingredients lead to (1) a new, much more runtime-efficient algorithm to exactly compute general higher-order $U$-statistics, and (2) a more streamlined characterization of runtime complexity of computing $U$-statistics. We develop an accompanying open-source package called \texttt{u-stats} in both Python (https://github.com/zrq1706/U-Statistics-Python) and R (https://github.com/cxy0714/U-Statistics-R). We demonstrate through three examples in statistics that \texttt{u-stats} achieves impressive runtime performance compared to existing benchmarks. This paper also aspires to achieve two goals: (1) to capture the interest of researchers in both statistics and other related areas to further advance the algorithmic development of $U$-statistics and (2) to lift the burden of implementing higher-order $U$-statistics from practitioners.
Food commodities and energy bills have experienced rapid undulating movements and hikes globally in recent times. This spurred this study to examine the possibility that the shocks that arise from fluctuations of one market spill over to the other and to determine how time-varying the spillovers were across a time. Data were daily frequency (prices of grains and energy products) from 1 July 2019 to 31 December 2022, as quoted in markets. The choice of the period was to capture the COVID pandemic and the Russian–Ukrainian war as events that could impact volatility. The returns were duly calculated using spreadsheets and subjected to ADF stationarity, co-integration, and the full BEKK-GARCH estimation. The results revealed a prolonged association between returns in the energy markets and food commodity market returns. Both markets were found to have volatility persistence individually, and time-varying bidirectional transmission of volatility across the markets was found. No lagged-effects spillover was found from one market to the other. The findings confirm that shocks that emanate from fluctuations in energy markets are impactful on the volatility of prices in food commodity markets and vice versa, but this impact occurs immediately after the shocks arise or on the same day such variation occurs.
Yulia Alexandr, Miles Bakenhus, Mark Curiel
et al.
In the last quarter of a century, algebraic statistics has established itself as an expanding field which uses multilinear algebra, commutative algebra, computational algebra, geometry, and combinatorics to tackle problems in mathematical statistics. These developments have found applications in a growing number of areas, including biology, neuroscience, economics, and social sciences. Naturally, new connections continue to be made with other areas of mathematics and statistics. This paper outlines three such connections: to statistical models used in educational testing, to a classification problem for a family of nonparametric regression models, and to phase transition phenomena under uniform sampling of contingency tables. We illustrate the motivating problems, each of which is for algebraic statistics a new direction, and demonstrate an enhancement of related methodologies.
This paper considers a family of linear two-point boundary value problems for systems of ordinary differential equations. The questions of existence of its solutions are investigated and methods of finding approximate solutions are proposed. Sufficient conditions for the existence of a family of linear two-point boundary value problems for systems of ordinary differential equations are established. The uniqueness of the solution of the problem under consideration is proved. Algorithms for finding an approximate solution based on modified of the algorithms of the D.S. Dzhumabaev parameterization method are proposed and their convergence is proved. According to the scheme of the parameterization method, the problem is transformed into an equivalent family of multipoint boundary value problems for systems of differential equations. By introducing new unknown functions we reduce the problem under study to an equivalent problem, a Volterra integral equation of the second kind. Sufficient conditions of feasibility and convergence of the proposed algorithm are established, which also ensure the existence of a unique solution of the family of boundary value problems with parameters. Necessary and sufficient conditions for the well-posedness of the family of linear boundary value problems for the system of ordinary differential equations are obtained.
This paper addresses the solution of a differential-difference type equation having an interior layer behaviour. A difference scheme is suggested to solve this equation using a non-standard finite difference method. Finite differences are derived from the first and second order derivatives. Using these approximations, the given equation is discretized. The discretized equation is solved using the algorithm for the tridiagonal system. The method is examined for convergence. Numerical examples are illustrated to validate the method. Maximum errors in the solution, in contrast to the other methods are organized to justify the method. The layer behaviour in the solution of the examples is depicted in graphs.
Linkage disequilibrium score regression (LDSC) has emerged as an essential tool for genetic and genomic analyses of complex traits, utilizing high-dimensional data derived from genome-wide association studies (GWAS). LDSC computes the linkage disequilibrium (LD) scores using an external reference panel, and integrates the LD scores with only summary data from the original GWAS. In this paper, we investigate LDSC within a fixed-effect data integration framework, underscoring its ability to merge multi-source GWAS data and reference panels. In particular, we take account of the genome-wide dependence among the high-dimensional GWAS summary statistics, along with the block-diagonal dependence pattern in estimated LD scores. Our analysis uncovers several key factors of both the original GWAS and reference panel datasets that determine the performance of LDSC. We show that it is relatively feasible for LDSC-based estimators to achieve asymptotic normality when applied to genome-wide genetic variants (e.g., in genetic variance and covariance estimation), whereas it becomes considerably challenging when we focus on a much smaller subset of genetic variants (e.g., in partitioned heritability analysis). Moreover, by modeling the disparities in LD patterns across different populations, we unveil that LDSC can be expanded to conduct cross-ancestry analyses using data from distinct global populations (such as European and Asian). We validate our theoretical findings through extensive numerical evaluations using real genetic data from the UK Biobank study.
Based on the asymmetric copula function, this paper analyzes the static and dynamic correlation between Shanghai Composite Index and Shenzhen Composite Index. Through the static analysis, it is found that the asymmetric copula function is better than Gumbel Copula in describing the distribution characteristics of the top tail dependence between the Shanghai Composite Index and the Shenzhen Composite Index, and the copula correlation coefficient definition based on the asymmetric copula function can well describe the asymmetric dependence between variables. In the time-varying analysis, the paper improves the traditional dynamic evolution equation of the tail-dependence coefficient. Through empirical analysis, the result shows that the improved dynamic evolution equation can better reflect the dynamic evolution process of the tail-dependence coefficient.
Yeong Kin Teoh, Nur Fatihah Hamdan, Suzanawati Abu Hasan
et al.
In late 2019, the unique severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), also known as COVID-19, first emerged in Wuhan City, Hubei Province, China and quickly spread throughout the world. Until June 30, 2022, a total of 4,566,055 cases of COVID-19 have been reported in Malaysia, with 35,765 deaths and 4,500,856 recovered cases. This study aims to generalise a deterministic SIR model with vital dynamics for understanding the proliferation of infectious diseases. The SIR model with vital dynamics is more realistic in mimicking reality than the basic SIR model because it can determine the dynamic behaviours of COVID-19 over a more extended period. The SIR model utilises vital dynamics with unequal birth and death rates. Furthermore, the SIR model with vital dynamics is rescaled with the total time-varying population and analysed according to its epidemic condition. The results indicated that the number of infected individuals would peak about 10 - 15 days and reach their steady state about 25 - 60 days. The findings of this research may help policymakers establish, plan, and implement effective COVID-19 pandemic response strategies.
M.T. Jenaliyev, A.S. Kassymbekova, M.G. Yergaliyev
et al.
This paper considers an initial boundary value problem for a one-dimensional Boussinesq-type equation in a domain, that is, a trapezoid. Using the methods of the theory of monotone operators, we establish theorems on their unique weak solvability in Sobolev classes.
The study of online decision-making problems that leverage contextual information has drawn notable attention due to their significant applications in fields ranging from healthcare to autonomous systems. In modern applications, contextual information can be rich and is often represented as a matrix. Moreover, while existing online decision algorithms mainly focus on reward maximization, less attention has been devoted to statistical inference. To address these gaps, in this work, we consider an online decision-making problem with a matrix context where the true model parameters have a low-rank structure. We propose a fully online procedure to conduct statistical inference with adaptively collected data. The low-rank structure of the model parameter and the adaptive nature of the data collection process make this difficult: standard low-rank estimators are biased and cannot be obtained in a sequential manner while existing inference approaches in sequential decision-making algorithms fail to account for the low-rankness and are also biased. To overcome these challenges, we introduce a new online debiasing procedure to simultaneously handle both sources of bias. Our inference framework encompasses both parameter inference and optimal policy value inference. In theory, we establish the asymptotic normality of the proposed online debiased estimators and prove the validity of the constructed confidence intervals for both inference tasks. Our inference results are built upon a newly developed low-rank stochastic gradient descent estimator and its convergence result, which are also of independent interest.
In this paper γµ -open sets and γµ -closed sets in a GTS ( X,µ ) have been studied, where γµ is an operation from µ to P( X ). In general, collection of γµ -open sets is smaller than the collection of µ -open sets. The condition under which both are same are also established here. Some properties of such sets have been discussed. Some closure like operators are also defined and their properties are discussed. The relation between similar types of closure operators on the GTS ( X,µ ) has been established. The condition under which the newly defined closure like operator is a Kuratowski closure operator is given. We have also defined a generalized type of closed sets termed as γµ -generalized closed set with the help of this newly defined closure operator and discussed some basic properties of such sets. As an application, we have introduced some weak separation axioms and discussed some of their properties. Finally, we have shown some preservation theorems of such generalized concepts.
Nyström approximation is a fast randomized method that rapidly solves kernel ridge regression (KRR) problems through sub-sampling the n-by-n empirical kernel matrix appearing in the objective function. However, the performance of such a sub-sampling method heavily relies on correctly estimating the statistical leverage scores for forming the sampling distribution, which can be as costly as solving the original KRR. In this work, we propose a linear time (modulo poly-log terms) algorithm to accurately approximate the statistical leverage scores in the stationary-kernel-based KRR with theoretical guarantees. Particularly, by analyzing the first-order condition of the KRR objective, we derive an analytic formula, which depends on both the input distribution and the spectral density of stationary kernels, for capturing the non-uniformity of the statistical leverage scores. Numerical experiments demonstrate that with the same prediction accuracy our method is orders of magnitude more efficient than existing methods in selecting the representative sub-samples in the Nyström approximation.
Bib Paruhum Silalahi, Silviana Eka Pertiwi, Hidayatul Mayyani
et al.
Marketing management is an activity to plan and organize marketing activities in order to achieve organizational or company goals efficiently and effectively. Problems arise when there are several or many different projects that can be implemented as company marketing projects. These projects are usually categorized by several objectives. These goals can be complementary or contradictory. In operation, decision-makers are required to choose and determine the right project to achieve the target. In this paper, we discuss a programming model using the zero-one goal programming approach, a selection of marketing projects to meet many objectives and constraints, and then give examples of its implementation. Discussion and implementation include goal programming categories: nonpreemptive goal programming and preemptive goal programming
While computing has become an important part of the statistics field, course offerings are still influenced by a legacy of mathematically centric thinking. Due to this legacy, Bayesian ideas are not required for undergraduate degrees and have largely been taught at the graduate level; however, with recent advances in software and emphasis on computational thinking, Bayesian ideas are more accessible. Statistics curricula need to continue to evolve and students at all levels should be taught Bayesian thinking. This article advocates for adding Bayesian ideas for three groups of students: intro-statistics students, undergraduate statistics majors, and graduate student scientists; and furthermore, provides guidance and materials for creating Bayesian-themed courses for these audiences. Supplementary files for this article are available on line.
Special aspects of education, Probabilities. Mathematical statistics
Within the field of causal inference, it is desirable to learn the structure of causal relationships holding between a system of variables from the correlations that these variables exhibit; a sub-problem of which is to certify whether or not a given causal hypothesis is compatible with the observed correlations. A particularly challenging setting for assessing causal compatibility is in the presence of partial information; i.e. when some of the variables are hidden/latent. This paper introduces the possible worlds framework as a method for deciding causal compatibility in this difficult setting. We define a graphical object called a possible worlds diagram, which compactly depicts the set of all possible observations. From this construction, we demonstrate explicitly, using several examples, how to prove causal incompatibility. In fact, we use these constructions to prove causal incompatibility where no other techniques have been able to. Moreover, we prove that the possible worlds framework can be adapted to provide a complete solution to the possibilistic causal compatibility problem. Even more, we also discuss how to exploit graphical symmetries and cross-world consistency constraints in order to implement a hierarchy of necessary compatibility tests that we prove converges to sufficiency.
A solution manifold is the collection of points in a $d$-dimensional space satisfying a system of $s$ equations with $s<d$. Solution manifolds occur in several statistical problems including hypothesis testing, curved-exponential families, constrained mixture models, partial identifications, and nonparametric set estimation. We analyze solution manifolds both theoretically and algorithmically. In terms of theory, we derive five useful results: the smoothness theorem, the stability theorem (which implies the consistency of a plug-in estimator), the convergence of a gradient flow, the local center manifold theorem and the convergence of the gradient descent algorithm. To numerically approximate a solution manifold, we propose a Monte Carlo gradient descent algorithm. In the case of likelihood inference, we design a manifold constraint maximization procedure to find the maximum likelihood estimator on the manifold. We also develop a method to approximate a posterior distribution defined on a solution manifold.
We consider a sequence of independent random variables X1,X2,…,Xm,…,Xnn≥3 exhibiting a change in the probability distribution of the data generating mechanism. We suppose that the distribution changes at some point, called a change point, to a second distribution for the remaining observations. We propose Bayes estimators of change point under symmetric loss functions and asymmetric loss functions. The sensitivity analysis of Bayes estimators are carried out by simulation and numerical comparisons with R-programming.
For $D$ a bounded domain in $\mathbb R^d, d \ge 2,$ with smooth boundary $\partial D$, the non-linear inverse problem of recovering the unknown conductivity $γ$ determining solutions $u=u_{γ, f}$ of the partial differential equation \begin{equation*} \begin{split} \nabla \cdot(γ\nabla u)&=0 \quad \text{ in }D, \\ u&=f \quad \text { on } \partial D, \end{split} \end{equation*} from noisy observations $Y$ of the Dirichlet-to-Neumann map \[f \mapsto Λ_γ(f) = {γ\frac{\partial u_{γ,f}}{\partial ν}}\Big|_{\partial D},\] with $\partial/\partial ν$ denoting the outward normal derivative, is considered. The data $Y$ consists of $Λ_γ$ corrupted by additive Gaussian noise at noise level $\varepsilon>0$, and a statistical algorithm $\hat γ(Y)$ is constructed which is shown to recover $γ$ in supremum-norm loss at a statistical convergence rate of the order $\log(1/\varepsilon)^{-δ}$ as $\varepsilon \to 0$. It is further shown that this convergence rate is optimal, up to the precise value of the exponent $δ>0$, in an information theoretic sense. The estimator $\hat γ(Y)$ has a Bayesian interpretation in terms of the posterior mean of a suitable Gaussian process prior and can be computed by MCMC methods.