Consistency Theorem Research Articles

Tail index estimation for tail adversarial stable time series with an application to high‐dimensional tail clustering

For stationary time series with regularly varying marginal distributions, an important problem is to estimate the associated tail index which characterizes the power‐law behavior of the tail distribution. For this, various results have been developed for independent data and certain types of dependent data. In this article, we consider the problem of tail index estimation under a recently proposed notion of serial tail dependence called the tail adversarial stability. Using the technique of adversarial innovation coupling and a martingale approximation scheme, we establish the consistency and central limit theorem of the tail index estimator for a general class of tail dependent time series. Based on the asymptotic normal distribution from the obtained central limit theorem, we further consider an application to cluster a large number of regularly varying time series based on their tail indices by using a robust mixture algorithm. The results are illustrated using numerical examples including Monte Carlo simulations and a real data analysis.

A partitioning Monte Carlo approach for consensus tasks in crowdsourcing

The planner’s role in crowdsourcing involves determining the time to stop collecting information (i.e., timed decision, TD). Previous studies have modeled the uncertain crowdsourcing environment as Partially Observable Markov Decision Processes (POMDPs) and utilized the value of information (VOI) to balance the utilities and costs of information collection. However, there is still room for optimization in solving the TD problem within single-agent POMDPs. In this paper, we propose a partitioning Monte Carlo approach for consensus tasks based on the option-candidate (OC) model partitioning. We simplify the state representation for the OC model and introduce a multi-agent POMDP representation. We establish a correspondence between the single-agent and multi-agent models using two consistency theorems. We propose three progressively improved partitioning Monte Carlo (PMC) algorithms to solve the TD problem within the multi-agent POMDP. We conducted experiments in synthetic domains and a citizen science project, demonstrating that the proposed algorithms exhibit continuous improvements in runtime advantages and consistently outperform the SOTA single-agent Monte Carlo sampling-based algorithm while providing nearly consistent output results.

Applications of fractional difference operators for new version of Brudno-Mazur Orlicz bounded consistency theorem

In this paper, we intend to prove that the modulus $\mathcal{A}-$lacunary statistical convergence of fractional difference double sequences and modulus lacunary fractional matrix of four-dimensions taken over the space of modulus $\mathcal{A}-$lacunary fractional difference uniformly integrable real sequences are equivalent. We represent another version of the Brudno-Mazur Orlicz bounded consistency theorem by using modulus function, lacunary sequence, and fractional difference operator. We show that the four-dimensional $RH-$ regular matrices $\mathcal{A}$ and $\mathcal{B}$ are modulus lacunary fractional difference consistent over the multipliers space of modulus fractional difference $\mathcal{A}-$summable sequences and an algebra $Z.$

Total effects with constrained features

Recent studies have emphasized the connection between machine learning feature importance measures and total order sensitivity indices (total effects, henceforth). Feature correlations and the need to avoid unrestricted permutations make the estimation of these indices challenging. Additionally, there is no established theory or approach for non-Cartesian domains. We propose four alternative strategies for computing total effects that account for both dependent and constrained features. Our first approach involves a generalized winding stairs design combined with the Knothe-Rosenblatt transformation. This approach, while applicable to a wide family of input dependencies, becomes impractical when inputs are physically constrained. Our second approach is a U-statistic that combines the Jansen estimator with a weighting factor. The U-statistic framework allows the derivation of a central limit theorem for this estimator. However, this design is computationally intensive. Then, our third approach uses derangements to significantly reduce computational burden. We prove consistency and central limit theorems for these estimators as well. Our fourth approach is based on a nearest-neighbour intuition and it further reduces computational burden. We test these estimators through a series of increasingly complex computational experiments with features constrained on compact and connected domains (circle, simplex), non-compact and non-connected domains (Sierpinski gaskets), we provide comparisons with machine learning approaches and conclude with an application to a realistic simulator.

A multi-step method to solve bipolar-fuzzy initial value problem

Bipolar-fuzzy differential equations applied for modeling in science, engineering, and social science. The aim of this paper is to provide a numerical method to determine the preferred level of accuracy with less computational cost. The proposed method is generated by combining three-step explicit and two-step implicit bipolar-fuzzy methods, which are obtained by the spline interpolation. For more illustration, the consistency, stability, and convergence theorems are discussed in detail. Finally, for show the performance and efficiency of the method, several numerical examples are presented.

A method for solving the limit of multivariate composite functions

Abstract In this paper, according to the concept of multivariate composite function, the limit and continuity conditions of multivariate composite function are given, and the continuity relationship between the inner function and the outer function of the composite function at the same point is investigated. The relationship between the limit and continuity of a composite function composed of two functions at the same point is investigated by examples, while some limit results are explained from the perspective of composite functions, and the limit calculation law of multivariate composite functions is proposed. Using the consistent central limit theorem in the limit calculation law, the absolute value and limit of the interval of the multivariate composite function are obtained, and the iterative limit of the variational reduction approximation method is confirmed by combining the absolute value and limit of the interval of the multivariate composite function. On the basis of the application of limit operation of a multivariate composite function, the limit distribution of high-dimensional likelihood ratio statistics of multivariate composite function is analyzed by using data simulation analysis. The results show that both limit distributions have better simulation results when p are small, such as p = 5 or p = 10, but the traditional limit is better than the normal limit, indicating that the results of multivariate composite function limit calculations have good accuracy. A method for solving the limits of multivariate composite functions is proposed in this study, which is of great theoretical and practical significance in the study of higher mathematics.

Formalizing the Semantics of a Classical-Quantum Imperative Language in Coq

In order to verify the functional correctness of quantum circuits or algorithms, a prominent approach is to specify them as quantum programs and semi-automatically deduce them in a theorem prover. It is indispensable to first formalize the semantics of the basic quantum language. We formalize in Coq an imperative language which allows for classical and quantum information interactions. We define small-step operational semantics and state-based denotational semantics. Then we prove a consistency theorem between these two semantics. A distribution-based denotational semantics is also defined and related to the state-based one. Finally, we describe a few typical quantum algorithms and utilize the distribution-based denotational semantics to verify their correctness.

Wave equation for Sturm-Liouville operator with singular potentials

The paper is denoted to the initial-boundary value problem for the wave equation with the Sturm-Liouville operator with irregular (distributive) potentials. To obtain a solution to the equation, the separation method and asymptotics of the eigenvalues and eigenfunctions of the Sturm-Liouville operator are used. Homogeneous and inhomogeneous cases of the equation are considered. Next, existence, uniqueness, and consistency theorems for a very weak solution of the wave equation with singular coefficients are proved.

Natural deduction calculi for classical and intuitionistic S5

We propose an indexed natural deduction system for the modal logic , ideally following Wansing's previous work in the context of tableaux sequents. The system, given both in the classical and intuitionistic versions (called and respectively), is designed to match as much as possible the structure and properties of the standard system of natural deduction for first-order logic, exploiting the formal analogy between modalities and quantifiers. We study a (syntactical) normalization theorem for both and and its main consequences, the sub-formula principle and the consistency theorem. In particular, we propose an intuitionistic encoding of classical (via a suitable extension of the Gödel translation for first-order classical logic). Moreover, via the BHK interpretation of intuitionistic proofs, we propose a suitable Curry–Howard isomorphism for . By translation into the natural deduction system given by Galmiche and Salhi in [(2010b). Label-free proof systems for intuitionistic modal logic is5. In E. M. Clarke & A. Voronkov (Eds.), Logic for programming, artificial intelligence, and reasoning (pp. 255–271). Springer Berlin Heidelberg.], we prove the equivalence of w.r.t. an Hilbert-style axiomatization of . However, when considering the sheer provability of labelled formulas, our system is comparable to the one presented by Simpson in [(1993). The proof theory and semantics of intuitionistic modal logic [PhD thesis], University of Edinburgh, UK.]. Nevertheless, it remains uncertain whether it is feasible to establish a translation between the corresponding derivations.

Fixed-Domain Posterior Contraction Rates for Spatial Gaussian Process Model with Nugget

Spatial Gaussian process regression models typically contain finite dimensional covariance parameters that need to be estimated from the data. We study the Bayesian estimation of covariance parameters including the nugget parameter in a general class of stationary covariance functions under fixed-domain asymptotics, which is theoretically challenging due to the increasingly strong dependence among spatial observations. We propose a novel adaptation of the Schwartz’s consistency theorem for showing posterior contraction rates of the covariance parameters including the nugget. We derive a new polynomial evidence lower bound, and propose consistent higher-order quadratic variation estimators that satisfy concentration inequalities with exponentially small tails. Our Bayesian fixed-domain asymptotics theory leads to explicit posterior contraction rates for the microergodic and nugget parameters in the isotropic Matérn covariance function under a general stratified sampling design. We verify our theory and the Bayesian predictive performance in simulation studies and an application to sea surface temperature data. Supplementary materials for this article are available online.

Posterior consistency for the spectral density of non‐Gaussian stationary time series

AbstractVarious nonparametric approaches for Bayesian spectral density estimation of stationary time series have been suggested in the literature, mostly based on the Whittle likelihood approximation. A generalization of this approximation involving a nonparametric correction of a parametric likelihood has been proposed in the literature with a proof of posterior consistency for spectral density estimation in combination with the Bernstein–Dirichlet process prior for Gaussian time series. In this article, we will extend the posterior consistency result to non‐Gaussian time series by employing a general consistency theorem for dependent data and misspecified models. As a special case, posterior consistency for the spectral density under the Whittle likelihood is also extended to non‐Gaussian time series. Small sample properties of this approach are illustrated with several examples of non‐Gaussian time series.

Consistency without compactness of the parameter space in spatial econometrics

When studying the consistency of an estimator without a closed-form solution for a spatial econometric model, we usually assume that the parameter space is compact. However, compactness assumptions are restrictive as we need to know the boundaries of parameter spaces. We establish a consistency theorem for concave objective functions. We apply this result to rebuild the consistency of the quasi maximum likelihood estimator (QMLE) of a spatial autoregressive (SAR) model and a SAR Tobit model. Their log-likelihood functions are not concave, but they can be concave after proper reparameterization as in Olsen (1978).

Finitely additive mixtures of probability measures

Let D be a linear space of real bounded functions and P:D→R a coherent functional. Also, let Q be a collection of coherent functionals on D. Under mild conditions, there is a finitely additive probability Π on the power set of Q such that P(f)=∫QQ(f)Π(dQ) for each f∈D. This fact has various consequences and such consequences are investigated in this paper. Three types of results are provided: (i) Existence of common extensions satisfying certain properties, (ii) Finitely additive mixtures of extreme points, (iii) Countably additive mixtures. Among other things, we obtain new versions of Kolmogorov's consistency theorem and de Finetti's representation theorem.

Checking for model failure and for prior-data conflict with the constrained multinomial model

Multinomial models can be difficult to use when constraints are placed on the probabilities. An exact model checking procedure for such models is developed based on a uniform prior on the full multinomial model. For inference, a nonuniform prior can be used and a consistency theorem is proved concerning a check for prior-data conflict with the chosen prior. Applications are presented and a new elicitation methodology is developed for multinomial models with ordered probabilities.

Консистентнiсть оцiнки найменших квадратiв параметрiв тригонометричної моделi регресiї у присутностi лiнiйного випадкового шуму

Регресiйний аналiз є iстотною частиною математичної та прикладної статистики. Нелiнiйний регресiйний аналiз є значним розширенням та ускладненням класичного лiнiйного регресiйного аналiзу, завдяки використанню нелiнiйних або частково нелiнiйних за параметрами моделей, якi адекватнiше описують, нiж лiнiйнi моделi, явища, що потребують статистичного аналiзу. Велика кiлькiсть прикладних проблем у численних наукових, технiчних та гуманiтарних галузях знань дають поштовх розвитку нелiнiйного регресiйного аналiзу. Задача оцiнювання векторного параметра сигналу в моделях спостереження «сигнал + шум» є добре вiдомою проблемою статистики випадкових процесiв, та у випадку нелiнiйного параметра сигналу – задачею нелiнiйного регресiйного аналiзу. Серед рiзноманiтностi задач нелiнiйного регресiйного аналiзу оцiнювання амплiтуд та кутових частот суми гармонiчних коливань, що спостерiгається на фонi випадкового шуму, займає значне мiсце, завдяки її численним застосуванням. Статистичнi моделi такого типу називаються тригонометричними моделями регресiї, а проблема статистичного оцiнювання її параметрiв називається задачею виявлення прихованих перiодичностей. Статтю присвячено вивченню тригонометричної моделi регресiї, в якiй випадковий шум є лiнiйним Левi-керованим стацiонарним четвертого порядку випадковим процесом iз нульовим середнiм, iнтегровную та iнтегровную з квадратом iмпульсною перехiдною функцiєю. Це припущення призводить до iнтегровностi коварiацiйної функцiї та кумулянтної функцiї 4-го порядку. Для оцiнювання амплiтуд та кутових частот такої тригонометричної моделi ми використовуємо оцiнки найменших квадратiв у сенсi Уолкера, тобто розглянуто спецiальну множину параметрiв, щоб розрiзнити належним чином рiзнi кутовi частоти в сумi гармонiчних коливань. У статтi доведено теорему про сильну консистентнiсть оцiнки найменших квадратiв за описаними вище припущеннями щодо випадкового шуму. Для отримання такого результату було доведено дуже важливу лему про рiвномiрну збiжнiсть майже напевно середнього значення перетворення Фурьє лiнiйного Левi-керованого випадкового процеса. Ця лема є головним iнструментом доведення теореми про сильну консистентнiсть. Для доведення теореми, по-перше, знаходимо деякi представлення оцiнок найменших квадратiв амплiтуд через вiдповiднi оцiнки кутових частот. По-друге, ми пiдставляємо цi формули у функцiонал методу найменших квадратiв. Останнiй крок доведення полягає у перетвореннi L2-норми рiзницi мiж емпiричною тригонометричною функцiєю регресiї та iстиною функцiєю регресiї таким чином, що ця норма прямує до нуля майже напевно тодi i тiльки тодi, коли оцiнки є сильно консистентними.

A minimal contrast estimator for the linear fractional stable motion

In this paper we present an estimator for the three-dimensional parameter $$(\sigma , \alpha , H)$$ of the linear fractional stable motion, where H represents the self-similarity parameter, and $$(\sigma , \alpha )$$ are the scaling and stability parameters of the driving symmetric Levy process L. Our approach is based upon a minimal contrast method associated with the empirical characteristic function combined with a ratio type estimator for the self-similarity parameter H. The main result investigates the strong consistency and weak limit theorems for the resulting estimator. Furthermore, we propose several ideas to obtain feasible confidence regions in various parameter settings. Our work is mainly related to Ljungdahl and Podolskij (A note on parametric estimation of Levy moving average processes, p 294, 2019) and Mazur et al. (Bernoulli 26(1): 226–252, 2020) in which parameter estimation for the linear fractional stable motion and related Levy moving average processes has been studied.

Application of consistent geometric decomposition theorem to dynamic finite element of 3D composite beam based on experimental and numerical analyses

Analyzing static and dynamic problems including composite structures has been of high significance in research efforts and industrial applications. In this article, equivalent single layer approach is utilized for dynamic finite element procedures of 3D composite beam as the building block of numerous composite structures. In this model, both displacement and strain fields are decomposed into cross-sectional and longitudinal components, called consistent geometric decomposition theorem. Then, the model is discretized using finite element procedures. Two local coordinate systems and a global one are defined to decouple mechanical degrees of freedom. Furthermore, from the viewpoint of consistent geometric decomposition theorem, the transformation and element mass matrices for those systems are introduced here for the first time. The same decomposition idea can be used for developing element stiffness matrix. Finally, comprehensive validations are conducted for the theory against experimental and numerical results in two case studies and for various conditions.

Spot estimation for fractional Ornstein–Uhlenbeck stochastic volatility model: consistency and central limit theorem

There has been an increasing interest for rough stochastic volatility models. However, little is known about the statistical inference for such models, especially for high frequency data. This paper investigates estimation of the fractional spot volatility from discrete observations of the price process on a grid with a time interval $$\Delta _n\rightarrow 0$$ as $$n\rightarrow \infty $$ . Namely, the model with fractional Ornstein–Uhlenbeck log-volatility and Ito-semimartingale log-price processes is considered. In this setup both consistency and central limit theorem are proven for truncated and non-truncated spot volatility estimators. Then, asymptotic confidence intervals are derived for a finite number of spot volatility estimators at different estimation times. Consequently, the highest possible rate of convergence achieved in the central limit theorem $$\Delta _n^{H/(2H+1)}$$ is a function of the Hurst parameter H of the fractional Brownian motion driving the volatility. This rate coincides with the already known highest convergence rate for the Brownian case when $$H=0.5$$ . Furthermore, simulations in this paper validate the consistency and central limit theorem numerically. Article class.

A consistency theorem for randomized singular value decomposition

The singular value decomposition (SVD) and the principal component analysis are fundamental tools and probably the most popular methods for data dimension reduction. The rapid growth in the size of data matrices has lead to a need for developing efficient large-scale SVD algorithms. Randomized SVD was proposed, and its potential was demonstrated for computing a low-rank SVD (Rokhlin et al., 2009). In this article, we introduce a consistency notion for random projections used in randomized SVD and provide a consistency theorem for it. We also present a numerical example to show how the random projections to low dimension affect the consistency.

Interval multiplicative pairwise comparison matrix: Consistency, indeterminacy and normality

To manifest human judgments, a long-established method called Pairwise Comparison (PC) has been successfully applied in the Analytic Hierarchy Process (AHP). In practice, human judgments are often made with uncertainty, and can be characterized by an Interval Multiplicative Pairwise Comparison Matrix (IMPCM). Since consistency is a key issue that has plagued decision makers and researchers for a long time, it is useful to propose a transformation that can effectively convert an inconsistent IMPCM into a consistent one, especially in group decision-making. However, a consistent IMPCM is not sufficient to be acceptable, indeterminacy should also be considered. Moreover, the interval priority weights should be normalized. To consider consistency, indeterminacy, and normality simultaneously, we put forward a new definition of acceptable IMPCM. To obtain such an acceptable IMPCM, we propose a theorem of consistency, a consistent transformation, and a normalized prioritization scheme. As a result, the proposed methods guarantee an inconsistent IMPCM can be directly converted into an acceptable IMPCM. Five theorems are proved to corroborate the proposed methods. A numerical example is presented to illustrate the validity and superiority of the proposed methods. Finally, discussion and conclusions are given.