Asymptotic Statistics - Science topic

A machine learning tasks from observations must encounter and process uncertainty and novelty, especially when it is expected to maintain performance when observing new information and to choose the best fitting hypothesis to the currently observed information. In this context, some key questions arise: what is information, how much information did...

This article proposes a new method of truncated estimation to estimate the tail index \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha (0<\alpha \le 2)$$\end{docu...

Non-Abelian evolution is a landmark in modern theoretical physics. But if non-commutative dynamics has a significant impact in the control of entanglement and transport in quantum systems is an open question. Here we propose to utilize non-Abelian Thouless pumping in one-dimensional discrete-time quantum walks in lattices with degenerate Bloch-band...

In this paper, we propose the concepts of asymptotic equivalence , asymptotic statistical equivalence, lacunary statistical equivalence of order (α, β) in sense of Wijsman. We also make an effort to define these concepts by using modulus function with respect to ideal I and examine some algebraic and topological properties related to these concepts...

Phase reduction is an important tool for studying coupled and driven oscillators. The question of how to generalize phase reduction to stochastic oscillators remains actively debated. In this work, we propose a method to derive a self-contained stochastic phase equation of the form dϕ = a(ϕ) dt + sqrt(2D(ϕ)) dW (t) that is valid not only for noise-...

In ophthalmology and otolaryngology, data collected from paired body parts are typically reformatted into categorical bilateral data structures for subsequent research. This article applies Donner’s equal correlation coefficient model and obtains nine simultaneous confidence intervals (SCI) of proportion ratios under three asymptotic statistical me...

The Hardy–Weinberg equilibrium (HWE) assumption is essential to many population genetics models. Multiple tests were developed to test its applicability in observed genotypes. Current methods are divided into exact tests applicable to small populations and a small number of alleles, and approximate goodness-of-fit tests. Existing tests cannot handl...

The present study deals with asymptotically equivalent sequences in partial metric spaces. We define the notions of strongly asymptotically lacunary equivalence, asymptotically statistical equivalence, and asymptotically lacunary statistical equivalence. We theoretically contribute to these notions and investigate some of their basic properties.

The Hardy-Weinberg Equilibrium (HWE) assumption is essential to many population genetics models. Multiple tests were developed to test its applicability in observed genotypes. Current methods are divided into exact tests applicable to small populations and a small number of alleles, and approximate goodness of fit tests. Existing tests cannot handl...

It is commonly necessary to perform inferences on the difference, ratio, and odds ratio of two proportions p1 and p2 based on two independent samples. For this purpose, the most common asymptotic statistics are based on the score statistics (S-type statistics). As these do not correct the bias of the estimator of the product pi (1-pi), Miettinen an...

The multivariate inverse hypergeometric (MIH) distribution is an extension of the negative multinomial (NM) model that accounts for sampling without replacement in a finite population. Even though most studies on longitudinal count data with a specific number of ‘failures’ occur in a finite setting, the NM model is typically chosen over the more ac...

We present a noise robust deep learning based aberration analysis method using 2-step phase shift measurement data. We first propose a realistic aberration pattern generation method to synthesize a sufficient amount of real-world-like aberration patterns for training a deep neural network by exploiting the asymptotic statistical distribution parame...

We study the one-parameter family of Fredholm determinants det ( I − ρ 2 K n , x ) , ρ ∈ R , where K n , x stands for the integral operator acting on L 2 ( x , + ∞ ) with the higher order Airy kernel. This family of determinants represents a new universal class of distributions which is a higher order analogue of the classical Tracy–Widom distribut...

We study the one-parameter family of Fredholm determinants $\det(I-\rho^2\mathcal{K}_{n,x})$, $\rho\in\mathbb{R}$, where $\mathcal{K}_{n,x}$ stands for the integral operator acting on $L^2(x,+\infty)$ with the higher order Airy kernel. This family of determinants represents a new universal class of distributions which is a higher order analogue of...

A Bernoulli scheme with unequal harmonic success probabilities is investigated, together with some of its natural extensions. The study includes the number of successes over some time window, the times to (between) successive successes and the time to the first success. Large sample asymptotics, statistical parameter estimation, and relations to Si...

The Fisher information matrix is a quantity of fundamental importance for information geometry and asymptotic statistics. In practice, it is widely used to quickly estimate the expected information available in a data set and guide experimental design choices. In many modern applications, it is intractable to analytically compute the Fisher informa...

The mean shift (MS) algorithm seeks a mode of the kernel density estimate (KDE). This study presents a convergence guarantee of the mode estimate sequence generated by the MS algorithm and an evaluation of the convergence rate, under fairly mild conditions, with the help of the argument concerning the {\L}ojasiewicz inequality. Our findings, which...

Let $\Gamma _{g}$ be the fundamental group of a closed connected orientable surface of genus $g\geq 2$ . We develop a new method for integrating over the representation space $\mathbb {X}_{g,n}=\mathrm {Hom}(\Gamma _{g},S_{n})$ , where $S_{n}$ is the symmetric group of permutations of $\{1,\ldots ,n\}$ . Equivalently, this is the space of all verte...

The paper introduces a novel non-parametric Riemannian regression method using Isometric Riemannian Manifolds (IRMs). The proposed technique, Intrinsic Local Polynomial Regression on IRMs (ILPR-IRMs), enables global data mapping between Riemannian manifolds while preserving underlying geometries. The ILPR method is generalized to handle multivariat...

For a diffusion process $X(t)$ of drift $\mu(x)$ and of diffusion coefficient $D=1/2$, we study the joint distribution of the two local times $A(t)= \int_{0}^{t} d\tau \delta(X(\tau)) $ and $B(t)= \int_{0}^{t} d\tau \delta(X(\tau)-L) $ at positions $x=0$ and $x=L$, as well as the simpler statistics of their sum $ \Sigma(t)=A(t)+B(t)$. Their asympto...

In medical clinical studies, we often encounter paired organs’ unilateral or bilateral data. For bilateral data, there exists an intraclass correlation between paired organs. Under an intraclass correlation model, this paper proposes asymptotic statistics for testing the equality of many-to-one relative risk ratios in combined unilateral and bilate...

The goal of this paper is to develop methodology for the systematic analysis of asymptotic statistical properties of data driven DRO formulations based on their corresponding non-DRO counterparts. We illustrate our approach in various settings, including both phi-divergence and Wasserstein uncertainty sets. Different types of asymptotic behaviors a...

Gwet’s first-order agreement coefficient (AC1) is widely used to assess the agreement between raters. This paper proposes several asymptotic statistics for a homogeneity test of stratified AC1 in large sample sizes. These statistics may have unsatisfactory performance, especially for small samples and a high value of AC1. Furthermore, we propose th...

In the present article, we introduce the concepts of strongly asymptotically lacunary equivalence, asymptotically statistical equivalence, asymptotically lacunary statistical equivalence for sequences in gmetric spaces. We investigate some properties and relationships among this new concepts.

We adopt an information-theoretic framework to analyze the generalization behavior of the class of iterative, noisy learning algorithms. This class is particularly suitable for study under information-theoretic metrics as the algorithms are inherently randomized, and it includes commonly used algorithms such as Stochastic Gradient Langevin Dynamics...

We extend the methodology in [Yang et al., 2021] to learn autonomous continuous-time dynamical systems from invariant measures. We assume that our data accurately describes the dynamics' asymptotic statistics but that the available time history of observations is insufficient for approximating the Lagrangian velocity. Therefore, invariant measures...

Let C≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\ge 2$$\end{document} be a positive integer. Consider the set of n×n\documentclass[12pt]{minimal} \usepackage{am...

In the present article, we introduce the concepts of strongly asymptotically lacunary equivalence, asymptotically statistical equivalence , and asymptotically lacunary statistical equivalence for sequences in g-metric spaces. We investigate some properties and relationships among these new concepts.

In this paper, we study the asymptotic properties (bias, variance, mean squared error) of Bernstein estimators for cumulative distribution functions and density functions near and on the boundary of the $d$-dimensional simplex. Our results generalize those found by Leblanc (2012), who treated the case $d=1$, and complement the results from Ouimet (...

In this short note, we prove an asymptotic expansion for the ratio of the Dirichlet density to the multivariate normal density with the same mean and covariance matrix. The expansion is then used to derive an upper bound on the total variation between the corresponding probability measures and rederive the asymptotic variance of the Dirichlet kerne...

Background High-throughput metagenomic sequencing technologies have shown prominent advantages over traditional pathogen detection methods, bringing great potential in clinical pathogen diagnosis and treatment of infectious diseases. Nevertheless, how to accurately detect the difference in microbiome profiles between treatment or disease conditions...

The main goal of this article is to present the notion of double asymptotically lacunary statistical equivalent of order α for sequences of fuzzy numbers by considering fuzzy numbers and Pringsheim limit. To accomplish this goal, we mainly investigate some fundamental properties of the newly introduced notion. Additionally, it should be note that s...

This article proposes a new method of truncated estimation to estimate the tail index $\alpha$ of the extremely heavy-tailed distribution with infinite mean or variance. We not only present two truncated estimators $\hat{\alpha}$ and $\hat{\alpha}^{\prime}$ for estimating $\alpha$ ($0<\alpha \leq 1$) and $\alpha$ ($1<\alpha \leq 2$) respectively, b...

For a diffusion process $X(t)$ of drift $\mu(x)$ and of diffusion coefficient $D=1/2$, we study the joint distribution of the two local times $A(t)= \int_{0}^{t} d\tau \delta(X(\tau)) $ and $B(t)= \int_{0}^{t} d\tau \delta(X(\tau)-L) $ at positions $x=0$ and $x=L$, as well as the simpler statistics of their sum $ \Sigma(t)=A(t)+B(t)$. Their asympto...

In this paper, we develop local expansions for the ratio of the centered matrix-variate T density to the centered matrix-variate normal density with the same covariances. The approximations are used to derive upper bounds on several probability metrics (such as the total variation and Hellinger distance) between the corresponding induced measures....

In this paper, we aim to estimate the prediction error of machine learning models under the true distribution of the data on hand. We consider the prediction model as a data-driven black-box function and quantify its statistical properties using non-parametric methods. We propose a novel sampling technique that takes advantage of the underlying pro...

In this paper, we prove a local limit theorem for the chi-square distribution with r > 0 degrees of freedom and noncentrality parameter λ ≥ 0. We use it to develop refined normal approximations for the survival function. Our maximal errors go down to an order of r −2 , which is significantly smaller than the maximal error bounds of order r −1/2 rec...

We consider the incompressible 2D Navier-Stokes equations on the torus driven by a deterministic time quasi-periodic force and a noise that is white in time and extremely degenerate in Fourier space. We show that the asymptotic statistical behavior is characterized by a uniquely ergodic and exponentially mixing quasi-periodic invariant measure. The...

The noncentral Wishart distribution has become more mainstream in statistics as the prevalence of applications involving sample covariances with underlying multivariate Gaussian populations as dramatically increased since the advent of computers. Multiple sources in the literature deal with local approximations of the noncentral Wishart distributio...

The hard edge Pearcey process is universal in random matrix theory and many other stochastic models. This paper deals with the gap probability for the thinned/unthinned hard edge Pearcey process over the interval $(0,s)$ by working on the relevant Fredholm determinants. We establish an integral representation of the gap probability via a Hamiltonia...

The marginal Bayesian predictive classifiers (mBpc), as opposed to the simultaneous Bayesian predictive classifiers (sBpc), handle each data separately and, hence, tacitly assume the independence of the observations. Due to saturation in learning of generative model parameters, the adverse effect of this false assumption on the accuracy of mBpc ten...

We introduce a new generalization of the Pseudo-Lindley distribution by applying alpha power transformation. The obtained distribution is referred as the Pseudo-Lindley alpha power transformed distribution (\textit{PL-APT}). Some tractable mathematical properties of the \textit{PL-APT} distribution as reliability, hazard rate, order statistics and...

Social discounting is a critical and contentious issue in evaluating the costs and benefits of climate policy, infrastructure projects, and other long-term public policies. In this paper, we look at the asymptotic statistical properties of the Weitzman (2001) gamma discounting in a dynamic stochastic continuous-time framework where individual disco...

In this paper, we introduce new definitions which are related to the notions \(\lambda \mu\)–double asymptotically statistically equivalent to multiple L and strongly \(\lambda \mu\)–double asymptotically equivalent to multiple L by using \(f(\varsigma ,\tau )\) and \(g(\varsigma ,\tau )\) bivariate measurable real valued functions on \(\left( 1,\i...

In this short note, we develop a local approximation for the log-ratio of the multivariate hypergeometric probability mass function over the corresponding multinomial probability mass function. In conjunction with the bounds from Carter [4] and Ouimet [14] on the total variation between the law of a multinomial vector jittered by a uniform on (−1/2...

Gaussian processes provide a framework for nonlinear nonparametric Bayesian inference widely applicable across science and engineering. Unfortunately, their computational burden scales cubically with the training sample size, which in the case that samples arrive in perpetuity, approaches infinity. This issue necessitates approximations for use wit...

The paper concerns convergence and asymptotic statistics for stochastic approximation driven by Markovian noise: $$ \theta_{n+1}= \theta_n + \alpha_{n + 1} f(\theta_n, \Phi_{n+1}) \,,\quad n\ge 0, $$ in which each $\theta_n\in\Re^d$, $ \{ \Phi_n \}$ is a Markov chain on a general state space X with stationary distribution $\pi$, and $f:\Re^d\times...

Abstract A new reduced‐rank (RR) space‐time adaptive processing (STAP) algorithm based on multistage selections of angle‐Doppler filters is proposed in the form of a generalised sidelobe canceller. First, two types of the RR auxiliary angle‐Doppler filters are designed based on the discrete Fourier basis functions. Then, a novel multistage method i...

In Mombeni et al. (2019), Birnbaum-Saunders and Weibull kernel estimators were introduced for the estimation of cumulative distribution functions (c.d.f.s) supported on the half-line [0, ∞). They were the first authors to use asymmetric kernels in the context of c.d.f. estimation. Their estimators were shown to perform better numerically than tradi...

Suppose (f,X,μ) is a measure preserving dynamical system and ϕ:X→R a measurable observable. Let Xi=ϕ∘fi−1 denote the time series of observations on the system, and consider the maxima process Mn:=max{X1,…,Xn}. Under linear scaling of Mn, its asymptotic statistics are usually captured by a three-parameter generalised extreme value distribution. This...

Direct comparison of bulk gene expression profiles is complicated by distinct cell type mixtures in each sample that obscure whether observed differences are actually caused by changes in the expression levels themselves or are simply a result of differing cell type compositions. Single-cell technology has made it possible to measure gene expressio...

Sparse Group LASSO (SGL) is a regularized model for high-dimensional linear regression problems with grouped covariates. SGL applies $l_1$ and $l_2$ penalties on the individual predictors and group predictors, respectively, to guarantee sparse effects both on the inter-group and within-group levels. In this paper, we apply the approximate message p...

In this paper, we prove a local limit theorem for the ratio of the Poisson distribution to the Gaussian distribution with the same mean and variance, using only elementary methods (Taylor expansions and Stirling's formula). We then apply the result to derive an upper bound on the Le Cam distance between Poisson and Gaussian experiments, which gives...

The primary goal of this article is to present the concepts of asymptotically equivalent function and asymptotic regular function transformations. Moreover, by using these definitions, we examine the bivariate function transformation of asymptotically statistical equivalent measurable functions.

In this paper, we introduce the concepts of asymptotically f‐statistical equivalence, asymptotically f‐lacunary statistical equivalence, and strong asymptotically f‐lacunary equivalence for non‐negative two delta measurable real‐valued functions defined on time scales with the aid of modulus function f. Furthermore, the relationships between these...

Time series data sets often contain heterogeneous signals, composed of both continuously changing quantities and discretely occurring events. The coupling between these measurements may provide insights into key underlying mechanisms of the systems under study. To better extract this information, we investigate the asymptotic statistical properties...

With a new proof approach we prove in a more general setting the classical convergence theorem that almost everywhere convergence of measurable functions on a finite measure space implies convergence in measure. Specifically, we generalize the theorem for the case where the codomain is a separable metric space and for the case where the limiting ma...

In this article, we introduce and investigate the concepts of Δ θ m ‐asymptotically statistical equivalent of order α ˜ and strong Δ θ m ‐asymptotically equivalent of order α ˜ of double sequences. Also, we give some relationships related to these concepts.

Phase-amplitude coupling (PAC) is the association of the amplitude of a high-frequency oscillation with the phase of a low-frequency oscillation. In neuroscience, this relationship provides a mechanism by which neural activity might be coordinated between distant regions. The dangers and pitfalls of assessing phase-amplitude coupling with existing...

The likelihood ratio test (LRT) is widely used for comparing the relative fit of nested latent variable models. Following Wilks’ theorem, the LRT is conducted by comparing the LRT statistic with its asymptotic distribution under the restricted model, a $$\chi ^2$$ χ 2 distribution with degrees of freedom equal to the difference in the number of fre...

This study proposed statistical distribution called Exponentiated distribution using one of the twelve families of burr, that is, Burr V (BV) distribution. The theoretical properties of this proposed distribution are proven such as the validity of the distribution with also a graphical representation of the probability density function (PDF) and th...

The rate coding response of a single peripheral sensory neuron in the asymptotic, near-equilibrium limit can be derived using information theory, asymptotic Bayesian statistics and a theory of complex systems. Almost no biological knowledge is required. The theoretical expression shows good agreement with spike-frequency adaptation data across diff...

In this paper we give a new sufficient condition for asymptotic periodicity of Frobenius-Perron operator corresponding to two--dimensional maps. The result of the asymptotic periodicity for strictly expanding systems, that is, all eigenvalues of the system are greater than one, in a high-dimensional dynamical systems was already known. Our new theo...

Sufficient conditions were recently established for the rates of thermodynamic entropy production and phase space volume contraction to be identically equal in thermostatted Hamiltonian systems in non-equilibrium steady states (Ramshaw 2017 Phys. Rev. E 96 052122). This equality has previously been interpreted as a statistical analogue of the secon...

Chaos is ubiquitous in physical systems. The associated sensitivity to initial conditions is a significant obstacle in forecasting the weather and other geophysical fluid flows. Data assimilation is the process whereby the uncertainty in initial conditions is reduced by the astute combination of model predictions and real-time data. This chapter re...

Direct comparison of bulk gene expression profiles is complicated by distinct cell type mixtures in each sample which obscure whether observed differences are actually due to changes in expression levels themselves or simply cell type compositions. Single-cell technology has made it possible to measure gene expression in individual cells, achieving...

The purpose of the research is to develop a generalized structural scheme of organizational and technical systems based on the general theory of management, which contains the necessary and sufficient number of modules and formalize on this basis the main management tasks that act as goals of the behavior of the management object. The main modules...

According to our current conception of physics, any valid physical theory is supposed to describe the objective evolution of a unique external world. However, this condition is challenged by quantum theory, which suggests that physical systems should not always be understood as having objective properties which are simply revealed by measurement. F...

Time series datasets often contain heterogeneous signals, composed of both continuously changing quantities and discretely occurring events. The coupling between these measurements may provide insights into key underlying mechanisms of the systems under study. To better extract this information, we investigate the asymptotic statistical properties...

We consider a single outbreak susceptible-infected-recovered (SIR) model and corresponding estimation procedures for the effective reproductive number $\mathcal{R}(t)$. We discuss the estimation of the underlying SIR parameters with a generalized least squares (GLS) estimation technique. We do this in the context of appropriate statistical models f...

Let $\Gamma_{g}$ be the fundamental group of a closed connected orientable surface of genus $g\geq2$. We develop a new method for integrating over the representation space $\mathbb{X}_{g,n}=\mathrm{Hom}(\Gamma_{g},S_{n})$ where $S_{n}$ is the symmetric group of permutations of $\{1,\ldots,n\}$. Equivalently, this is the space of all vertex-labeled,...

Modal linear regression (MLR) is a method for obtaining a conditional mode predictor as a linear model. We study kernel selection for MLR from two perspectives: "which kernel achieves smaller error?" and "which kernel is computationally efficient?". First, we show that a Biweight kernel is optimal in the sense of minimizing an asymptotic mean squar...

Permutations of correlated sequences of random variables appear naturally in a variety of applications such as graph matching and asynchronous communications. In this paper, the asymptotic statistical behavior of such permuted sequences is studied. It is assumed that a collection of random vectors is produced based on an arbitrary joint distributio...

In order to improve the multicast transmissions, a full-duplex (FD) user cooperation scheme is proposed. In this paper, the transmitter sends the common messages to two FD users, and each user forwards its received signals to its counterpart by amplify-and-forward (AF) scheme. Considering the imperfect self-interference (SI) cancellation at the use...

In Siotani and Fujikoshi (1984), a precise local limit theorem for the multinomial distribution is derived by inverting the Fourier transform, where the error terms are explicit up to order N−1. In this paper, we give an alternative (conceptually simpler) proof based on Stirling’s formula and a careful handling of Taylor expansions, and we show how...

In this study, we introduce and examine the concepts of asymptotically ?-statistical equivalent sequences of order ? in probability and strong asymptotically ?-equivalent sequences of order ? in probability. We give some relations connected to these concepts.

Co-array-based Direction of Arrival (DoA) estimation using Sparse Linear Arrays (SLAs) has recently gained considerable interest in array processing thanks to its capability of providing enhanced degrees of freedom. Although the literature presents a variety of estimators in this context, none of them are proven to be statistically efficient. This...

Survivorship analysis allows to statistically analyze situations that can be modeled as waiting times to an event. These waiting times are characterized by the cumulative hazard rate, which can be estimated by the Nelson-Aalen estimator or diverse confidence estimators based on asymptotic statistics. To better understand the small sample properties...

Circular KPZ interfaces spreading radially in the plane have GUE Tracy-Widom (TW) height distribution (HD) and Airy$_2$ spatial covariance, but what are their statistics if they evolve on the surface of a different background space, such as a bowl, a mountain, or any surface of revolution? To give an answer to this, we report here extensive numeric...

In the spirit of recent asymptotic works on the General Poverty Index (GPI) in the field of Welfare Analysis, the asymptotic statistical representation of the non-decomposable Takayama’s index, which has failed to be incorporated in the unified GPI approach, is addressed and established here. This representation also allows to extend to it, recent...

A discrete time stochastic model for a multicomponent system is presented, which consists of two random vectors representing a multivariate cumulative damage and their corresponding failure times. The times of occurrence of some events, for the system components, are correlated and their associate cumulative damages are assumed to be additive. Sinc...

Circular KPZ interfaces spreading radially in the plane have GUE Tracy-Widom (TW) height distribution (HD) and Airy$_2$ spatial covariance, but what are their statistics if they evolve on the surface of a different background space, such as a bowl, a cup, or any surface of revolution? To give an answer to this, we report here extensive numerical an...

Inverse synthetic aperture radar (ISAR) imaging of target with complex motion is very important in the radar signal processing domain. In this case, the received signal can be characterized as multi-component cubic phase signal (CPS), and the high quality instantaneous ISAR images can be obtained by the parameters estimation approach. The match Fou...

Exchange of location and sensor data among connected and automated vehicles will demand accurate global referencing of the digital maps currently being developed to aid positioning for automated driving. This paper explores the limit of such maps’ globally-referenced position accuracy when the mapping agents are equipped with low-cost Global Naviga...