In this paper, we mainly investigate the nonparametric regression model based on widely orthant dependent samples. The weak consistency, strong consistency, complete consistency, and the rates of strong consistency and complete consistency for the wavelet estimator are established under some mild conditions, which generalize and improve the corresponding ones for other dependent cases. Some numerical simulations are presented to verify the validity of the theoretical results based on finite samples.
In this paper, we study complete f-moment convergence, complete sth moment convergence and complete convergence for m-widely orthant dependent linear processes with random coefficients under some general conditions. The results presented extend and improve some known results. As an application, we present conclusions on complete f-moment consistency and complete consistency for a weighted estimator in a nonparametric regression model based on widely orthant dependent errors. Moreover, we perform a numerical simulation to verify the validity of the theoretical results. 2020 Mathematics Subject Classification: Primary 60F15; Secondary 62G20.
This paper establishes an exponential inequality for a sequence of widely orthant dependent (WOD) random variables. Leveraging this result, we prove the strong consistency of a non-parametric regression estimator under a general moment condition. We further derive the convergence rate of this estimator, highlighting its alignment with classical results under specific parameter choices. As an application, we examine the nearest neighbor estimator and support our theoretical findings with a numerical study.
In the paper we consider estimation of the regression coefficients in a linear regression model Y = X3 + & varepsilon;, where 3 is assumed to lie in a given ellipsoid Theta. The estimation is performed using a weighted squared error loss function. Under specific conditions, we find an explicit formula for an estimator 3bM of 3, which is minimax in the class D of all decision rules, not limited to linear estimators. We generalize the results of Wilczynski (2007), who found a minimax decision rule 3bM under considerably more restrictive conditions.
The so-called commitment capacity of a discrete memoryless channel is given by the maximum of the conditional Shannon entropy H(X|Y ) over all input distributions. We examine in detail this optimization problem, motivated by its relevance in information-theoretic cryptography. In particular, we study the role of the channel's connected components in attaining the commitment capacity, and the questions of uniqueness and support of the optimal (capacity-achieving) input and output distributions. We also describe an iterative algorithm for computing the commitment capacity and the optimal input distribution.
The paper applies the Rao-Blackwell technique to construct an unbiased estimator of the reliability function in the exponential distribution under left censored data. The resulting estimator is a function of a sufficient statistic. However, we cannot conclude that it is a minimal variance estimator because the sufficient statistic is not complete. We also give an asymptotic confidence interval for reliability based on the estimator obtained. The performance of this estimator is compared to the maximum likelihood estimator via Monte Carlo simulation.
We provide a necessary and sufficient condition on a sequence of random variables for the Boolean and Fermi central limit theorems to hold true. The result allows us to connect the classical and the non-commutative analogues of the central limit theorems. We show that the Boolean domain of attraction of the symmetric Bernoulli distribution, the free domain of attraction of the semicircle law and the classical domain of attraction of the Gaussian distribution coincide with each other.
Consider the model Y-ni = g(x(ni))+sigma(ni)epsilon(ni), i = 1, ... , n, where sigma(2)(ni) = f (u(ni)), the design points (x(ni), u(ni)) are known and nonrandom, g() and f() are unknown functions defined on [0, 1], and the random errors {epsilon(ni), 1 <= i <= n} are assumed to have the same distribution as {theta(i), 1 <= i <= n}, which is a sequence of identically distributed alpha-mixing random variables with mean zero. Estimators of f() and g() are constructed by the G-M method and their rth (r > 2) mean consistency and strong consistency are obtained under appropriate conditions. To demonstrate the validity of theoretical results, finite sample behaviors of the estimators are considered via a simulation study.
We investigate a nested balls-in-boxes scheme in a random environment. The boxes follow a nested hierarchy, with infinitely many boxes in each level, and the hitting probabilities of boxes are random and obtained by iterated fragmentation of a unit mass. The hitting probabilities of the first-level boxes are given by a stick-breaking model PW1W2..... Wk-1(1Wk) for k is an element of N, where W-1, W-2... are independent copies of a random variable W taking values in (0, 1). The infinite balls-in-boxes scheme in the first level is known as a Bernoulli sieve. We assume that the mean of log W is infinite and the distribution tail of vertical bar log W vertical bar is regularly varying at o infinity. Denote by K-n (j) the number of occupied boxes in the jth level provided that there are n balls and call the level j intermediate if j = j(n) and j(n) o((logn)(a)) as n ->infinity for some a > 0 We prove that, for some intermediate levels j, finite-dimensional distributions of the pro-cess (K-n ( lfloor j(n) u rfloor)) (u > 0), properly normalized, converge weakly as n ->infinity to those of a pathwise Lebesgue-Stieltjes integral, with the integrand being an exponential function and the integrator being an inverse stable subordinator. The present paper continues the line of investigation initiated in the articles of Buraczewski, Dovgay and Iksanov (2020) and Iksanov, Marynych and Samoilenko (2022) in which the random variable vertical bar log W vertical bar has a finite second moment, and of Iksanov, Marynych and Rashytov (2022) in which log W has a finite mean and an infinite second moment.
We study Replica Mean Field limits for a neural system of infinitely many neurons with both inhibitory and excitatory interactions. As a result we obtain an analytical characterisation of the invariant state. In particular we focus on the Galves-L\"ocherbach model with interactions beyond the Uniform Summability Principle.
The asymptotic behavior of intermediate order statistics and record values for folded distributions is studied. The relation between the weak convergence of a distribution function FX and of its folded distribution function, F|X|, is revealed in the cases of intermediate order statistics and record values. Additionally, a few illustrative examples are provided.
We investigate the asymptotic distribution of odd graphs in a deformed vacuum state, focusing on the spectral analysis of these graphs. We explore the adjacency matrices of odd graphs and derive explicit expressions for their mean and variance in the deformed vacuum state. Our main results provide the probability measures and the corresponding coherent states for the distribution of these graphs. We calculate the Jacobi coefficients and Cauchy transforms related to these distributions, which have not been addressed explicitly in the existing literature. Our findings contribute to a deeper understanding of the probabilistic and spectral properties of odd graphs in quantum state frameworks.
We prove a Bernstein-type bound for the difference between the average of the negative log-likelihoods of independent categorical variables with infinitely many levels - that is, a countably infinite number of categories, and its expectation - namely, the Shannon entropy. The result holds for the class of discrete random variables with tails lighter than or of the same order as a discrete power-law distribution. Most commonly used discrete distributions, such as the Poisson distribution, the negative binomial distribution, and the power-law distribution itself, belong to this class. The bound is effective in the sense that we provide a method to compute the constants within it. The new technique we develop allows us to obtain a uniform concentration inequality for categorical variables with a finite number of levels with the same optimal rate as in the literature, but with a much simpler proof.
We prove that quantiles are best predictors in a special metric. The best predictor turns out to coincide with the notions of generalized arithmetic mean, exponential barycenter and certainty equivalent. We also show that the computation of tail value at risk (TVaR) reduces to the computation of a quantile with a higher level of confidence. This point of view makes the analysis of the statistical properties of TVaR easier.
This article gives a formula for associated Stirling numbers of the second kind based on the moment of a sum of independent random variables having a beta distribution. From this formula we deduce lower and upper bounds for these numbers, using a probabilistic approach.
We study the complete f-moment convergence for arrays of row- wise random variables satisfying a Rosenthal type moment inequality, and then establish general results on the complete moment convergence and complete convergence for partial sums and weighted sums of arrays of row- wise random variables. As applications, we further describe the statistical properties of complete f-moment convergence in both semiparametric regression models and simple linear errors-in-variables models. The asymptotic properties for estimators are established. We also provide some simulations to verify the validity of the theoretical results. 2020 Mathematics Subject Classification: Primary 60F15; Secondary 62F12, 62G20.
In this paper, we derive the asymptotic distributions for the maxima of two types of Gaussian functions, including a chi-random sequence and a Gaussian order statistics sequence subject to missing observations, where the Gaussian functions are generated by stationary Gaussian sequences with covariance functions r n satisfying r n log n- gamma is an element of [0, infinity) as n- infinity.
We use the Stein-Chen method to prove new explicit inequalities for the total variation, Wasserstein and local distances between the distribution of a random diagonal sum of a Bernoulli matrix and a Poisson distribution. Approximation results using a finite signed measure of higher order are given as well. Some of our total variation bounds improve existing results in the literature.
We give two-sided estimates of a ground state for Schr & ouml;dinger operators with confining potentials. We propose a semigroup approach, based on resolvent and the Feynman-Kac formula, which leads to a new, rather short and direct proof. Our results take the sharpest form for slowly varying, radial and increasing potentials.
