scholarly journals Generic Uniform Convergence

1992 ◽  
Vol 8 (2) ◽  
pp. 241-257 ◽  
Author(s):  
Donald W.K. Andrews
Keyword(s):  
Uniform Convergence ◽  
Asymptotic Properties ◽  
Convergence In Probability ◽  
Test Statistics ◽  
Parameter Spaces ◽  
Totally Bounded ◽  
Convergence Results ◽  
Large Numbers ◽  
Uniform Law ◽  
Convergence Almost Surely

This paper presents several generic uniform convergence results that include generic uniform laws of large numbers. These results provide conditions under which pointwise convergence almost surely or in probability can be strengthened to uniform convergence. The results are useful for establishing asymptotic properties of estimators and test statistics.The results given here have the following attributes, (1) they extendresults of Newey to cover convergence almost surely as well as convergence in probability, (2) they apply to totally bounded parameter spaces (rather than just to compact parameter spaces), (3) they introduce a set of conditions for a generic uniform law of large numbers that has the attribute of giving the weakest conditions available for i.i.d. contexts, but which apply in some dependent nonidentically distributed contexts as well, and (4) they incorporate and extend themain results in the literature in a parsimonious fashion.

scholarly journals Robust Statistical Inference in Generalized Linear Models Based on Minimum Renyi’s Pseudodistance Estimators

Entropy ◽  
2022 ◽  
Vol 24 (1) ◽  
pp. 123
Author(s):  
María Jaenada ◽  
Leandro Pardo
Keyword(s):  
Generalized Linear Models ◽  
Influence Function ◽  
Regression Models ◽  
Linear Models ◽  
Asymptotic Properties ◽  
Significant Loss ◽  
Superior Performance ◽  
Test Statistics ◽  
Linear Regression Models ◽  
Type Test

Minimum Renyi’s pseudodistance estimators (MRPEs) enjoy good robustness properties without a significant loss of efficiency in general statistical models, and, in particular, for linear regression models (LRMs). In this line, Castilla et al. considered robust Wald-type test statistics in LRMs based on these MRPEs. In this paper, we extend the theory of MRPEs to Generalized Linear Models (GLMs) using independent and nonidentically distributed observations (INIDO). We derive asymptotic properties of the proposed estimators and analyze their influence function to asses their robustness properties. Additionally, we define robust Wald-type test statistics for testing linear hypothesis and theoretically study their asymptotic distribution, as well as their influence function. The performance of the proposed MRPEs and Wald-type test statistics are empirically examined for the Poisson Regression models through a simulation study, focusing on their robustness properties. We finally test the proposed methods in a real dataset related to the treatment of epilepsy, illustrating the superior performance of the robust MRPEs as well as Wald-type tests.

scholarly journals Subcritical Sevastyanov branching processes with nonhomogeneous Poisson immigration

2017 ◽  
Vol 54 (2) ◽  
pp. 569-587 ◽  
Author(s):  
Ollivier Hyrien ◽  
Kosto V. Mitov ◽  
Nikolay M. Yanev
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Cell Populations ◽  
Subcritical Case ◽  
Immigration Process ◽  
Large Numbers

Abstract We consider a class of Sevastyanov branching processes with nonhomogeneous Poisson immigration. These processes relax the assumption required by the Bellman–Harris process which imposes the lifespan and offspring of each individual to be independent. They find applications in studies of the dynamics of cell populations. In this paper we focus on the subcritical case and examine asymptotic properties of the process. We establish limit theorems, which generalize classical results due to Sevastyanov and others. Our key findings include a novel law of large numbers and a central limit theorem which emerge from the nonhomogeneity of the immigration process.

A clustering law for some discrete order statistics

2003 ◽  
Vol 40 (01) ◽  
pp. 226-241 ◽  
Author(s):  
Sunder Sethuraman
Keyword(s):  
Distribution Function ◽  
Positive Integer ◽  
Order Statistics ◽  
Law Of Large Numbers ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Large Numbers ◽  
The Law ◽  
Sample Maximum ◽  
Independent Identically Distributed

Let X 1, X 2, …, X n be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

Convergence in Lp Norm

2021 ◽  
pp. 418-437
Author(s):  
James Davidson
Keyword(s):  
Law Of Large Numbers ◽  
Almost Sure Convergence ◽  
Convergence In Probability ◽  
Martingale Differences ◽  
Lp Norm ◽  
Large Numbers

This chapter looks in detail at proofs of the weak law of large numbers (convergence in probability) using the technique of establishing convergence in Lp‐norm. The extension to a proof of almost‐sure convergence is given, and then special results for martingale differences, mixingales, and approximable processes. These results are proved in array notation to allow general forms of heterogeneity.

scholarly journals Gaps in Discrete Random Samples

2009 ◽  
Vol 46 (04) ◽  
pp. 1038-1051 ◽  
Author(s):  
Rudolf Grübel ◽  
Paweł Hitczenko
Keyword(s):  
Analysis Of Algorithms ◽  
Sufficient Conditions ◽  
Borderline Case ◽  
Convergence In Probability ◽  
Enumerative Combinatorics ◽  
Random Samples ◽  
The Family ◽  
Heavy Tailed ◽  
Convergence Almost Surely ◽  
Necessary And Sufficient

Let (X i ) i∈ℕ be a sequence of independent and identically distributed random variables with values in the set ℕ0 of nonnegative integers. Motivated by applications in enumerative combinatorics and analysis of algorithms we investigate the number of gaps and the length of the longest gap in the set {X 1,…,X n } of the first n values. We obtain necessary and sufficient conditions in terms of the tail sequence (q k ) k∈ℕ0 , q k =P(X 1≥ k), for the gaps to vanish asymptotically as n→∞: these are ∑ k=0 ∞ q k+1/q k <∞ and limk→∞ q k+1/q k =0 for convergence almost surely and convergence in probability, respectively. We further show that the length of the longest gap tends to ∞ in probability if q k+1/q k → 1. For the family of geometric distributions, which can be regarded as the borderline case between the light-tailed and the heavy-tailed situations and which is also of particular interest in applications, we study the distribution of the length of the longest gap, using a construction based on the Sukhatme–Rényi representation of exponential order statistics to resolve the asymptotic distributional periodicities.

scholarly journals Hermite-Fejér type interpolation and Korovkin's theorem

1990 ◽  
Vol 42 (3) ◽  
pp. 383-390
Author(s):  
H.-B. Knoop ◽  
F. Locher
Keyword(s):  
Uniform Convergence ◽  
Jacobi Polynomials ◽  
Korovkin Theorem ◽  
Type Interpolation ◽  
Special Values ◽  
Convergence Results ◽  
Interpolation Operators ◽  
Positive Functionals ◽  
Interpolation Polynomials ◽  
Korovkin’S Theorem

In this note we consider Hermite-Fejér interpolation at the zeros of Jacobi polynomials and with additional boundary conditions. For the associated Hermite-Fejér type operators and special values of α, β it was proved by the first author in recent papers that one has uniform convergence on the whole interval [−1,1]. The second author could show by introducing the concept of asymptotic positivity how to get the known convergence results for the classical Hermite-Fejér interpolation operators. In the present paper we show, using a slightly modified Bohman-Korovkin theorem for asymptotically positive functionals, that the Hermite-Fejér type interpolation polynomials , converge pointwise to f for arbitrary α, β > −1. The convergence is uniform on [−1 + δ,1 − δ].

GRAPHICAL METHODS FOR DETERMINING/ESTIMATING OPTIMAL REPAIR-LIMIT REPLACEMENT POLICIES

2000 ◽  
Vol 07 (01) ◽  
pp. 43-60 ◽  
Author(s):  
TADASHI DOHI ◽  
KENTARO TAKEITA ◽  
SHUNJI OSAKI
Keyword(s):  
Steady State ◽  
Asymptotic Properties ◽  
Repair Time ◽  
Graphical Methods ◽  
Test Statistics ◽  
Numerical Examples ◽  
Time Limits ◽  
Nonparametric Estimators

In this paper, we consider two kinds of repair-limit replacement models and develop the corresponding graphical methods to estimate the optimal repair-time limits which minimize the expected costs per unit time in the steady state. Then, both the total time on test statistics and the Lorenz statistics play important roles to develop nonparametric estimators of the optimal repair-time limits. Numerical examples are devoted to illustrate asymptotic properties of nonparametric estimators for the optimal repair-limit policies.

scholarly journals Convergence Results for the Gaussian Mixture Implementation of the Extended-Target PHD Filter and Its Extended Kalman Filtering Approximation

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Feng Lian ◽  
Chongzhao Han ◽  
Jing Liu ◽  
Hui Chen
Keyword(s):  
Uniform Convergence ◽  
Kalman Filtering ◽  
Gaussian Mixture ◽  
Extended Kalman Filtering ◽  
Probability Hypothesis Density ◽  
Target Probability ◽  
Convergence Results ◽  
Extended Target ◽  
Nonlinear Condition

The convergence of the Gaussian mixture extended-target probability hypothesis density (GM-EPHD) filter and its extended Kalman (EK) filtering approximation in mildly nonlinear condition, namely, the EK-GM-EPHD filter, is studied here. This paper proves that both the GM-EPHD filter and the EK-GM-EPHD filter converge uniformly to the true EPHD filter. The significance of this paper is in theory to present the convergence results of the GM-EPHD and EK-GM-EPHD filters and the conditions under which the two filters satisfy uniform convergence.

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