Weak Convergence of $U$-Statistics and Von Mises' Differentiable Statistical Functions
R. G. Miller,Pranab Kumar Sen +1 more
TLDR
For partial cumulative sums of independent and identically distributed random variables with a finite (positive) variance, weak convergence to Brownian motion processes has been established by Donsker (1951, 1952).Abstract:
For partial cumulative sums of independent and identically distributed random variables (i.i.d.r.v.) with a finite (positive) variance, weak convergence to Brownian motion processes has been established by Donsker (1951, 1952). The result is extended here to differentiable statistical functions of von Mises (1947) and $U$-statistics of Hoeffding (1948). Along with the extension to generalized $U$-statistics, a few applications are briefly sketched.read more
Citations
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Journal ArticleDOI
On U-statistics and v. mise’ statistics for weakly dependent processes
Manfred Denker,Gerhard Keller +1 more
TL;DR: In this article, the authors extend these results considerably and prove central limit theorems and their rate of convergence (in the Prohorov metric and a Berry Esseen type theorem), functional central limit theorem and as approximation by a Brownian motion.
Journal ArticleDOI
Some Invariance Principles Relating to Jackknifing and Their Role in Sequential Analysis
TL;DR: For a broad class of jackknife statistics, it was shown in this article that the Tukey estimator of the variance converges almost surely to its population counterpart, and that the usual invariance principles (relating to the Wiener process approximations) usually filter through jackknifing under no extra regularity conditions.
Sequential point estimation of estimable parameters based on u-statistics
Pranab Kumar Sen,Malay Ghosh +1 more
TL;DR: In this paper, an asymptotically risk-efficient sequential point estimation of regular functionals of distribution functions based on U-statistics is considered under appropriate regularity conditions.
References
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Book
Convergence of Probability Measures
TL;DR: Weak Convergence in Metric Spaces as discussed by the authors is one of the most common modes of convergence in metric spaces, and it can be seen as a form of weak convergence in metric space.
Book ChapterDOI
A Class of Statistics with Asymptotically Normal Distribution
TL;DR: In this article, the authors considered the problem of estimating a U-statistic of the population characteristic of a regular functional function, where the sum ∑″ is extended over all permutations (α 1, α m ) of different integers, 1 α≤ (αi≤ n, n).
Journal ArticleDOI
Limiting Behavior of Posterior Distributions when the Model is Incorrect
TL;DR: In this paper, the authors examined the large sample behavior of posterior distributions without the assumption that the model is correct and showed that the posterior distribution for a parameter is confined to a set (called the asymptotic carrier) which may, in general, contain more than one point.
Journal ArticleDOI
Consistency and Unbiasedness of Certain Nonparametric Tests
TL;DR: In this paper, it was shown that there exist strictly unbiased and consistent tests for the univariate and multivariate two-and fc-sample problem, for the hypothesis of independence, and for the hypotheses of symmetry with respect to a given point.