R. G. Miller

Bio: R. G. Miller is an academic researcher. The author has contributed to research in topics: Independent and identically distributed random variables & Differentiable function. The author has an hindex of 1, co-authored 1 publications receiving 75 citations.

Weak Convergence of $U$-Statistics and Von Mises' Differentiable Statistical Functions

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.

On U-statistics and v. mise’ statistics for weakly dependent processes

Abstract: Some probabilistic limit theorems for Hoeffding's U-statistic [13] and v Mises' functional are established when the underlying processes are not necessarily independent We consider absolutely regular processes [24] and processes (X n)n≧1 which are uniformly mixing [14] as well as their time reversal (X −n )n≦−1, called uniformly mixing in both directions of time Many authors have weakened the hypothesis of independence in statistical limit theorems and considered m-dependent, Markov or weakly dependent processes; in particular for U statistics under weak dependence Sen [22] has considered *-mixing processes and derived a central limit theorem and a law of the iterated logarithm, while Yoshihara [26] proved central limit theorems and as results in the absolutely regular and uniformly mixing case Here we 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 theorems and as approximation by a Brownian motion Extensions to multisample versions and other extensions are briefly discussed

Some Invariance Principles Relating to Jackknifing and Their Role in Sequential Analysis

Abstract: For a broad class of jackknife statistics, it is shown that the Tukey estimator of the variance converges almost surely to its population counterpart. Moreover, the usual invariance principles (relating to the Wiener process approximations) usually filter through jackknifing under no extra regularity conditions. These results are then incorporated in providing a bounded-length (sequential) confidence interval and a preassigned-strength sequential test for a suitable parameter based on jackknife estimators.

Sequential point estimation of estimable parameters based on u-statistics

Abstract: Asymptotically risk-efficient sequential point estimation of regular functionals of distribution functions based on U-statistics is considered under appropriate regularity conditions. Some auxiliary results on U-statistics are also considered in this context.