Variance Estimation Based on Invariance Principles

Consistent variance estimators for certain stochastic processes are suggested using the fact that (weak or strong) invariance principles may be available. Convergence rates are also derived, the latter being essentially determined by the approximation rates in the corresponding invariance principles. As an application, a change point test in a simple AMOC renewal model is briefly discussed, where variance estimators possessing good enough convergence rates are required.     

On Non-Normal Invariance Principles for Multi-Response Permutation Procedures

A non-normal invariance principle is established for a restricted class of univariate multi-response permutation procedures whose distance measure is the square of Euclidean distance. For observations from a distribution with finite second moment, the test statistic is found asymptotically to have a centered chi-squared distribution. Spectral expansions are used to determine the asymptotic distribution for more general distance measures d, and it is shown that if d(x, y) = |x — y|^u, u? 2, the asymptotic distribution is not invariant (i.e. it is dependent on the distribution of the observations).     

Invariance and factorial models

Two factors having the same set of levels are said to be homologous. This paper aims to extend the domain of factorial models to designs that include homologous factors. In doing so, it is necessary first to identify the characteristic property of those vector spaces that constitute the standard factorial models. We argue here that essentially every interesting statistical model specified by a vector space is necessarily a representation of some algebraic category. Logical consistency of the sort associated with the standard marginality conditions is guaranteed by category representations, but not by group representations. Marginality is thus interpreted as invariance under selection of factor levels ( I -representations), and invariance under replication of levels ( S -representations). For designs in which each factor occurs once, the representations of the product category coincide with the standard factorial models. For designs that include homologous factors, the set of S -representations is a subset of the I -representations. It is shown that symmetry and quasi-symmetry are representations in both senses, but that not all representations include the constant functions (intercept). The beginnings of an extended algebra for constructing general I -representations is described and illustrated by a diallel cross design.     

An NMF-framework for Unifying Posterior Probabilistic Clustering and Probabilistic Latent Semantic Indexing

In document clustering, a document may be assigned to multiple clusters and the probabilities of a document belonging to different clusters are directly normalized. We propose a new Posterior Probabilistic Clustering (PPC) model that has this normalization property. The clustering model is based on Nonnegative Matrix Factorization (NMF) and flexible such that if we use class conditional probability normalization, the model reduces to Probabilistic Latent Semantic Indexing (PLSI). Systematic comparison and evaluation indicates that PPC is competitive with other state-of-art clustering methods. Furthermore, the results of PPC are more sparse and orthogonal, both of which are highly desirable.     

Invariance principles in probability for stable processes generated by a class of dependent sequences

We prove an invariance principle in probability for φ-mixing sequences of random variables taking values in a separable Banach space. The conditions on the dependence structure required for this approximation result are known to be necessary and sufficient for weak convergence of the partial-sum process to the appropriate limiting process.     

Probabilistic Properties of Parametric Dual and Inverse Time Series Models Generated by ARMA Models

For the class of autoregressive-moving average (ARMA) processes, we examine the relationship between the dual and the inverse processes. It is demonstrated that the inverse process generated by a causal and invertible ARMA (p, q) process is a causal and invertible ARMA (q, p) model. Moreover, it is established that this representation is strong if and only if the generating process is Gaussian. More precisely, it is derived that the linear innovation process of the inverse process is an all-pass model. Some examples and applications to time reversibility are given to illustrate the obtained results.     

Invariance principles for jackknifing U-statistics for finite population sampling and some applications

For simple random sampling (without replacement) from a finite population, suitable stochastic processes are constructed from the entire sequence of jackknife estimators based on smooth functions of U-statistics and these are approximated (in distributions) by some Brownian bridge processes. Strong convergence of the Tukey estimator of the variance of a jackknife U-statistic has been interpreted suitably and established. Some applications of these results in sequential analysis relating to finite population sampling are also considered.     

Invariance of linearity of regression and related characterizations of some classical discrete distributions

Rao (1963) introduced what we call an additive damage model. In this model, original observation is subjected to damage according to a specified probability law by the survival distribution. In this paper, we consider a bivariate observation with second component subjected to damage. Using the invariance of linearity of regression of the first component on the second under the transition of the second component from the original to the damaged state, we obtain the characterizations of the Poisson, binomial and negative binomial distributions within the framework of the additive damage model.     

Invariance,Optimality, and a 1-Observation Confidence Interval for a Normal Mean

In a 1965 Decision Theory course at Stanford University, Charles Stein began a digression with “an amusing problem”: is there a proper confidence interval for the mean based on a single observation from a normal distribution with both mean and variance unknown? Stein introduced the interval with endpoints ± c|X| and showed indeed that for c large enough, the minimum coverage probability (over all values for the mean and variance) could be made arbitrarily near one. While the problem and coverage calculation were in the author’s hand-written notes from the course, there was no development of any optimality result for the interval. Here, the Hunt–Stein construction plus analysis based on special features of the problem provides a “minimax” rule in the sense that it minimizes the maximum expected length among all procedures with fixed coverage (or, equivalently, maximizes the minimal coverage among all procedures with a fixed expected length). The minimax rule is a mixture of two confidence procedures that are equivariant under scale and sign changes, and are uniformly better than the classroom example or the natural interval X ± c|X|?.     

Wald检验不一致性的微分几何解释及改进

Wald检验对于等价的零假设中不同形式的表达式在有限样本的情况下缺乏一致性,而从微分几何的角度来解释这一现象,并发现由于Wald统计量是一个混杂的不恰当的几何量,从而对不同的含参数的等价表达式不具有一致性。同时还展示了芬斯拉（Finsler）测地统计量如何能较为简便的计算出来、它在线性回归模型中的非线性约束条件下如何应用以及两者在什么情况下保持一致,并提出了一种解决Wald检验不一致性的思路。