Asymptotically Optimal Tests and Optimal Designs for Testing the Mean in Regression Models with Applications to Change-Point Problems

Let a linear regression model be given with an experimental region [a, b] R and regression functions f ₁, ..., f _d+1 : [a, b] R. In practice it is an important question whether a certain regression function f _d+1, say, does or does not belong to the model. Therefore, we investigate the test problem H ₀ : "f _d+1 does not belong to the model" against K : "f _d+1 belong to the model" based on the least-squares residuals of the observations made at design points of the experimental region [a, b]. By a new functional central limit theorem given in Bischoff (1998, Ann. Statist. 26, 1398–1410), we are able to determine optimal tests in an asymptotic way. Moreover, we introduce the problem of experimental design for the optimal test statistics. Further, we compare the asymptotically optimal test with the likelihood ratio test (F-test) under the assumption that the error is normally distributed. Finally, we consider real change-point problems as examples and investigate by simulations the behavior of the asymptotic test for finite sample sizes. We determine optimal designs for these examples.     

Detecting and interpreting clusters of economic activity in rural areas using scan statistic and LISA under a unified framework

The primary aim of this paper is to expose the use and the value of spatial statistical analysis in business and especially in designing economic policies in rural areas. Specifically, we aim to present under a unified framework, the use of both point and area‐based methods, in order to analyze in‐depth economic data, as well as, to drive conclusions through interpreting the analysis results. The motivating problem is related to the establishment of women‐run enterprises in a rural area of Greece. Moreover, in this article, the spatial scan statistic is successfully applied to the spatial economic data at hand, in order to detect possible clusters of small women‐run enterprises in a rural mountainous and disadvantaged region of Greece. Then, it is combined with Geographical Information System based on Local Indicator of Spatial Autocorrelation scan statistic for further exploring and interpreting the spatial patterns. The rejection of the random establishment of women‐run enterprises and the interpretation of the clustering patterns are deemed necessary, in order to assist government in designing policies for rural development. Copyright © 2014 John Wiley & Sons, Ltd.     

Second-order optimality of randomized estimation and test procedures

Let P_{(Θ, τ)} 6 $\text{A}$, θ ∈ Θ ? $\text{R}$, τ ∈ T ? $\text{R}$^p denote a family of probability measures, where τ denotes the vector of nuisance parameters. Starting from randomized asymptotic maximum likelihood (as. m. l.) estimators for (θ, τ) we construct randomized estimators which are asymptotically median unbiased up to $\text{o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$ resp. test procedures which are as. similar of level $\text{α + o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$ (for testing θ = θ₀, τ ∈ T against one sided alternatives). The estimation procedures are second-order efficient in the class of estimators which are median unbiased up to $\text{o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$ and the test procedures are second-order efficient in the class of tests which are as. of level $\text{α + o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$. These results hold without any continuity condition on the family of probability measures.     

Two step-down tests for equality of covariance matrices

The classical problem of testing the equality of the covariance matrices from k ? 2 p-dimensional normal populations is reexamined. The likelihood ratio (LR) statistic, also called Bartlett’s statistic, can be decomposed in two ways, corresponding to two distinct component-wise decompositions of the null hypothesis in terms of the covariance matrices or precision matrices, respectively. The factors of the LR statistic that appear in these two decompositions can be interpreted as conditional and unconditional LR statistics for the component-wise null hypotheses, and their mutual independence under the null hypothesis allows the determination of the overall significance level.     

A topological reflection principle equivalent to Shelah's strong hypothesis

We notice that Shelah's Strong Hypothesis is equivalent to the following reflection principle:

Suppose is a first-countable space whose density is a regular cardinal, . If every separable subspace of is of cardinality at most , then the cardinality of is .

     

基于SVt-EVT-Vine Copula的投资组合VaR研究

针对现有风险度量模型不能准确的模拟高维金融资产收益率风险,以上证指数、沪深300指数和股指期货指数为例,首先利用SVt和EVT对各序列的边缘分布进行建模,然后采用Vine Copula方法分析多序列之间的秩相关关系和极大似然值估计法估计参数,得到RVine,CVine和DVine三种不同树结构的分解模型,通过Monte Carlo模拟法计算出在同一边缘分布不同Vine Copula方法下和在不同边缘分布同一Vine Copula方法下单资产和投资组合的金融风险VaR.经实证检验并分析对比,VaR和返回式检验均表明SVt和EVT相结合对边缘分布有较好的拟合效果,再运用RVine描述资产间的相依结构在度量投资组合金融风险方面更准确合理.     

Planning accelerated life tests with random effects of test chambers

In accelerated life tests (ALTs), test units are often tested in multiple test chambers along with different stress conditions. The nonhomogeneity of test chambers precludes the complete randomized experiment and may affect the life‐stress relationship of the test product. The chamber‐to‐chamber variation should be taken into account for ALT planning so as to obtain more accurate test results. In this paper, planning ALTs under a nested experimental design structure with random test chamber effects is studied. First, by a 2‐phase approach, we illustrate to what extent different test chamber assignments to stress conditions may impact the estimation of unknown parameters. Then, D‐optimal test plans with 2 test chambers are considered. To construct the optimal design, we establish the generalized linear mixed model for failure‐time data and apply a quasi‐likelihood method, where test chamber assignments, as well as other decision variables that are required for planning ALTs, are simultaneously determined.     

On certain character sums over

Let be the finite field with elements and let denote the ring of polynomials in one variable with coefficients in . Let be a monic polynomial irreducible in . We obtain a bound for the least degree of a monic polynomial irreducible in ( odd) which is a quadratic non-residue modulo . We also find a bound for the least degree of a monic polynomial irreducible in which is a primitive root modulo .

     

Some Simple Test Procedures for Normal Mean Vector with Incomplete Data

The problem of testing normal mean vector when the observations are missing from subsets of components is considered. For a data matrix with a monotone pattern, three simple exact tests are proposed as alternatives to the traditional likelihood ratio test. Numerical power comparisons between the proposed tests and the likelihood ratio test suggest that one of the proposed tests is indeed comparable to the likelihood ratio test and the other two tests perform better than the likelihood ratio test over a part of the parameter space. The results are extended to a nonmonotone pattern and illustrated using an example.