Best affine unbiased response decomposition

Given two linear regression models y₁=X₁β₁+u₁ and y₂=X₂β₂+u₂ where the response vectors y₁ and y₂ are unobservable but the sum y=y₁+y₂ is observable, we study the problem of decomposing y into components and , intended to be close to y₁ and y₂, respectively. We develop a theory of best affine unbiased decomposition in this setting. A necessary and sufficient condition for the existence of an affine unbiased decomposition is given. Under this condition, we establish the existence and uniqueness of the best affine unbiased decomposition and provide an expression for it.     

Statistical Estimation of the Shannon Entropy

The behavior of the Kozachenko–Leonenko estimates for the(differential) Shannon entropy is studied when the number of i.i.d. vector-valued observations tends to infinity. The asymptotic unbiasedness and L~2-consistency of the estimates are established. The conditions employed involve the analogues of the Hardy–Littlewood maximal function. It is shown that the results are valid in particular for the entropy estimation of any nondegenerate Gaussian vector.     

Admissibility of unbiased tests for a composite hypothesis with a restricted alternative

This paper discusses -admissiblility and d-admissiblity which are important concepts in studying the performance of statistical tests for composite hypotheses. A sufficient condition for -admissibility is presented. When =1/m, the Nomakuchi-Sakata test, which is uniformly more powerful than the likelihood ratio test for hypotheses min (₁, ₁) = 0 versus min (₁, ₁) > 0, is generalized for a class of distributions in an exponential family, and its unbiasedness and -admissibility are shown. Finally, the case of 1/m is discussed in brief.     

Addendum to “a third-order optimum property of the maximum likelihood estimator”

A third-order optimum property of the maximum likelihood estimator is extended to not necessarily symmetric loss functions under an appropriate restriction on the class of competing estimators.     

无重复因析试验中散度效应的ML估计

在无重复因析试验中,若因子A,B的散度效应显著,则不论其交互效应AB的散度效应是否显著,其散度效应的现有估计常常是有偏的,从而导致其被错误地识别为显著效应.提出了散度效应的一种新的估计方法(称为ML估计),并给出了ML估计的方差的精确表达形式,证明了在一类模型中,交互效应AB的散度效应的ML估计是无偏的.最后,将ML估计与现有的常用估计进行了比较.     

On the second order local comparison between perturbed maximum likelihood estimators and Rao's statistic as test statistics

A bias-adjusted maximum likelihood estimator (mle), which has been shown to possess certain optimality criteria as an estimate of θ (see Ghosh, J. K., Sinha, B. K., and Joshi, S. N. (1982). In Statistical Decision Theory and Related Topics III (S. S. Gupta and J. O. Berger, Eds.), Vol. 1, pp. 403–456. Academic Press, Orlando, FL) is compared with Rao's statistic as a test statistic for the standard two-sided testing problem. It is shown that Rao's statistic is locally superior to any bias-adjusted mle in the sense of Chandra and Joshi (1983, Sankhy Ser. A 45 226–246). A second interpretation of a conjecture of C. R. Rao is proposed and Rao's statistic is shown to be superior as a test statistic according to the new interpretation as well. The last fact provides an interesting supplement to the results of Chandra and Joshi (1983, Sankhy Ser. A 45 226–246). Furthermore, a partial reason for the inferiority of the likelihood ratio and Wald's statistic to Rao's statistic is supplied and certain regularity assumptions of the last paper are eliminated. Finally, the local powers of certain modified versions of Rao's and Wald's statistics (see Skovgaard, I. M. (1985). Ann. Statist. 13 534–551) are studied.     

关于MLE在渐近中位无偏意义下的渐近有效性

本文讨论正则条件下极大似然估计在渐近中位无偏意义下的渐近有效性。     

关联实验数据的4种χ^2形式及其性质

研究了关联实验数据的四种因子化形式的χ^2表达式及其极小化性质，同时讨论了极小化估计值的有偏性与无偏性．利用简化的R值测量定量地检验了相关的理论结果．     

Unbiased multi-index Monte Carlo

We introduce a new class of Monte Carlo-based approximations of expectations of random variables such that their laws are only available via certain discretizations. Sampling from the discretized versions of these laws can typically introduce a bias. In this paper, we show how to remove that bias, by introducing a new version of multi-index Monte Carlo (MIMC) that has the added advantage of reducing the computational effort, relative to i.i.d. sampling from the most precise discretization, for a given level of error. We cover extensions of results regarding variance and optimality criteria for the new approach. We apply the methodology to the problem of computing an unbiased mollified version of the solution of a partial differential equation with random coefficients. A second application concerns the Bayesian inference (the smoothing problem) of an infinite-dimensional signal modeled by the solution of a stochastic partial differential equation that is observed on a discrete space grid and at discrete times. Both applications are complemented by numerical simulations.     

关联实验数据线性函数拟合情况下无偏x~2估计量的构造(英文)

针对线性函数拟合的情况构造了一种无偏的因子化形式的x~2估计量,用以处理关联数据的极小化.详细比较并说明了有偏性与无偏性x~2拟合的区别.利用简化的R值测量数据对相关的结论进行了定量的检验.