Another characterization of the weibull distribution

A necessary and sufficient condition is given that an absolutely continuous positive random variable follow a Weibull distribution. The condition Is based on some functions of order statistics.     

On the estimation of restricted mean

In sampling from a continuous distribution with unknown mean μ and variance σ² the problem of estimation of μ, when it is known that μ∈(a, ∞) (or μ∈(-∞, b)), is considered. The estimators proposed here lie in the interval (a, ∞) (or (-∞, b)) almost surely. The performance of these estimators is compared to that of some known estimators in the case of sampling from a normal, exponential and a weighted difference of two independent chi-square distributions.     

Optimum designs in complete two-way layouts

In the usual two-way layout of ANOVA (interactions are admitted) let nij ? 1 be the number of observations for the factor-level combination(i, j). For testing the hypothesis that all main effects of the first factor vanish numbers $\text{n}{}^1\text{ij}$ are given such that the power function of the F-test is uniformly maximized (U-optimality), if one considers only designs (nij) for which the row-sums ni are prescribed. Furthermore, in the (larger) set of all designs for which the total number of observations is given, all D-optimum designs are constructed.     

Some complex variable transformations and exact power comparisons of two-sided tests of equality of two Hermitian covariance matrices

Power studies of tests of equality of covariance matrices of two p-variate complex normal populations σ₁ = σ₂ against two-sided alternatives have been made based on the following five criteria: (1) Roy's largest root, (2) Hotelling's trace, (4) Wilks' criterion and (5) Roy's largest and smallest roots. Some theorems on transformations and Jacobians in the two-sample complex Gaussian case have been proved in order to obtain a general theorem for establishing the local unbiasedness conditions connecting the two critical values for tests (1)–(5). Extensive unbiased power tabulations have been made for p=2, for various values of n₁, n₂, λ₁ and λ₂ where n₁ is the df of the SP matrix from the ith sample and λ₁ is the ith latent root of σ₁σ^-1₂ (i=1, 2). Equal tail areas approach has also been used further to compute powers of tests (1)–(4) for p=2 for studying the bias and facilitating comparisons with powers in the unbiased case. The inferences have been found similar to those in the real case. (Chu and Pillai, Ann. Inst. Statist. Math. 31.     

Estimation of a linear transformation and an associated distributional problem

A multivariate “errors in variables” regression model is proposed which generalizes a model previously considered by Gleser and Watson (1973). Maximum likelihood estimators [MLE's] for the parameters of this model are obtained, and the consistency properties of these estimators are investigated. Distribution of the MLE of the “error” variance is obtained in a simple case while the mean and the variance of the estimator are obtained in this case without appealing to the exact distribution.     

10-14世纪北方少数民族政权的经学

宋元时期,北方各少数民族在黄河流域建立政权以后,即开始自觉地学习和吸纳汉文化,实行崇儒尊经的文教政策,并开始了对经学的研习和传播。由于北方与南方的文化环境不同,因而经学研究也表现出明显的地域差异。北方长达４００年处于游牧民族的统治之下,杂染胡风,重实际,轻玄想,故其经学研究以六经为主,注重经史结合,从中寻求治国的道理;虽然倡扬程朱理学,但仍不废汉唐注疏,具有经世致用、平实质朴的特点。     

Efficiency of weighted averages

When combining estimates of a common parameter (of dimension d?1$d?1$) from independent data sets—as in stratified analyses and meta analyses—a weighted average, with weights ‘proportional’ to inverse variance matrices, is shown to have a minimal variance matrix (a standard fact when d=1$d=1$)—minimal in the sense that all convex combinations of the coordinates of the combined estimate have minimal variances. Minimum variance for the estimation of a single coordinate of the parameter can therefore be achieved by joint estimation of all coordinates using matrix weights. Moreover, if each estimate is asymptotically efficient within its own data set, then this optimally weighted average, with consistently estimated weights, is shown to be asymptotically efficient in the combined data set and avoids the need to merge the data sets and estimate the parameter in question afresh. This is so whatever additional non-common nuisance parameters may be in the models for the various data sets. A special case of this appeared in Fisher [1925. Theory of statistical estimation. Proc. Cambridge Philos. Soc. 22, 700–725.]: Optimal weights are ‘proportional’ to information matrices, and he argued that sample information should be used as weights rather than expected information, to maintain second-order efficiency of maximum likelihood. A number of special cases have appeared in the literature; we review several of them and give additional special cases, including stratified regression analysis—proportional-hazards, logistic or linear—, combination of independent ROC curves, and meta analysis. A test for homogeneity of the parameter across the data sets is also given.