The Likelihood Ratio Test for the Presence of Immunes in a Censored Sample

Models for populations with immune or cured individuals but with others subject to failure are important in many areas, such as medical statistics and criminology. One method of analysis of data from such populations involves estimating an immune proportion 1 ? p and the parameter(s) of a failure distribution for those individuals subject to failure. We use the exponential distribution with parameter λ for the latter and a mixture of this distribution with a mass 1 ? p at infinity to model the complete data. This paper develops the asymptotic theory of a test for whether an immune proportion is indeed present in the population, i.e., for H ₀:p = 1. This involves testing at the boundary of the parameter space for p. We use a likelihood ratio test for H ₀. and prove that minus twice the logarithm of the likelihood ratio has as an asymptotic distribution, not the chi-square distribution, but a 50–50 mixture of a chi-square distribution with 1 degree of freedom, and a point mass at 0. The result is proved under an independent censoring assumption with very mild restrictions.     

史前浙江——一个区域性的文化格局

诞生于钱塘江上游地区的上山文化、跨湖桥文化,是浙江史前史的立足点.从河姆渡文化开始,钱塘江史前文化与以江淮文化为主体的北域文明交流、碰撞,发展出马家浜文化,成就了一个以挑战、接受、壮大为建构的文化格局,最终创造出在中华文明中占据一席之地的良渚文明和吴越文明.交流、碰撞需要有自己的力量,良渚文明和吴越文明的形成依托于原生文化的质量与生命力.     

罪名与罪行——江阴县私立青阳中学校长谢家玉破坏土改案解读

1950年10月23日,江阴县私立青阳中学校长谢家玉突然被逮捕.政府方面宣布他犯有包庇地主、出卖田地、隐瞒田亩、逃避秋征、开地主会议5项罪名.青阳中学校董会由当地在上海的工商界及文教界人士组成.谢家玉被逮捕后,青阳中学方面即与政府交涉,但最终,谢家玉仍然被判刑.通过对谢家玉的所谓5项罪状进行分析后发现,政府方面提出的罪名都不成立.这桩错案的产生是由于政府方面有意在土改过程中消灭乡村中原有地方精英的政治影响,可见当时的法制化水平受到那个时代社会革命的总体条件的限制.     

高校二级学院党建工作的研究与探索

二级学院党建工作是高校党建工作的重要组成部分。随着高校的扩招、高校体制的改革和社会的变革,二级学院党建工作面临着诸多新问题,积极研究探索二级学院党建工作的新形式、新思路,使党建工作更有成效,是当前高校党建工作者面临的一个重要课题。文章分析了高校二级学院出现的新情况,研究了二级学院党建工作面临的新问题,提出加强二级学院党建工作的几点建议。     

Estimation in one–way classification with heteroscedastic error

For an unbalanced one way calssification under the random effect model the problem of estimation of the fixed effect parameter and the variance is considered. Tje error variance which are funtionally related to the above set of parameters are assumed to fall into k classes with constant error varaince for a class. The asymptotic properties of the proposed estimate is established for increasing number of classes k, assuming the number of observations in the classes form a fixed sequence     

Improved orthogonal polynomial density estimates

Density estimates that are expressible as the product of a base density function and a linear combination of orthogonal polynomials are considered in this paper. More specifically, two criteria are proposed for determining the number of terms to be included in the polynomial adjustment component and guidelines are suggested for the selection of a suitable base density function. A simulation study reveals that these stopping rules produce density estimates that are generally more accurate than kernel density estimates or those resulting from the application of the Kronmal–Tarter criterion. Additionally, it is explained that the same approach can be utilized to obtain multivariate density estimates. The proposed orthogonal polynomial density estimation methodology is applied to several univariate and bivariate data sets, some of which have served as benchmarks in the statistical literature on density estimation.     

Second order asymptotic and non-asymptotic optimality properties of combined tests

Let p independent test statistics be available to test a null hypothesis concerned with the same parameter. The p are assumed to be similar tests. Asymptotic and non-asymptotic optimality properties of combined tests are studied. The asymptotic study centers around two notions. The first is Bahadur efficiency. The second is based on a notion of second order comparisons. The non-asymptotic study is concerned with admissibility questions. Most of the popular combining methods are considered along with a method not studied in the past. Among the results are the following: Assume each of the p statistics has the same Bahadur slope. Then the combined test based on the sum of normal transforms, is asymptotically best among all tests studied, by virtue of second order considerations. Most of the popular combined tests are inadmissible for testing the noncentrality parameter of chi-square, t, and F distributions. For chi-square a combined test is offered which is admissible, asymptotically optimal (first order), asymptotically optimal (second order) among all tests studied, and for which critical values are obtainable in special cases. Extensions of the basic model are given.     

Fitting a multiple regression function

Consider the p-dimensional unit cube [0,1]^p, p≥1. Partition [0, 1]^p into n regions, R_1,n,…,R_n,n such that the volume Δ(R_j,n) is of order n^?1,j=1,…,n. Select and fix a point in each of these regions so that we have x⁽ⁿ⁾₁,…,x⁽ⁿ⁾_n. Suppose that associated with the j-th predictor vector x⁽ⁿ⁾_j there is an observable variable Y⁽ⁿ⁾_j, j=1,…,n, satisfying the multiple regression model $\text{Y}{}^{\mathrm{(n)}}{}_{\mathrm{j}}\text{=g(}\text{x}{}^{\mathrm{(n)}}{}_{\mathrm{j}}\text{)+e}{}^{\mathrm{(n)}}{}_{\mathrm{j}}$, where g is an unknown function defined on [0, 1]^pand {e⁽ⁿ⁾_j} are independent identically distributed random variables with Ee⁽ⁿ⁾₁=0 and Var e⁽ⁿ⁾₁=σ²<∞. This paper proposes as an estimator of g(x), where k(u) is a known p-dimensional bounded density and {a_n} is a sequence of reals converging to 0 asn→∞. Weak and strong consistency of g_n(x) and rates of convergence are obtained. Asymptoticnormality of the estimator is established. Also proposed is $\text{σ}{}^2{}_{\mathrm{n}}\text{=n}{}^{\mathrm{?1}}\text{Σ}{}^{\mathrm{n}}{}_{\mathrm{j=1}}\text{(Y}{}^{\mathrm{(n)}}{}_{\mathrm{j}}\text{?g}{}_{\mathrm{n}}\text{(}\text{x}{}^{\mathrm{(n)}}{}_{\mathrm{j}}\text{))}{}^2$ as a consistent estimate of σ².     

On the influence curve for quantal bioassay

Influence curves are defined for estimators of the ED50 in quantal bioassay, assuming a symmetic tolerance distribution. A definition of influence-curve robustness is suggested. Influence curves are obtained for L-,M- and R-estimators, and their robustness is investigated. The influence curve of the logistic maximum likelihood estimator is studied.     

Linear discrimination for three known normal populations

A random vector is assumed to have one of three known multivariate normal distributions with equal covariance matrices. It is desired to separate the three distributions by means of a single linear discriminant function. Such a function can lead to a classification rule. The function whose classification rule minimizes the average of the three probabilities of misclassification is found. Also the function is found whose rule minimizes the maximum of the three probabilities of misclassification.