Augmenting lehmann alternatives for improving asymptotic efficiency of rank tests

Locally most powerful tests for augmented simple Lehmann alternatives are obtained. These tests turn out to be linearcombinations of the Savage and Mann-Whitney-Wilcoxon test criteria. We study their performance in terms of the asymptotic efficiency relative to their parametric competitors against location and scale alternatives. For small sample sizes, critical points of a couple of test procedures are given.     

Linear signed rank tests for hultivariate location

Previously proposed linear signed rank tests for multivariate location are not invariant under linear transformations of the observations, The asymptotic relative efficiencies of the tests 2 with respect to Hotelling's T²test depend on the direction of shift and the covariance matrix of the alternative distributions. For distributions with highly correlated components, the efficiencies of some of these tests can be arbitrarily low; they approach zero for certain multivariate normal alternatives, This article proposes a transformation of the data to be performed prior to standard linear signed rank tests, The resulting procedures have attractive power and efficiency properties compared to the original tests, In particular, for elliptically symmetric contiguous alternafives, the efficiencies of the new tests equal those of corresponding univariate linear signed rank tests with respect to the t test.     

QUALITY IMPROVEMENT FROM DISASSEMBLY-REASSEMBLY EXPERIMENTS

The goal of achieving high quality products has led to an emphasis on reducing variation in performance characteristics. It may often happen that one of the product's components is responsible for much of the observed variation. This research is stimulated by the problem of detecting a component that impairs quality by systematically inflating the variance in a product that is assembled from “interchangeable components.” We consider the class of “disassembly-reassembly” experiments, in which components are swapped among assemblies. The specific units used in the experiment are sampled from a large population of units, so it is natural to measure the influence of each factor by its variance component. We present the model for these experiments as a special case of the mixed linear model, compare several estimators for the variance components and consider the problem of testing hypotheses to identify troublesome components.     

Minimax estimation of a lower-bounded scale parameter of a gamma distribution for scale-invariant squared-error loss

Let X have a gamma distribution with known shape parameter θr;aL and unknown scale parameter θ. Suppose it is known that θ ≥ a for some known a > 0. An admissible minimax estimator for scale-invariant squared-error loss is presented. This estimator is the pointwise limit of a sequence of Bayes estimators. Further, the class of truncated linear estimators C = {θ_ρ|θ_ρ(x) = max(a, ρ), ρ > 0} is studied. It is shown that each θ_ρ is inadmissible and that exactly one of them is minimax. Finally, it is shown that Katz's [Ann. Math. Statist., 32, 136–142 (1961)] estimator of θ is not minimax for our loss function. Some further properties of and comparisons among these estimators are also presented.     

Edgeworth expansions for signed linear rank statistics under near location alternatives

Edgeworth expansions with the uniform remainder of order o(N⁻¹) are established for signed linear rank statistics with regression constants under near location alternatives. The results are obtained both with exact scores and with approximate scores, normalized with natural parameters as well as with simple constants.     

Estimation of a distribution function by extrapolating upper and lower bounds

Constrained optimization is proposed as a practical solution to the problem of estimating a distribution function at each point in a given set from monotone sequences of upper and lower bounds. The proposed solution employs least absolute value estimation and, hence, has a linear programming formulation. The special structure inherent in this formulation is exploited and an efficient computational method is discussed. The procedure is illustrated by two examples.     

法律上的“唯一正解”——从德沃金的学说谈起

司法判断中是否存在“唯一正解”的问题长期聚讼纷纭。当代法理学大师德沃金针对哈特的司法裁量理论而提出在包括疑难案件在内的任何案件中都存在一个“唯一正解”,因而受到了波斯纳从经济分析法学立场出发的批评,而更有力的挑战则可能来自当代的“不可公度”理论,但各家的学说实际上乃立足于不同的底盘。在价值多元流动的困境中,人类依据目前的智识和智慧尚无法对这个问题得出肯定的回答与否定的回答,为此只能将求得解决问题的理想委之于法的实践理性。     

Parameter estimation under orthant restrictions

The authors consider the estimation of linear functions of a multivariate parameter under orthant restrictions. These restrictions are considered both for location models and for the Poisson distribution. For these models, situations are characterized for which the restricted maximum likelihood estimator dominates the unrestricted one for the estimation of any linear function of the parameter. The results obtained point directly to the importance of the dimension of the parameter space, the central direction of the cone and its vertex in these cases. Special attention is given to examples, such as the one‐way analysis of variance, where the estimation of individual interesting linear functions of the parameter, as the coordinates and the differences between them, is also treated.     

Inference for multivariate normal hierarchical models

This paper provides a new method and algorithm for making inferences about the parameters of a two-level multivariate normal hierarchical model. One has observed J p -dimensional vector outcomes, distributed at level 1 as multivariate normal with unknown mean vectors and with known covariance matrices. At level 2, the unknown mean vectors also have normal distributions, with common unknown covariance matrix A and with means depending on known covariates and on unknown regression coefficients. The algorithm samples independently from the marginal posterior distribution of A by using rejection procedures. Functions such as posterior means and covariances of the level 1 mean vectors and of the level 2 regression coefficient are estimated by averaging over posterior values calculated conditionally on each value of A drawn. This estimation accounts for the uncertainty in A , unlike standard restricted maximum likelihood empirical Bayes procedures. It is based on independent draws from the exact posterior distributions, unlike Gibbs sampling. The procedure is demonstrated for profiling hospitals based on patients' responses concerning p =2 types of problems (non-surgical and surgical). The frequency operating characteristics of the rule corresponding to a particular vague multivariate prior distribution are shown via simulation to achieve their nominal values in that setting.     

Truncated linear estimation of a bounded multivariate normal mean

We consider the problem of estimating the mean θ$\theta$ of an _Np(θ,_Ip)${\mathrm{N}}_{\mathrm{p}}(\theta ,I_p)$ distribution with squared error loss ∥δ−θ∥²$\parallel \delta -\theta {\parallel }^2$ and under the constraint ∥θ∥≤m$\parallel \theta \parallel \le m$, for some constant m>0$m>0$. Using Stein's identity to obtain unbiased estimates of risk, Karlin's sign change arguments, and conditional risk analysis, we compare the risk performance of truncated linear estimators with that of the maximum likelihood estimator _δmle${\delta }_{\mathrm{mle}}$. We obtain for fixed (m,p)$(m,p)$ sufficient conditions for dominance. An asymptotic framework is developed, where we demonstrate that the truncated linear minimax estimator dominates _δmle${\delta }_{\mathrm{mle}}$, and where we obtain simple and accurate measures of relative improvement in risk. Numerical evaluations illustrate the effectiveness of the asymptotic framework for approximating the risks for moderate or large values of p.