The Structure of a Linear Model: Sufficiency, Ancillarity, Invariance, Equivariance, and the Normal Distribution

Consider a general linear model Y=Xβ+Z where Cov Z may be known only partially. We investigate carefully the notions of sufficiency, ancillarity, invariance, and equivariance and related notions for projectors in a general linear model. In this way we can prove a Basu type theorem. This result can be used to give the relation between the sufficiency of the generalized least-squares estimator and the assumption that Z is normally distributed. So we can generalize the well-known result that the generalized least-squares estimator is sufficient for β if Z is normally distributed. Further we can solve the converse problem as well.     

套重复测量模型的充分性和估计

当每一个体有相同的子个体,并且每一子个体的处理水平是成对的时候,我们使用套重复测量模型.令Yi=(Yilll,…Yimrc)′是第i个个体的观测向量.假设Yi为相互独立的正态分布,均值为μi,协方差阵为∑＞0.假设可简化为所有测量值的方差为σ2;相同个体的不同子个体之间的成对测量值之间的关系如下(1)不同列不同行的观测值;(2)相同列不同行的测量值;(3)相同行不同列的测量值,它们的协方差分别为ρ2σ2,ρ3σ2,ρ4σ2.我们假设试验是给定的,用坐标自由(coordinate-free)的方法研究了套重复测量模型的完备充分统计量,最小方差无偏估计(MVUE)和极大似然估计(MLE).     

Sufficiency and fuzziness in random experiments

In previous papers, the consequences of the “presence of fuzziness” in the experimental information on which statistical inferences are based were discussed. Thus, the intuitive assertion «fuzziness entails a loss of information» was formalized, by comparing the information in the “exact case” with that in the “fuzzy case”. This comparison was carried out through different criteria to compare experiments (in particular, that based on the “pattern” one, Blackwell's sufficiency criterion). Our purpose now is slightly different, in the sense that we try to compare two “fuzzy cases”. More precisely, the question we are interested in is the following: how will different “degrees of fuzziness” in the experimental information affect the sufficiency? In this paper, a study of this question is carried out by constructing an alternative criterion (equivalent to sufficiency under comparability conditions), but whose interpretation is more intuitive in the fuzzy case. The study is first developed for Bernoulli experiments, and the coherence with the axiomatic requirements for measures of fuzziness is also analyzed in such a situation. Then it is generalized to other random experiments and a simple example is examined.     

Sufficiency conditions for multiple integral control problems

The construction of multiple integrals which are independent, in the sense that they depend solely on the values of their integrands on the boundary of the domain of integration is described. These integrals are applied to the derivation of a sufficiency condition for multiple integral optimal control problems.This research was supported in part by NSF Grant No. GP-32830.     

Premium principles and translation invariance

Premium principles τ are examined with respect to the property of translation invariance, that means τ(X + c) = τ(X) + c holds for all real c (and all risks X). It is shown that this condition of translation invariance is highly overdetermined: For all known principles the range of c can be reduced to a large extent from all the reals to only two (or one) single values of c respectively.     

Modular invariance and anomaly cancellation formulas

By studying modular invariance properties of some characteristic forms, we get some new anomaly cancellation formulas. As an application, we derive some results on divisibilities on spin manifolds and congruences on spin ^c manifolds.     

Theorems of large deviations in the multivariate invariance principle

Let B be a real separable Banach space with norm |ß|_B, X, X₁, X₂, … be a sequence of centered independent identically distributed random variables taking values in B. Let s_n = s_n(t), 0 ≤ t ≤ 1 be the random broken line such that s_n(0) = 0, s_n(k/n) = n^−1/2 Σ_i=1^k X_i for n = 1, 2, … and k = 1, …, n. Denote |s_n|_B = sup_{0 ≤ t ≤ 1} |s_n(t)|_B and assume that w(t), 0 ≤ t ≤ 1 is the Wiener process such that covariances of w(1) and X are equal. We show that under appropriate conditions P(|s_n|_B > r) = P(|w|_B > r)(1 + o(1)) and give estimates of the remainder term. The results are new already in the case of B having finite dimension.     

An invariance theorem for mappings

If a continuous mapping f carries the boundary of a set D into the closure of D, it may not be true that f maps D into its closure, even if f is injective on D. Examples are discussed, and conditions are given under which it is true. A simplified application is given to a biological migration–selection model.     

Solution structures of tensor complementarity problem

We introduce two new types of tensors called the strictly semimonotone tensor and the range column sufficient tensor and explore their structure properties. Based on the obtained results, we make a characterization to the solution of tensor complementarity problem.     

Scoring indices,top-truncated preferences,and splitting invariance

We present a new characterization of the class of weight-based scoring indices for ranking problems with top-truncated preferences. The main novel axiom is Splitting Invariance: if an alternative is split up into a number of distinct yet unranked alternatives, then the total score of these alternatives should increase by the score of the original alternative, and the scores of the other alternatives should not change.     

Gauge invariance and asymptotic behavior for the Ginzburg-Landau equations of superconductivity

In this paper we study the gauge invariance of the time-dependent Ginzburg-Landau equations through the introduction of a model which uses observable variables. We observe that the various choices of gauge lead to a different representation of such variables and therefore to a different definition of the weak solution of the problem. With a suitable decomposition of the unknown fields, related to the choice of London gauge, we examine the Ginzburg-Landau equations and deduce some energy estimates which prove the existence of a maximal attractor for the system.     

On the domain invariance of countably condensing vector fields

Using homotopy theory, we give the domain invariance theorem for countably condensing vector fields, where the notion of countably condensing maps is due to Väth. A starting point of this investigation is that there is a symmetric characteristic set for a countably condensing map.     

边界提取和几何不变性相结合的标定方法研究

通过边界提取和几何不变性研究一种新的快速标定方法．首先设计了一个基于闽值的边界提取算法用于获得物体相片的轮廓;然后给出了标定过程中关于几何不变性的几个命题,并用于确定充分多的输入输出对应关系;最后建立了一个非线性优化问题求解视觉系统．数值求解结果显示了本文方法的准确性和有效性．     

Gauge invariance of the dimension-two condensate in the Yang-Mills theory

We demonstrate the gauge independence of the dimension-two vacuum condensate in the Yang-Mills theory using a noncommutative theory technique.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 1, pp. 3–8, April, 2005.     

Uniform invariance principle of discontinuous systems with parameter variation

An extension of the invariance principle for a class of discontinuous righthand sides systems with parameter variation in the Filippov sense is proposed. This extension allows the derivative of an auxiliary function V, also called a Lyapunov-like function, along the solutions of the discontinuous system to be positive on some sets. The uniform estimates of attractors and basin of attractions with respect to parameters are also obtained. To this end, we use locally Lipschitz continuous and regular Lyapunov functions, as well as Filippov theory. The obtained results settled in the general context of differential inclusions, and through a uniform version of the LaSalle invariance principle. An illustrative example shows the potential of the theoretical results in providing information on the asymptotic behavior of discontinuous systems.     

Power Utility Maximization Problems Under Partial Information and Information Sufficiency in a Brownian Setting

We consider the problem of expected power utility maximization from terminal wealth in diffusion market models under partial information. After obtaining novel neat expressions for the value-process and for the optimal strategy, the issue of information sufficiency is addressed. In particular, necessary and sufficient conditions that guarantee that the partial information optimal strategy is still optimal when having access to all market information, are provided.     

Determination of means by invariance

If M is a mean on and M(f(x₁),f(x₂),…,f(xn))=f(M(x₁,x₂,…,xn)) then we say that M is invariant under f. The problem is to find a class of functions that by invariance determines a mean uniquely. We focus on the geometric mean, which can be transformed to obtain results for other means.     

On extensions of the domain invariance theorem

If f maps continuously a compact subset X of Rⁿ into Rⁿ and x is a point whose distance from the boundary ∂X is greater than double diameter of the fibres of the points in f(∂X) then f(x) is in the interior of f(X). This theorem extends some results due to Borsuk and Sitnikov.