Locally minimax tests for multiple correlations

Let X be a normally distributed p-dimensional column vector with mean μ and positive definite covariance matrix σ. and let X α, α = 1,…, N, be a random sample of size N from this distribution. Partition X as ( X ₁, X ₍₂₎', X '₍₃₎)', where X₁ is one-dimension, X₍₂₎ is p₂- dimensional, and so 1 + p₁ + p₂ = p. Let ρ₁ and ρ be the multiple correlation coefficients of X₁ with X₍₂₎ and with ( X '₍₂₎, X '₍₃₎)', respectively. Write ρ2/2 = ρ² - ρ2/1. We shall cosider the following two problems     

Preference Aggregation With Incomplete Information

We show in an environment of incomplete information that monotonicity and the Pareto property applied only when there is common knowledge of Pareto dominance imply (i) there must exist a common prior over the smallest common knowledge event, and (ii) aggregation must be ex ante and ex post utilitarian with respect to that common prior and individual von Neumann–Morgenstern utility indices.     

关于勾股定理的起源

本文对勾股定理的起源作了初步的探讨，纠正了一些错误说法，并给出了一些新的见解。     

论马克思创立的实践人类学哲学

马克思实现了哲学视域的转换1.从“一切存在”转换到“人的社会存在”;2.从“人的类本质”转换到人类的生产交往共同体;3.确认“人类活动”是实践人类学哲学的思考框架;4.明确了哲学的目标理念人类的个体自由、生存合理性与发展规律性;5.强调了哲学的理论旨归从实践上改变“非人世界”,走向“自由人的联合体”;6.创立了哲学的实践批判原则以实践作为观察世界和社会历史批判的本源;7.确立了实践人类学哲学的边际关系与中心法则等。这几方面共同形成了马克思“改变世界”的实践人类学哲学,这是马克思一生理论关怀的总体特质。     

Games without Rules

We introduce the notion of an outcome space, in which strategic interactions are embedded. This allows us to investigate the idea that one strategic interaction might be an expanded version of another interaction. We then characterize the Nash equilibria arising in such extensions and demonstrate a folk-type theorem stating that any individually rational element of the outcome space is a Nash equilibrium.     

Choices Under Ambiguity With Familiar And Unfamiliar Outcomes

This paper considers a decision-making process under ambiguity in which the decision-maker is supposed to split outcomes between familiar and unfamiliar ones. She is assumed to behave differently with respect to unfamiliar gains, unfamiliar losses and customary (familiar) outcomes. In particular, she is supposed to be pessimistic on gains, optimistic on losses and ambiguity neutral on the familiar outcomes. A generalization of the usual Choquet Integral is formalized when the decision maker holds capacities and probabilities. A characterization of the decision-maker’s behavior is provided for a specific subset of capacities, in which it is shown that the decision-maker underestimates the unfamiliar outcomes while is linear in probabilities on customary ones.     

Two applications of a theorem of Dvoretsky, Wald, and Wolfovitz to cake division

In this note, we show that a partition of a cake is Pareto optimal if and only if it maximizes some convex combination of the measures used by those who receive the resulting pieces of cake. Also, given any sequence of positive real numbers that sum to one (which may be thought of as representing the players' relative entitlements), we show that there exists a partition in which each player receives either more than, less than, or exactly his or her entitlement (according to his or her measure), in any desired combination, provided that the measures are not all equal.     

Estimation of regression vectors in linear mixed models with Dirichlet process random effects

The Dirichlet process has been used extensively in Bayesian non parametric modeling, and has proven to be very useful. In particular, mixed models with Dirichlet process random effects have been used in modeling many types of data and can often outperform their normal random effect counterparts. Here we examine the linear mixed model with Dirichlet process random effects from a classical view, and derive the best linear unbiased estimator (BLUE) of the fixed effects. We are also able to calculate the resulting covariance matrix and find that the covariance is directly related to the precision parameter of the Dirichlet process, giving a new interpretation of this parameter. We also characterize the relationship between the BLUE and the ordinary least-squares (OLS) estimator and show how confidence intervals can be approximated.