On two market equilibrium theorems

Two theorems are presented which have as corollaries known market equilibrium theorems. The first theorem is for correspondences defined on sets which are compact and convex and the second theorem is for correspondences defined on the unit sphere. The proofs of both theorems rely upon one lemma.     

“二维LP问题的一个直接算法”中定理反例

针对"二维LP问题的一个直接算法"一文中的算法基本定理,给出了两个反例,分别说明其中的引理和定理都是错误的,建立在这些结论基础上的求解一般线性规划问题的代数算法无法求出一般线性规划问题的最优解。数值试验表明,随着方程个数的增加,用他们的方法求得正确解的概率将越来越低。给出了一个二维线性规划问题最优解的性质定理,由该定理可导出文[1]中代数算法有效的条件。     

A lemma on relative stability with an application to an economic case study

In the existing literature relative stability of trajectories is obtained under particular assumptions. In this paper a lemma is stated which provides a technique to prove relative stability in most general cases. Such a technique permits to replace the study of the stability of a balanced growth path by the study of the stability of a fixed point for an associated function. The lemma is applied to a case study, which could not be treated by means of usual techniques.     

Recovering the production technology from the cost function

An alternative technique for recovering the technology from the cost function is investigated. The recovery method uses an optimization problem with a finite number of constraints that, under appropriate conditions, has, the equations of Shephard's lemma among the first order conditions. Hence, a justification is provided for the technique of recovering the production function from the cost function by solving the system of equations generated by Shephard's lemma. The alternative technique also recovers the output distance function from the cost function in the multioutput case.The editor for this paper was R. Robert Russell.     

Über lokal optimale Rang-Tests

Zusammenfassung Ausgehend von einem bekannten Lemma von Hoeffding werden verschiedene lokale Optimalit?tseigenschaften einseitiger Rang-Tests rekapituliert und die korrespondierenden Optimalit?tseigenschaften der zugeh?rigen zweiseitigen Rang-Tests bewiesen (u.a. die Optimalit?t des zweiseitigen Wilcoxon-Tests). Dabei wird ein Optimalit?tsbegriff zu Grunde gelegt, der sich aus der Auffassung des zweiseitigen Tests als kombiniertes Test-und Klassifikations-Verfahren ergibt.

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Summary Starting from a well known lemma of Hoeffding, several local optimum properties of one sided rank tests are recapitulated. In addition, the corresponding optimum properties for the related two sided rank tests are established (including for example the optimality of the two-sided Wilcoxon test), using a concept of optimality which arises from a combined test and classification process.

     

On testing the correlation coefficient of a bivariate normal distribution

Summary Here we propose two tests for testing in the bivariate normal population assuming that the ratio of the variances in it is known. The first test (U.M.P.U.) is derived by using the Neyman-Pearson lemma, whereas the second test is obtained through testing the scale parameter of the Cauchy distribution. The powers of the first and second tests are compared with a well-known test, based on the sample correlation coefficient for small and large samples respectively.     

On a lemma associated with Box,Jenkins and Granger

The lemma that the aggregate of independent moving average processes is also moving average is discussed, and a simple proof presented. This proof depends on the fact that, for any order q, the vectors of MA autocorrelations (?₁…?_q) occupy a convex space.     

实物期权理论中的数学基础

资产价值运动路径的模型的随机过程处理及关于随机过程函数的工具就是Ito引理和评价投资价值的基础方法即实物期权评价方法。设F(S t,t)为资产的价格,它是S t的函数。这两种效应之和就是随机微分dF(S t,t)令S t是连续型随机过程。     

Some results on classifier selection with missing covariates

Methods are proposed to choose classifiers from a given collection of classifiers when there are missing covariates in the data. Two situations are considered: (i) the case where the new observation (to be classified) has no missing covariates and (ii) the case where the new observation is also allowed to have missing covariates. Using arguments from the empirical process theory, exponential performance bounds will be derived for the resulting classifiers. Such bounds, together with the Borel-Cantelli lemma, yield various strong consistency results.     

Equilibria in a large production economy with an infinite dimensional commodity space and price dependent preferences

We prove the existence of a competitive equilibrium in a production economy with infinitely many commodities and a measure space of agents whose preferences are price dependent. We employ a saturated measure space for the set of agents and apply recent results for an infinite dimensional separable Banach space such as Lyapunov’s convexity theorem and an exact Fatou’s lemma to obtain the result.     

Addendum to L. Lauwers and L. Van Liedekerke, “Ultraproducts and aggregation” [J. Math. Econ. 24 (3) (1995) 217–237]

This addendum fills a minor gap in the key lemma of a paper by Lauwers and Van Liedekerke [Lauwers, L., Van Liedekerke, L., 1995. J. Math. Econ. 24 (3), 217–237].     

Existence of equilibria in atomless production economies with price dependent preferences

The existence of a competitive equilibrium for an economy with a measure space of agents is proved by reducing the economy to a three-person game and then applying Debreu's lemma (1952). In addition to the relative simplicity and shortness of the proof, we generalize known results [Aumann (1966), Hildenbrand (1970), Schmeidler (1969)] by allowing both price dependent preferences and satiation.     

Pyrrho's lemma,or have it your way

A simple lemma is derived to support the claim that regression models can be manipulated to a very large extent: by simply adding one regressor one can obtain essentially every set of desired regression coefficients and predictions as well ast-values and standard errors. Consequently, if the product of a specification search is not shown to be generalizable, a sceptical attitude towards its validity is well-founded.     

General conditions for the existence of maximal elements via the uncovered set

This paper disentangles the topological assumptions of classical results (e.g.,Walker, 1977 on the existence of maximal elements from rationality conditions. It is known from the social choice literature that under the standard topological conditions—with no rationality assumptions on preferences—there is an element such that the upper section of strict preference at that element is minimal in terms of set inclusion, i.e., the uncovered set is nonempty. Assuming the finite subordination property, a condition that weakens known acyclicity and convexity assumptions, each such uncovered alternative is in fact maximal. Implications are a generalization of a result of Yannelis and Prabhakar (1983) on semi-convexity, an extension of Fan’s (1961) lemma on KKM correspondences, and the existence of fixed points for subordinate convex correspondences generalizing the work of Browder (1968).     

Existence of equilibrium actions and of equilibrium : A note on the ‘new’ existence theorems

A lemma by Ky Fan, which asserts the existence of a maximal element relative to some ordering (an equilibrium action) in a compact choice set, is extended to a larger class of orderings. This extension allows one to generalize slightly several existence theorems and seems to be useful for the applications. Kakutani's fixed-point theorem and the existence of an equilibrium in a generalized game are used to exemplify this. In the last case the method of proof may be of some independent interest involving, as it does, the reduction of an n-person game to a corresponding 1-person game.     

Cyclically monotone equilibrium problems and Ekeland’s principle

We study the existence of equilibria and approximate equilibria avoiding any assumption of convexity both for the domain and for the bifunction. Our approach is based on the concept of cyclic monotonicity for bifunctions. First, we exploit this notion to obtain an Ekeland’s variational principle for bifunctions which leads to the existence of approximate solutions of the so-called Minty equilibrium problem. Then, we prove the existence of equilibria in compact and noncompact settings. We introduce a new notion as a key tool for deriving a Minty’s lemma avoiding the use of convexity.     

On the existence of equilibria in economies with an infinite dimensional commodity space

We prove an infinite dimensional extension of the Gale–Nikaido–Debreu lemma which includes all necessary limiting processes and allows a proof of the existence of equilibria under standard assumptions in an economy with infinitely many commodities which exactly parallels the proof of Debreu (1959) for the finite dimensional case.     

Pure strategy equilibria in games with countable actions

In this paper, we extend the Radner–Rosenthal theorem with finite action spaces on the existence of a pure strategy equilibrium for a finite game to the case that the action space is countable and complete. We also prove the existence of a pure strategy equilibrium for a game with a continuum of players of finite types and with a countable and complete action space. To work with the countably infinite action spaces, we prove some regularity properties on the set of distributions induced by the measurable selections of a correspondence with a countable range by using the Bollobás–Varopoulos extension of the marriage lemma.     

The women of Cairo: Equilibria in large anonymous games

In this article, I characterize Nash equilibria of large anonymous games by providing the following neccessary and sufficient condition for an equilibrium distribution: for no subset K of actions more players play actions in K than have a best response in K to the given distribution. While neccessity is trivial the proof for sufficiency relies on a theorem by [Math. Proc. Camb. Philos. Soc. 78 (1974) 323] which is an extension of Hall’s theorem or the marriage lemma well known from graph theory. The veiling problem for the women of Cairo serves as an illustrating heuristic example explaining the nature of the result.     

On the generalized Poisson distribution

We use Euler's difference lemma to prove that, for θ > 0 and 0 ≤λ < 1, the function P _n defined on the non-negative integers by
P _n (θ, λ) = [θ(θ + n λ) ^{n −1}/ n !]e^{− n λ−θ}
defines a probability distribution, known as the Generalized Poisson Distribution.