The non-existence of smooth demand in general banach spaces

We study the questions of existence and smoothness of demand functions with an infinite number of commodities. The main result obtained, under some hypothesis, is: if a C¹ demand exists in a commodity space B, then B can be given an inner product structure. For example, if B is L_p, 1p∞, and if there exists a C¹ demand function defined on B then p must be 2. Another result is: if a demand function exists, defined for all prices p and income, then the commodity space must be reflexive. For example, if B is L_p and a demand function exists on B, defined for all prices and incomes then 1<p<∞. We also study the cases L_∞ and L₁ with weaker assumptions. We finish the paper proving that the demand function is always defined for a dense set of prices and convenient incomes.     

Continua of stochastic dominance relations for unbounded probability distributions

Let P = {F,G,…} be the set of all probability distribution functions with support (0, ∞). An unrestricted stochastic dominance relation>_∞ is defined on P for each real ∞ 1, where F >_∞ G means that ^x_y = 0 (x - y)^{∞ - 1} dG(y) ^x_{n = 0}(x − y)^∞−1 dG(y) for all ∞ 0, with < for some x. These relations are partial orders that increase as ∞ increases with limit relation>_∞. A class U_∞ of utility functions u on (0, ∞) is defined in such a way that F >_∞ G iff udF > udG for all u ε U_∞. The U_∞ decrease as ∞ increases and have a non-empty intersection U_∞. Each u ε U_∞ is an increasing function that has derivatives of all orders that alternate in sign. Criteria which tell when F eventually dominates G in the sense of F >_∞ G are noted. Comparisons with bounded stochastic dominance results are made in several places.     

What Is the Economic Meaning of FDH?

The central feature of the FDH model is the lack of convexity for its production possibility set, TF. Starting with n observed (distinct) decision making units DMU_k , each defined by an input-output vector p _k = [y _k -x _k], domination is defined by ordinary vector inequalities. DMU_k is said to dominate DMU_j if p _k ≥ p _j , p _k ≠ p _j . The FDH production possibility set TF consists of the observed DMU_j together with all input-output vectors p=[y_k,?x_k] with y ≥ 0, x ≥ 0, y ≠ 0, x ≠ 0 which are dominated by at least one of the observed DMU_j. DMU_k is defined as “FDH efficient” if no DMU_j dominates it. In the BCC (or variable return to scale) DEA model the production possibility set TB consists of the observed DMU_k together with all input-output vectors dominated by any convex combination of them and DMU_k is DEA efficient if it is not dominated by any p in TB. In the DEA model, economic meaning is established by the introduction of (non negative) multiplier (price) vectors w = [u,v]. If DMU_k is undominated (in TB) then there exists a positive multiplier vector w for which (a) w ^T p _k = u ^T y _k ? v ^T x _k ≥ w ^T p for every p ∈ TB. In everyday language, the net return (or profit) for DMU_k relative to the given multiplier vector w is at least as great as that for any production possibility p. On the other hand, if DMU_k is FDH but not DEA efficient then it is proved that there exists no positive multiplier vector >w for which (a) holds, i.e. for any positive w there exists at least one DMU_j for which w ^T p _j > w^T p _k . Since, therefore, FDH efficiency does not guarantee price efficiency what is its economic significance? Without economic significance, how can FDH be considered as being more than a mathematical system however logically soundly it may be conceived?

     

Finitely additive exchange economies

It is shown that the core and the set of Walras allocations of a non-atomic exchange economy are equal, if the set A of agents is either countable or a continuum, and even if all subsets of A are admitted as coalitions. The set of Walras allocations is shown to be not empty. These results are obtained by use of finitely additive measures defined on the algebra of all subsets of A.     

Generalized entropies in coding theory

Equivalence of the generalized entropyH ^β (P, Φ _t ) defined in this paper andKapur’s entropy of orderα and typeβ, ie.H _α ^β (P), is established. The results given recently byCampbell follow as special cases. International Conference on System Sciences, Honolulu, January 1968.     

On the non-null distribution ofWilcoxon's signed rank test

Summary The properties of theWilcoxon signed rank sum testW ⁺ [Wilcoxon, 1945] for the hypothesis of symmetryH _o are discussed for alternativesH toH _o. The probability generating function and cumulant generating function ofW ⁺ are derived and a limiting form of the distribution is determined.     

Continua of stochastic dominance relations for bounded probability distributions

Let P={F,G,…} be the set of probability distribution functions on [0,b]. For each αε[1, ∞), F_αG means that ∫^x_o(x−y^α−1dF(y)∫^x_o(x−y)^α−1dG(y) for all xε[0, b], and F>_αG means that F_αG and F≠G. Each _α is reflexive and transitive and each>_α is asymmetric and transitive. Both _α and>_α increase as α increases but their limits are not complete. A class U_α of utility functions is defined to give F>_αG iff ∫udF>∫udG for all uεU_α. These classes decrease as α increases, and their limit is empty. Similar decreasing classes are defined for each _α, and their limit is essentially the constant functions on (0, b].     

Two families of kurtosis measures

Two families of kurtosis measures are defined as K ₁(b)=E[ab ^−|z|] and K ₂(b)=E[a(1−|z|^b)] where z denotes the standardized variable and a is a normalizing constant chosen such that the kurtosis is equal to 3 for normal distributions. K ₂(b) is an extension of Stavig's robust kurtosis. As with Pearson's measure of kurtosis β₂=E[z ⁴], both measures are expected values of continuous functions of z that are even, convex or linear and strictly monotonic in ℜ⁻ and in ℜ⁺. In contrast to β₂, our proposed kurtosis measures give more importance to the central part of the distribution instead of the tails. Tests of normality based on these new measures are more sensitive with respect to the peak of the distribution. K ₁(b) and K ₂(b) satisfy Van Zwet's ordering and correlate highly with other kurtosis measures such as L-kurtosis and quantile kurtosis. RID="*" ID="*"  The authors thank the referees for their insightful comments that significantly improved the clarity of the article.     

Notes and Comments: Sup-convolutions of HARA utilities in the affine term structure

Abstract In the financial literature, the problem of maximizing the expected utility of the terminal wealth has been investigated extensively (for a survey, see, e.g., Karatzas and Shreve (1998), p. 153, and references therein) by using different approaches. In this paper, we extend the existing literature in two directions. First, we let the utility function U(.) of the financial agent (who is a price taker) be implicitly defined through I(.)=(U ^′ (.))^–1, which is assumed to be additively separable, i.e., I(.)=∑ _k=1 ^N I _k (.). Second, we solve the investment problem in the general affine term structure model proposed by Duffie and Kan (1996) in which the functions I _k (.), k=1,...,N are associated to HARA utility functions (with possibly different risk aversion parameters), and we show that the utility maximization problem leads to a Riccati ODE. Moreover, we extend to the multi-factor framework the stability result proved in Grasselli (2003), namely, the almost-sure convergence of the solution with respect to the parameters of the utility function. Mathematics Subject Classification (2000): 91B28 Journal of Economic Literature Classification: G11     

Estimating process capability index C′′pmk for asymmetric tolerances: Distributional properties

Pearn et al. (1999) considered a capability index C ^′′ _pmk, a new generalization of C _pmk, for processes with asymmetric tolerances. In this paper, we provide a comparison between C ^′′ _pmk and other existing generalizations of C _pmk on the accuracy of measuring process performance for processes with asymmetric tolerances. We show that the new generalization C ^′′ _pmk is superior to other existing generalizations of C _pmk. Under the assumption of normality, we derive explicit forms of the cumulative distribution function and the probability density function of the estimated index . We show that the cumulative distribution function and the probability density function of the estimated index can be expressed in terms of a mixture of the chi-square distribution and the normal distribution. The explicit forms of the cumulative distribution function and the probability density function considerably simplify the complexity for analyzing the statistical properties of the estimated index . Received April 2000     

Zur Definition von Zufallsstichproben aus endlichen Grundgesamtheiten

Summary Suppose there is a universe which containsN elements,M of which are marked by a special property. From this universe a sample of sizen is drawn without replacement. From the Laplace-condition about the ( _n ^N ) possible outcomes of this experiment it follows that the number of marked elements in the sample is distributed hypergeometrically. It is shown that an inverse of this theorem is true, whereas the analogue inverse for the with-replacement-case is not true.Further some other evident properties of samples are derived from the Laplace-condition for both cases of the drawing scheme. For all cases of possible implications between these properties and the Laplace-condition it is decided if they are true or not.     

Essential supremum and essential maximum with respect to random preference relations

In the first part of the paper, we study concepts of supremum and maximum as subsets of a topological space X$X$ endowed by preference relations. Several rather general existence theorems are obtained for the case where the preferences are defined by countable semicontinuous multi-utility representations. In the second part of the paper, we consider partial orders and preference relations “lifted” from a metric separable space X$X$ endowed by a random preference relation to the space L⁰(X)$L^0\left(X\right)$ of X$X$-valued random variables. We provide an example of application of the notion of essential maximum to the problem of the minimal portfolio super-replicating an American-type contingent claim under transaction costs.     

Variation diminishing property of densities of uniform generalized order statistics

Let f _*,r , r ≥ 1, denote the density function of rth uniform generalized order statistics as defined by Kamps (1995) or Cramer and Kamps (2003). We prove the following variation diminishing property: the number of zeros in (0,1) of any linear combination does not exceed the number of sign changes in the sequence (a ₁, . . . ,a _r ). This result is applied to study monotonicity and convexity properties of f _*,r .     

A method to obtain new copulas from a given one

Given a strictly increasing continuous function φ from [0, 1] to [0, 1] and its pseudo-inverse φ^[−1], conditions that φ must satisfy for C_φ(x₁, . . . ,x_n)=φ^[−1](C(φ(x₁), . . . ,φ(x_n))) to be a copula for any copula C are studied. Some basic properties of the copulas obtained in this way are analyzed and several examples of generator functions φ that can be used to construct copulas C_φ are presented. In this manner, a method to obtain from a given copula C a variety of new copulas is provided. This method generalizes that used to construct Archimedean copulas in which the original copula C is the product copula, and it is related with mixtures     

Representing preferences with nontransitive indifference by a single real-valued function

Let be an interval order on a topological space (X, τ), and let x _˜^* y if and only if [y z x z], and x _˜^** y if and only if [z x z y]. Then _˜^* and _˜^** are complete preorders. In the particular case when is a semiorder, let x _˜⁰ y if and only if x _˜^* y and x _˜^** y. Then _˜⁰ is a complete preorder, too. We present sufficient conditions for the existence of continuous utility functions representing _˜^*, _˜^** and _˜⁰, by using the notion of strong separability of a preference relation, which was introduced by Chateauneuf (Journal of Mathematical Economics, 1987, 16, 139–146). Finally, we discuss the existence of a pair of continuous functions u, υ representing a strongly separable interval order on a measurable topological space (X, τ, μ, ).     

Existence of balanced sampling plans avoiding cyclic distances

This article considers balanced sampling plans avoiding adjacent units. It is shown, using linear algebraic and design theoretic tools, that such plans exist for all sufficiently large populations. More generally, if any set of m circular distances is to be avoided in balanced sampling with subsets of size k, then such a plan exists, provided the population size N is at least mk ³. Research of the Peter J. Dukes is supported by NSERC.     

Further remarks on totally ordered representable subsets of Euclidean space

We introduce the property of ? -norm-boundedness on totally ordered subsets of Euclidean spaces. We show that when a closed subset X of the Euclidean space ⁿ, endowed with a continuous total order ?, is ? -norm-bounded, the order topology and the induced Euclidean topology must coincide on X. This generalizes a recent result by Beardon, proved on connected totally ordered subsets of Euclidean space, because on totally ordered closed subsets of ⁿ connectedness is a particular case of ? -norm-boundedness. We also analyze necessary and sufficient conditions for the coincidence of both topologies, and discuss some extension to the infinite-dimensional context.     

Estimation of a regression coefficient after two preliminary tests of significance

Summary The estimation of the regression coefficient of a population, defined byE (y)= +x, incorporating two preliminary tests of significance has been discussed. The experimenter has two random samples of different sizes from two such populations, as defined above, with regression coefficients ₁ and ₂ respectively, where ₂ may possibly be equal to ₁. Besides this, it is also conjectured that the common conditional variance ² of the two populations has a specified value ₀ ² . The two preliminary tests are used to resolve these two uncertainties.The author has rejoined Lucknow University, Lucknow India on Oct. 4, 1976 after availing leave for two years.     

The Perron-Frobenius theorem without additivity

We prove the following non-linear generalization of the Perron-Frobenius theorem. Let A:R^m₊→R^m₊ be continuous, homogeneous of degree 1 and primitive (i.e., for some integer l, x≥yA^lxA^ly); then A has a positive eigenvector x₀, unique up to multiplication by a positive scalar, and for all x0, Aⁿx/|Aⁿx| converges to x₀/|x₀|.