Structural instability of the core

Let σ be a q-rule, where any coalition of size q, from the society of size n, is decisive. Let w(n,q)= 2q-n+1 and let W be a smooth ‘policy space’ of dimension w. Let U(W)^N be the space of all smooth profiles on W, endowed with the Whitney topology. It is shown that there exists an ‘instability dimension’ w^*(σ) with 2w^*(σ)w(n,q) such that:

1. (i) if ww^*(σ), and W has no boundary, then the core of σ is empty for a dense set of profiles in U(W)^N (i.e., almost always),

2. (ii) if ww^*(σ)+1, and W has a boundary, then the core of σ is empty, almost always,

3. (iii) if ww^*(σ)+1 then the cycle set is dense in W, almost always,

4. (iv) if ww^*(σ)+2 then the cycle set is also path connected, almost always.

The method of proof is first of all to show that if a point belongs to the core, then certain generalized symmetry conditions in terms of ‘pivotal’ coalitions of size 2q−n must be satisfied. Secondly, it is shown that these symmetry conditions can almost never be satisfied when either W has empty boundary and is of dimension w(n,q) or when W has non-empty boundary and is of dimension w(n,q)+1.     

On optimal operation of a jointly owned enterprise

An enterprise is owned jointly by m agents, the ith agent's share being θ_i > 0 where ∑_iθ_i=1. The enterprise is able to produce some non-negative n-vector x of goods where x lies in some convex production set X. An operation consists of choosing a vector from X and distributing it among the agents. The problem is to find an operations such that the value of the ith agent's bundle measured in a given price system is proportional to θ_i and such that the operation is (Pareto) optimal with respect to the agent's preferences. It is shown under standard assumptions that operations which are both optimal and proportional always exist. It is also shown that these operations are unique if (a) X is given by a separable production function, and (b) when X represents production of a single good over n time periods.     

Two-sample rank tests with truncated populations

LetX ₁,…,X _m andY ₁,…,Y _n be two independent samples from continuous distributionsF andG respectively. Using a Hoeffding (1951) type theorem, we obtain the distributions of the vector S=(S ₍₁₎,…,S _(n)), whereS _(j)=# (X _i ’s≤Y _(j)) andY _(j) is thej-th order statistic ofY sample, under three truncation models: (a)G is a left truncation ofF orG is a right truncation ofF, (b)F is a right truncation ofH andG is a left truncation ofH, whereH is some continuous distribution function, (c)G is a two tail truncation ofF. Exploiting the relation between S and the vectorR of the ranks of the order statistics of theY-sample in the pooled sample, we can obtain exact distributions of many rank tests. We use these to compare powers of the Hajek test (Hajek 1967), the Sidak Vondracek test (1957) and the Mann-Whitney-Wilcoxon test. We derive some order relations between the values of the probagility-functions under each model. Hence find that the tests based onS ₍₁₎ andS _(n) are the UMP rank tests for the alternative (a). We also find LMP rank tests under the alternatives (b) and (c).     

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?

     

Characterization of the multivariate normal distribution by conditional normal distributions

Summary LetX andY be two random vectors with values in ℝ ^k and ℝ∝, respectively. IfZ=(X ^T,Y ^T) ^T is multivariate normal thenX givenY=y andY givenX=x are (multivariate) normal; the converse is wrong. In this paper simple additional conditions are stated such that the converse is true, too. Furthermore, the case is treated that the random vectorZ=(X ₁ ^T , …,X _t ^T ) ^T is splitted intot≥3 partsX ₁, …,X _t.     

Testing whetherF is more IFR thanG

A distributionF is said to be “more IFR” than another distributionG ifG ⁻¹ F is convex. WhenF(0) =G(0) = 0, the problem of testingH ₀ :F(x) =G (θx) for someθ > 0 andx ⩾ 0, against the alternativeH _A:F is more IFR thanG, is considered in this paper. Both cases, whenG is completely specified (one-sample case) and when it is not specified but a random sample form it is available (two-sample case) are considered. The proposed tests are based onU-statistics. The asymptotic relative efficiency of the tests are compared with several other tests and the test statistics remain asymptotically normal under certain dependency assumptions. Research supported in part by a grant from the US Air Force Office of Scientific Research.     

On a linear stochastic differential equation

Summary Stochastic differential (s. d.) equations had been considered in [Nasr, 1960] and [Nasr]. We consider here, the s. d. equationf(D)x(t)=m(t)+v(t)z(t) wherem(t),v(t) are real functions oft,f(D) is a polynomial inD withD=d/dt, andz(t) is a random function. In particular,z(t) is assumed here, to be of the stationary type, while other types namely whenz(t) is of theGaussian or of thePoisson type, are considered in [Nasr]. A particular integral of the stated equation, and an associated covariance function of this integral are given in the form of generalized (g-)functions; [Nasr, 1965]. The equationdx/dt=v(t)z(t) wherez(t) is stationary in the wide sense is considered as a special case.     

A general class of univariate skew distributions considering Stein’s lemma and infinite divisibility

In this article, we consider a general form of univariate skewed distributions. We denote this form by GUS(λ; h(x)) or GUS with density s(x|λ, h(x)) = 2f(x)G(λ h(x)), where f is a symmetric density, G is a symmetric differentiable distribution, and h(x) is an odd function. A special case of this general form, normal case, is derived and denoted by GUSN(λ; h(x)). Some representations and some main properties of GUS(λ; h(x)) are studied. The moments of GUSN(λ; h(x)) and SN(λ), the known skew normal distribution of Azzalini (1985), are compared and the relationship between them is given. As an application, we use it to construct a new form for skew t-distribution and skew Cauchy distribution. In addition, we extend Stein’s lemma and study infinite divisibility of GUSN(λ; h(x)).     

Kosten-optimale Prüfpläne für die Gut-Schlecht-Prüfung

Summary A lot is accepted if the number of defective units in a sample of sizen does not exceed the acceptance numberc. The usefulness of the sampling plan (n, c) is described by the regret function. This regret functionR(p), depending on the proportionp of defective units in the lot, is the expectation of the avoidable costs. There always exists an optimum sampling plan which minimizes the maximum ofR(p). The dependence of the maxima ofR(p) onn andc is studied and some theorems are given which are useful for calculating the minimax solution, that is the optimum sampling plan.      

Approximation of contractible valued correspondences by functions

Let X and Y be absolute neighborhood retracts (this is a large class of spaces) with X compact, and let F:X→Y be an upper hemicontinuous correspondence whose values are compact and contractible. It is shown that any neighborhood of the graph of F contains the graph of a continuous function f:X→Y. The relevance of this result to fixed point theory is indicated. It is also shown that if X is ‘locally infinite’, then F can be approximated in the stronger sense of the graph of f being close to the graph of F and every point in the graph of F being close to some point in the graph of f. A conjectured generalization of the main result is stated.     

Asymptotic expansion of the minimax value in survey sampling

Summary Suppose that a real numbery _u is associated with each unitu of a populationU and that the functiony:u →y _u onU is known to be an element of the parameter space Θ. The statistician has to select a samples ⊂U ofn units and to employy _u;u ∈s to estimate the arithmetic mean of ally _u,u ∈U. The performance of such a strategy is assessed by its mean square error or, more simply, by the supremum of the mean square error. This supremum cannot be determined exactly for the parameter space of Scott/Smith (1975). We propose, therefore, an asymptotic approximation; this approximation is based on the assumption, that the sample sizen is fixed and that linear estimators have to be used.     

Bayesian Hypothesis Testing: a Reference Approach

For any probability model M={p(x|θ, ω), θεΘ, ωεΩ} assumed to describe the probabilistic behaviour of data xεX, it is argued that testing whether or not the available data are compatible with the hypothesis H₀={θ=θ₀} is best considered as a formal decision problem on whether to use (a₀), or not to use (a₀), the simpler probability model (or null model) M₀={p(x|θ₀, ω), ωεΩ}, where the loss difference L(a₀, θ, ω) –L(a₀, θ, ω) is proportional to the amount of information δ(θ₀, ω), which would be lost if the simplified model M₀ were used as a proxy for the assumed model M. For any prior distribution π(θ, ω), the appropriate normative solution is obtained by rejecting the null model M₀ whenever the corresponding posterior expectation ∫∫δ(θ₀, θ, ω)π(θ, ω|x)dθdω is sufficiently large. Specification of a subjective prior is always difficult, and often polemical, in scientific communication. Information theory may be used to specify a prior, the reference prior, which only depends on the assumed model M, and mathematically describes a situation where no prior information is available about the quantity of interest. The reference posterior expectation, d(θ₀, x) =∫δπ(δ|x)dδ, of the amount of information δ(θ₀, θ, ω) which could be lost if the null model were used, provides an attractive nonnegative test function, the intrinsic statistic, which is invariant under reparametrization. The intrinsic statistic d(θ₀, x) is measured in units of information, and it is easily calibrated (for any sample size and any dimensionality) in terms of some average log‐likelihood ratios. The corresponding Bayes decision rule, the Bayesian reference criterion (BRC), indicates that the null model M₀ should only be rejected if the posterior expected loss of information from using the simplified model M₀ is too large or, equivalently, if the associated expected average log‐likelihood ratio is large enough. The BRC criterion provides a general reference Bayesian solution to hypothesis testing which does not assume a probability mass concentrated on M₀ and, hence, it is immune to Lindley's paradox. The theory is illustrated within the context of multivariate normal data, where it is shown to avoid Rao's paradox on the inconsistency between univariate and multivariate frequentist hypothesis testing.     

Some properties of the minimum and the maximum of random variables with joint logconcave distributions

It is shown that if (X ₁, X ₂, . . . , X _n ) is a random vector with a logconcave (logconvex) joint reliability function, then X _P = min _i∈P X _i has increasing (decreasing) hazard rate. Analogously, it is shown that if (X ₁, X ₂, . . . , X _n ) has a logconcave (logconvex) joint distribution function, then X ^P  = max _i∈P X _i has decreasing (increasing) reversed hazard rate. If the random vector is absolutely continuous with a logconcave density function, then it has a logconcave reliability and distribution functions and hence we obtain a result given by Hu and Li (Metrika 65:325–330, 2007). It is also shown that if (X ₁, X ₂, . . . , X _n ) has an exchangeable logconcave density function then both X _P and X ^P have increasing likelihood ratio.     

Convergence of a branching type recursion with non-stationary immigration

The asymptotic distribution of a branching type recursion with non-stationary immigration is investigated. The recursion is given by , where (X _l ) is a random sequence, (L _n ⁻¹⁽¹⁾ ) are iid copies ofL _n−1,K is a random number andK, (L _n ⁻¹⁽¹⁾ ), {(X _l ),Y _n } are independent. This recursion has been studied intensively in the literature in the case thatX _l ≥0,K is nonrandom andY _n =0 ∀n. Cramer, Rüschendorf (1996b) treat the above recursion without immigration with starting conditionL ₀=1, and easy to handle cases of the recursion with stationary immigration (i.e. the distribution ofY _n does not depend on the timen). In this paper a general limit theorem will be deduced under natural conditions including square-integrability of the involved random variables. The treatment of the recursion is based on a contraction method. The conditions of the limit theorem are built upon the knowledge of the first two moments ofL _n . In case of stationary immigration a detailed analysis of the first two moments ofL _n leads one to consider 15 different cases. These cases are illustrated graphically and provide a straight forward means to check the conditions and to determine the operator whose unique fixed point is the limit distribution of the normalizedL _n .     

Combining information for prediction in linear regression

This article deals with the prediction problem in linear regression where the measurements are obtained using k different devices or collected from k different independent sources. For the case of k=2, a Graybill-Deal type combined estimtor for the regression parameters is shown to dominate the individual least squares estimators under the covariance criterion. Two predictors ŷ _c and ŷ _p are proposed. ŷ _c is based on a combined estimator of the regression coefficient vector, and ŷ _p is obtained by combining the individual predictors from different models. Prediction mean square errors of both predictors are derived. It is shown that the predictor ŷ _p is better than the individual predictors for k≥2 and the predictor ŷ _c is better than the individual predictors for k=2. Numerical comparison between ŷ _c and ŷ _p shows that the former is superior to the latter for the case k=2.     

Locally lipschitz cooperative games

A locally Lipschitz cooperative generalized game is described by its coalition worth function v defined on the set [0, 1]ⁿ of generalized (or fuzzy) coalitions of n players. We assume that v is positively homogeneous and locally Lipschitz. We propose the Clarke's generalized gradient ∂v(c^N) of v at the coalition c^N=(1,…,1) of all players as a set of solutions, and we study its property. We point out that it coincides with the core when v is super–additive and to the Shapley value when v is smooth.     

Vergleich der hypergeometrischen mit der Binomial-Verteilung

Zusammenfassung Es seiL die Operations-Charakteristik beim Testen einer Hypothese über eine unbekannte Wahrscheinlichkeitp eines EreignissesA. L ist also die Wahrscheinlichkeit dafür, da? h?chstensc Elemente mitA in der Stichprobe auftreten. Wird die Stichprobe ohne Zurücklegen gezogen, so h?ngtL vom UmfangN der Grundgesamtheit ab. Es wird untersucht, für welchep die Operations-CharakteristikL monoton vonN abh?ngt. Den Abschlu? bilden Vergleiche der Operations-Charakteristik bei der Binomial-Verteilung mitL und deren Ann?herung durch die Poisson-Verteilung.

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Summary LetL be the operating characteristic for testing a hypothesis concerning an unknown probabilityp of an eventA. ThenL is the probability of being no more thanc elements withA in the sample. When the sample is drawn without replacementL depends on the sizeN of the population. Investigation is made for whichp the operating characteristicL depends monotone onN. Finally the operating characteristic of the binomial distribution is compared withL and with the approximation by the Poisson distribution.

     

Continuous semiorder representations

Given X,, where X is a topologically connected space and is an asymmetric binary relation, necessary and sufficient conditions are presented for the existence of a continuous representation of the form, u, δ; that is, for x,yX, xy if and only if u(x)>u(y)+δ where u:X→ is continuous and δ is a strictly positive real number. The results are related to existing results for numerical representations of interval orders and semiorders.     

Some Determinants of Property Crime: Economic Factors Influence Criminal Behavior But Cannot Completely Explain the Syndrome

Abstract . An empirical analysis of the property crimes, robbery, burglary and larceny, is presented for all 120 counties in Kentucky. While this analysis is based on an economic model of crime, certain sociological and legal variables are included as well in the system of equations. Overall, the empirical results support prior studies’findings with the exception that a quadratic relationship is found to exist between urbanization and each of the property crimes. Furthermore, neither the economic nor the non economic influences measured appear more important for affecting crime rates. Specifically, results indicate that the level of poverty, the degree of tourism, the presence of police, the unemployment rate and the apprehension rate all affect property crimes. In contrast, the length of sentence, the degree of industrialization, the level of public assistance payments and the proportion of youth in the county have no affect on property crime rates in these areas.     

The association between two random elements: A complete characterization and odds ratio models

For random elements X and Y (e.g. vectors) a complete characterization of their association is given in terms of an odds ratio function. The main result establishes for any odds ratio function and any pre-specified marginals the unique existence of a corresponding joint distribution (the joint density is obtained as a limit of an iterative procedure of marginal fittings). Restricting only the odds ratio function but not the marginals leads to semi-parmetric association models for which statistical inference is available for samples drawn conditionally on either X or Y. Log-bilinear association models for random vectors X and Y are introduced which generalize standard (regression) models by removing restrictions on the marginals. In particular, the logistic regression model is recognized as a log-bilinear association model. And the joint distribution of X and Y is shown to be multivariate normal if and only if both marginals are normal and the association is log-bilinear.Acknowledgements The author thanks both referees for their helpful comments which improved the first draft of the paper.