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).     

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, ).     

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.     

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)).     

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.     

A uniqueness theorem for convex-ranged probabilities

A finitely additive probability measure P defined on a class Σ of subsets of a space Ω is convex-ranged if, for all P(A)>0 and all 0 < α < 1, there exists a set, Σ∋B⊆A, such that P(B)=αP(A).?Our main result shows that, for any two probabilities P and Q, with P convex-ranged and Q countably additive, P=Q whenever there exists a set A∈Σ, with 0 < P(A) < 1, such that (P(A)=P(B)?Q(A)=Q(B)) for all B∈Σ. Received: 18 December 1999 / Accepted: 17 July 2000     

Economic Development and Crime: The Two May Be Associated as an Adaptation to Industrialism in Social Revolution

Crime and economic development are discussed as part of a third major human adaptation to a new form of civilized life called industrialism. Most humans have lived the lives of hunters and gatherers while a much smaller minority has lived as members of societies based on agriculture and animal domestication. A much smaller group is now embarked on an adaptation to urbanization, technology and industrialization. Within this paradigm the process of economic development is viewed in stages, as suggested by W. W. Rostow. This allows a more extensive consideration of the social, cultural, and historicalcontexts of crime. Clusters of circumstances distinctive of specific developmental stages, or criminogenic contexts, are suggested as means of characterizing such variables. Two such contexts are briefly described. The first is a product of the unplanned drift of rural populations to urban areas in search of a better life. The second is a product of the changeover of elite groups in societies seeking to make the transition to modernization and sustained economic growth. Policy options in the interest of minimizing crime in the developmental process are briefly considered.     

The Facts About Small Business and the Real Values of Its 'Life World': A Social Philosophical Interpretation of This Sector of the Modem Economy

Abstract . According to a line of European social philosophers from Marx to Habermas, the development of society is characterized by a process of rationalization, accompanied by alienation and objectification of labor and human relations, and more generally a degradation of what Jürgen Habermashas called the person's “life world.” But the facts about small and medium sized business, as disclosed by empirical investigation, offer a different picture, one of small and medium size enterprises (SME) with a number of attributes associated with social organization on a small scale, of e.g., personality and independence. These can provide a counterweight to its disadvantages and limitations and therefore can be seen as values from a social philosophical point of view. Economically the same attributes make SME vulnerable to the market power and competition of big business. The new technology offers threats to SME, but also opportunities.     

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.     

Semiparametric estimation of a censored regression model with an unknown transformation of the dependent variable

This paper presents a method for estimating the model Λ(Y)=min(β′X+U, C), where Y is a scalar, Λ is an unknown increasing function, X is a vector of explanatory variables, β is a vector of unknown parameters, U has unknown cumulative distribution function F, and C is a censoring threshold. It is not assumed that Λ and F belong to known parametric families; they are estimated nonparametrically. This model includes many widely used models as special cases, including the proportional hazards model with unobserved heterogeneity. The paper develops n^1/2-consistent, asymptotically normal estimators of Λ and F. Estimators of β that are n^1/2-consistent and asymptotically normal already exist. The results of Monte Carlo experiments illustrate the finite-sample behavior of the estimators.     

Explaining the American Dream: Telling Chinese Academics What'Liberty and Justice For All'Means Taxes Even an Eager Audience

Abstract . For more than a generation, contemporary mainland Chinese have lived under first a military dictatorship and now a political party dictatorship. How to explain to an eager Chinese academic audience what the American dream of ‘liberty and justice for all’—realized only in part in some areas but approached progressively nearer in others—really means? The U.S. has predominantly a capitalist system but its people are committed to equality of opportunity. It tempers ‘rugged individualism’ by concern for the poor, the handicapped and the unfortunate. It has many serious economic, social and cultural problems but its citizens, drawn from most of the peoples of the world—not the special interest groups trying to benefit at the expense of others—are determined to solve them equitably and rationally. The trade and budget deficits are related to government instrumentalities.     

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.     

Two ge neral classes of search designs for factor screening experiments with factors at three levels

In this paper, we are presenting general classes of factor screening designs for identifying a few important factors from a list of m (≥ 3) factors each at three levels. A design is a subset of 3^m possible runs. The problem of finding designs with small number of runs is considered here. A main effect plan requires at least (2m + 1) runs for estimating the general mean, linear and quadratic effects of m factors. An orthogonal main effect plan requires, in addition, the number of runs as a multiple of 9. For example, when m=5, a main effect plan requires at least 11 runs and an orthogonal main effect plan requires 18 runs. Two general factor screening designs presented here are nonorthogonal designs with (2m− 1) runs. These designs, called search designs permit us to search for and identify at most two important factors out of m factors under the search linear model introduced in Srivastava (1975). For example, when m=5, the two new plans given in this paper have 9 runs, which is a significant improvement over an orthogonal main effect plan with 18 runs in terms of the number of runs and an improvement over a main effect plan with at least 11 runs. We compare these designs, for 4≤m≤ 10, using arithmetic and geometric means of the determinants, traces, and maximum characteristic roots of certain matrices. Two designs D1 and D2 are identical for m=3 and this design is an optimal design in the class of all search designs under the six criteria discussed above. Designs D1 and D2 are also identical for m=4 under some row and column permutations. Consequently, D1 and D2 are equally good for searching and identifying one important factor out of m factors when m=4. The design D1 is marginally better than the design D2 for searching and identifying one important factor out of m factors when m=5, … , 10. The design D1 is marginally better than the D2 for searching and identifying two important factors out of m factors when m=5, 7, 9. The design D2 is somewhat better than the design D1 for m=6, 8. For m=10, D1 is marginally better than D2 w.r.t. the geometric mean and D2 is marginally better than D1 w.r.t. the arithmetic mean of the maximum characteristic roots.     

On Schumpeter, Services and Economic Change

Abstract . Joseph A. Schumpeter, the Austrian-American economist, worried about “Can capitalism survive” a generation ago, and so far it has. But his work poses these questions: How do growth and change take place in a free enterprise economy? Can his model for expansion through innovation and creative destruction through the competitive process be a basis for understanding present day industrial economies? He emphasized that capitalism was an evolutionary system and that the very need for newness or rejuventation should insure continual change in it. With respect to declining industries and firms the task of economics was to turn what might have been a rout into orderly retreat. And so he gave a limited blessing to certain monopoly practices deriving from a broader strategy, risk avoidance. Schumpeter's entrepreneurs have disappeared but innovative ability still is exercised, largely through issue-specific consulting services in the burgeoning service sector, services that foster stability as well as change     

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 .     

Corruption and economic resilience: recovery from the financial crisis in western economies

We consider the resilience of a group of 20 Western economies after the financial crisis of the late 2000s. We measure resilience by the growth of real GDP between 2007, the previous peak level, and 2015. The countries exhibit a broad range of experience, from a rise in GDP of 18 per cent in Australia to a fall of 26 per cent in Greece. A substantial proportion of the differences in growth rates can be accounted for by just two variables: the perceived level of corruption and membership of the Eurozone. The euro did have a negative impact on the recovery paths of the Mediterranean economies (Greece, Italy, Portugal and Spain), but their perceived corruption scores in 2007 had a bigger impact, especially on the first three of these economies.     

Slow convergence of the number of near-maxima for Burr XII distributions

A near-maximum is an observation which falls within a distance a of the maximum observation in an independent and identically distributed sample of size n. Subject to some conditions on the tail thickness of the population distribution, the number K _n (a) of near-maxima is known to converge in probability to one or infinity, or in distribution to a shifted geometric law. In this paper we show that for all Burr XII distributions K _n (a) converges almost surely to unity, but this convergence property may not become clear under certain cases even for very large n. We explore the reason of such slow convergence by studying a distributional continuity between Burr XII and Weibull distributions. We have also given a theoretical explanation of slow convergence of K _n (a) for the Burr XII distributions by showing that the rate of convergence in terms of P{K _n (a) > 1} tending to zero changes very little with the sample size n. Illustrations of the limiting behaviour K _n (a) for the Burr XII and the Weibull distributions are given by simulations and real data. The study also raises an important issue that although the Burr XII provides overall better fit to a given data set than the Weibull distribution, cautions should be taken for the extrapolation of the upper tail behaviour in the case of slow convergence.      

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.     

China's Approach to Social Market Economics: The Chinese variant of Market Socialism Seeks to Escape from the Difficulties of Central Command Planning

Abstract . The paradigm of a social market economy provides a rational basis for resolving structural socioeconomic changes, as presently evidenced in the People's Republic of China. China's approach to socioeconomic change, labeled as “Socialism with Chinese Characteristics,” resembles a modified version of Eucken's paradigm of a social market economy.¹ The Chinese paradigm, leaning on that of Walter Eucken, provides for a “limited”free market, which not only accommodates economic growth and change, but which also makes allowance for some degree of human, especially economic freedom, and dignity. Eucken's model is based on a sound institutional and legal framework which he considered to be essential for a viable modern market economy. China's new socioeconomic program is moving the country in this direction. It takes into account people's desire for a useful and Just socioeconomic order (however defined), characterized by stable socioeconomic and political policies and by a flexible price mechanism, to enable them to live a life in dignity, free of economic deprivation.