Distributions of exact tests in the exponential family

The aim of this paper is to give some results on the exact density of the I-divergence in the exponential family with gamma distributed observations. It is shown in particular that the I-divergence can be decomposed as a sum of two independent variables with known distributions. Since the considered I-divergence is related to the likelihood ratio statistics, we apply the method to compute the exact distribution of the likelihood ratio tests and discuss the optimality of such exact tests. One of these tests is the exact LR test of the model which is asymptotically optimal in the Bahadur sense. Numerical examples are provided to illustrate the methods discussed. Received: January 2002 Acknowledgements. I am grateful to Prof. Andrej Pázman for helpful discussions during the setup and the preparation of the paper and to the referees for constructive comments on earlier versions of the paper. Research is supported by the VEGA grant (Slovak Grant Agency) No 1/7295/20     

On the linear statistical relationship

The fitting of lines to data, when the relative precision of the measured data is a known constant, is a problem which is of some interest, for example in comparison of measurement procedures. We examine some ways of treating this problem with a view to clarifying the relationship of solutions with the Maximum Likelihood Estimate. Acknowledgement. I am grateful to Hans Schneeweiss for suggesting a strengthened form of the definition of f in Section 4.     

Manipulation via merging and splitting in claims problems

In claims problems, we study coalitional manipulations via claims merging and splitting. We characterize (division) rules that are non-manipulable via (pairwise) splitting and that also satisfy standard axioms of equal treatment of equals, consistency, and continuity. And we obtain a similar result for non-manipulability via (pairwise) merging.Received: 22 January 2002, Accepted: 5 February 2003, JEL Classification: C70, C71, D70I am grateful to William Thomson for many helpful comments and suggestions. I also thank Murat Sertel and an anonymous referee for many helpful suggestions for the revision. All remaining errors are mine.     

He who must not be named

Landsberger et al. have studied a sealed bid first price auction with two players in which the ranking of the valuations is known. They argue that such a situation can arise in a sequential auction where only the name of the winner is revealed. In this paper we consider sequential auctions where two identical goods are sold sequentially to N players who are interested in both objects. In sealed bid auctions, no information is a priori revealed by the mechanism, but the seller can in principle reveal whatever he wants. We restrict our attention to the case where only the name of the winner is revealed to be in the context of Landsberger et al. for the second auction. The aim of the paper is to compare such a sequential auction with a simultaneous auction where both goods are sold as a bundle or equivalently with a sequential auction where no information is revealed. We first show that there exists an equilibrium of the sequential game in pure and monotone strategies. Then, the comparison of the seller's expected revenue in the two cases allows us to conclude that contrary to Landsberger et al.'s predictions, the seller can not use the information to increase his revenue. This result is obtained using simulations for a large class of distribution functions. The seller must not reveal the name of the winner between the two auctions and instead sell both goods using a simultaneous auction.Received: 31 July 2001, Accepted: 5 February 2003, JEL Classification: B44I wish to thank Laurent Linnemer, Thomas Ricke, Michael Visser and Shmuel Zamir for helpful comments and suggestions. I am very grateful to the referees and the associated editors in charge of my paper.     

A note on the maximum deviation of the scale-contaminated normal to the best normal distribution

In this paper we consider the case of the scale-contaminated normal (mixture of two normals with equal mean components but different component variances: (1−p)N(μ,σ²)+pN(μ,τ²) with σ and τ being non-negative and 0≤p≤1). Here is the scale error and p denotes the amount with which this error occurs. It's maximum deviation to the best normal distribution is studied and shown to be montone increasing with increasing scale error. A closed-form expression is derived for the proportion which maximizes the maximum deviation of the mixture of normals to the best normal distribution. Implications to power studies of tests for normality are pointed out. Received May 2001     

Knightian decision theory. Part I

A theory of choice under uncertainty is proposed which removes the completeness assumption from the Anscombe–Aumann formulation of Savage's theory and introduces an inertia assumption. The inertia assumption is that there is such a thing as the status quo and an alternative is accepted only if it is preferred to the status quo. This theory is one way of giving rigorous expression to Frank Knight's distinction between risk and uncertainty. Received: 8 October 2001 / Accepted: 8 November 2001 The author is grateful to Martin Shubik for encouragement and long discussions. He is also grateful to Donald Brown, Edward Leamer, David Schmeidler, Martin Shubik, Sidney Winter, Asad Zaman and Arnold Zellner for calling his attention to relevant literature. Note of the Editor: Except for a few minor editorial changes, this article publishes content formerly circulated as Discussion Paper no.~807 of the Cowles Foundation at Yale University, November 1986.     

sk−p Fractional factorial designs in sb blocks

Draper and Guttman (1997) shows that for basic 2^k−p designs, p≥0, k − p replicates of blocks designs of size two are needed to estimate all the usual (estimable) effects. In this work, we provide an algebraic formal proof for the two-level blocks designs results and present results applicable to the general case; that is, for the case of s ^k factorial (p=0) or s ^k−p fractional factorial (p >0) designs in s ^b blocks, where 0<b<k− p, at least replicates are needed to clear up all possible effects. Through the theoretical development presented in this work, it can provide a clearer view on why those results would hold. We will also discuss the estimation equations given in Draper and Guttman (1997).  Research supported in part by the National Science Council of Taiwan, R.O.C., Grant No. NSC 89-2118-M110-010. Acknowledgement. The authors would like to thank the referee for very helpful comments.     

Choice of optimal initial designs in sequential experiments

Combined-optimal designs (Li and Lin, 2003) are obviously the best choices for the initial designs if we partition the experiment into two parts with equal size to obtain some information about the process, especially for the case not considering the blocking factor. In this paper, the definition of combined-optimal design is extended to the case when blocking factor is significant, and this new class of designs is called blocked combined-optimal designs. Some general results are obtained which relate 2^k−p_III initial designs with their complementary designs when , where n=2^k−p. By applying these results, we are able to characterize 2^k−p_III combined-optimal designs or blocked combined-optimal designs in terms of their complementary designs. It is also proved that both 2^k−p_III combined-optimal and blocked combined-optimal designs are not minimum aberration designs when and n−1−k > 2. And some combined-optimal and blocked combined-optimal designs with 16 and 32 runs are constructed for illustration. 2000 Mathematics Subject Classifications: 62K15, 62K05     

Book reviews

Books reviewed: Alan Campbell, Nina Fishman, and John McIlroy, British Trade Unions and Industrial Politics Vol I: The Post‐War Compromise, 1945–64. Vol II: The High Tide of Trade Unionism, 1964–79 Jack Eaton, Comparative Employment Relations Robert Taylor, The TUC: From the General Strike to New Unionism Stephen Bach, Lorenzo Bordogna, Guiseppe Della Rocca and David Winchester, (eds) Public Service Employment Relations in Europe: Transformation, modernization or inertia?     

An elementary core equivalence theorem¶in a countable economy

We consider an infinite exchange economy with countably many traders, which can be regarded as a natural extension of finite exchange economies to an infinite one. In our countable economy the core defined in the traditional manner would be empty. To avoid this unwanted situation we have to strengthen the notion of “improves upon”. We will achieve this based on the idea that forming coalitions involve costs. Received: 23 August 2000 / Accepted: 12 April 2002 I am very grateful to István Dancs, Péter Medvegyev, Miklós Pintér, Csaba Sándor, András Simonovits, László Szili, Péter Tallos and the anonymous referees for comments and suggestions. Parts of this research were done during the author's Bolyai János Research Fellowship provided by the Hungarian Academy of Sciences (MTA). Of course, all errors remain the author's responsibility.     

Pareto Density Distributions

Building on new insights into the genesis ofPareto-Distributions,(“Kopp” effect etc.) as publishedearlier in “Quality and Quantity”, the author gives at least oneauthentic/definitive Pareto-Formula. A practical example of the synthetic generation of Pareto Distributions by means of spreadsheets. A working D.I.Y-method for fine-fitting Pareto-curvesto scattergrams with spreadsheets using interalia an indirect method of the least squares of residuals is fully demonstrated. A comparative test-fit to a cumulative Pareto- Distribution example, where a simulative curve-formula evolved by Prof. B. Arnold/Ucla is used for demonstration. Easy to absorb and to retain graphical tableaux are employed to visualize the chain of descent and interconnections between normal distributions, log-normal distributions and Pareto- Distributions. A quasi-dichotomy of the Pareto-formulae is presented in tableau-form. One innovative formula for Pareto-distribution is given as: F(x)= k^*e― [((ln(Integral(In(x)))) ‐ (ln(Integral(ln(μ)))))² / 2^*(ln(Integral(ln(σ))))²} Readers e-mailed constructive opinions &/or inputs are encouraged and welcomed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Social identity,inequality, and conflict: correction and extension

This paper corrects and extends the analysis in Social Identity, Inequality, and Conflict by James Robinson (Economics of Governance, 2(1), 2001). For conflict along class lines, Robinson finds the total impact of mobility on conflict to be ambiguous. Contrary to his result, I show that, under his assumptions, the effect of social mobility on class conflict is unambiguous. Higher mobility always decreases conflict. In my extension to Robinsons model, I explore mobilitys impact on class conflict in a society where the tax rate is not fixed. I demonstrate that if the tax rate is proportional to the population of the group in power relative to the total population of the society, then the effects of social mobility on class conflict are indeed ambiguous.Submitted: February 2002, Accepted: July 2003,I am grateful to an anonymous referee and Amihai Glazer for their comments. I would also like to thank Herschel Grossman for his help along the way.     

Multi-stage point and interval estimation of the largest mean ofK normal populations and the associated second-order properties

Summary We havek independent normal populations with unknown meansμ ₁, …,μ _k and a common unknown varianceσ ². Both point and interval estimation procedures for the largest mean are proposed by means of sequential and three-stage procedures. For the point estimation problem, we require that the maximal risk be at mostW, a preassigned positive number. For the other problem, we wish to construct a fixed-width confidence interval having the confidence coefficient at least 1-α, a preassigned number between zero and one. Asymptotic second order expansions are provided for various characteristics, such as average sample size, associated risks etc., for the suggested multi-stage estimation procedures.     

A generalized maximum likelihood characterization of the normal distribution

LetP be a probability measure on ℝ andI _x be the set of alln-dimensional rectangles containingx. If for allx ∈ ℝⁿ and θ ∈ ℝ the inequality holds,P is a normal distributioin with mean 0 or the unit mass at 0. The result generalizes Teicher’s (1961) maximum likelihood characterization of the normal density to a characterization ofN(0, σ²) amongall distributions (including those without density). The m.l. principle used is that of Scholz (1980).     

The proportion of blocking coalitions in atomless economies

In this paper we investigate the number of coalitions that block a given non-competitive allocation. In an atomless economy with a finite number of types we identify coalition with its profile. Considering profiles π that represent coalitions with the same proportions of types as in the whole society, we prove that there is a ball B_π with π as its center so that ‘almost half’ of the profiles in B_π are blocking. This result is an analogous result to that of Mas-Colell (1978) who dealt with large finite markets.     

Multivariate t and beta distributions associated with the multivariate F distribution

Relationships between F, skew t and beta distributions in the univariate case are in this paper extended in a natural way to the multivariate case. The result is two new distributions: a multivariate t/skew t distribution (on ℜ^m) and a multivariate beta distribution (on (0,1)^m). A special case of the former distribution is a new multivariate symmetric t distribution. The new distributions have a natural relationship to the standard multivariate F distribution (on (ℜ⁺)^m) and many of their properties run in parallel. We look at: joint distributions, mathematically and graphically; marginal and conditional distributions; moments; correlations; local dependence; and some limiting cases. Received: March 2001     

Pre-assigned rents and bureaucratic friction

In this paper we study the endogenous determination of bureaucratic friction in a bureaucratic contest with () and without (n = 1) rent contestability. When n= 1 bureaucratic impediments induce the individual to undertake rent-securing activities at the same level as in the two-player rent-seeking contest. However, under rent contestability the bureaucracy no longer serves as a means of extracting resources from the public. The paper concludes with the study of the effect of ‘net costs’ on bureaucratic friction. It turns out that under cotestability the only reason for creating bureaucratic friction is the ‘negative costs’ it incurs while when n = 1 the effect of the bureaucrat's net costs of generating bureaucratic friction on the optimal degree of such friction is ambiguous. Received: October 30, 2000 / Accepted: December 28, 2001 RID="*" ID="*" The authors are grateful to two anonymous referees for their valuable comments     

On the generalized principal-agent problem: a comment

Faynzilberg and Kumar (Rev Econ Design 5(1):23–58, 2000) show that the usual Mirrlees–Rogerson conditions to validate the first-order approach in moral hazard agency models are no longer valid in the generalized agency case. They consider the risk-averse agent case and identify one set of technological conditions, where the production technology satisfies the linear distribution function condition in actions and types, that validates ex-ante the first-order approach. With the usefulness of their decomposition approach, we show that the first-order approach in the generalized agency case can be checked ex-ante under the Mirrlees–Rogerson conditions when the agent is risk-neutral but there is a binding limited liability constraint on the agent’s wage.      

A free-rider problem with a free-riding principal

In a moral hazard problem caused purely by joint production and not by uncertainty, we examine the problem faced by a principal who actively participates in production along with a group of agents. We show that, when designing the optimal output sharing rule, the principal need not look for anything more complicated than the frequently observed simple linear or piecewise linear rules. We also confirm the presence of a friction between the principal’s residual claimant role and her incentive to free-ride in the production process that prohibits her from completely mitigating the moral hazard problem. This paper is from the first chapter of my Ph.D. dissertation at The University of British Columbia, Canada, 1993. I would like to thank my thesis supervisor John Weymark for his many helpful comments and suggestions. I have also benefitted from the comments of Charles Blackorby, David Donaldson, Mukesh Eswaran, Kenneth Hendricks, Ashok Kotwal and Guofu Tan. An anonymous referee and an associate editor provided helpful suggestion. I am also grateful for the hospitality of the Indian Statistical Institute, Delhi Centre, and the Centre for Development Economics, Delhi School of Economics, where parts of the paper were revised while I was a visitor. I am solely responsible for any remaining errors and omissions.     

Endogenous growth, the Solow model and human capital

Both endogenous growth theory and the (augmented) Solow model propose a role for human capital in the growth process though each is based on different conceptual arguments. Since both approaches can justify the inclusion of human capitallevels andgrowth rates in an output growth regression the two theories cannot readily be distinguished empirically. This paper argues that the variable most commonly used in empirical studies to proxy human capital (levels or growth) — school enrolment rates (SERs) — may capture bothstock andaccumulation effects, butchanges in SERs can provide useful additional dynamic information on the contribution of human capital to growth. Empirical evidence from samples of developed and less developed countries during 1960–85 suggests important growth effects associated both with initial levels of, and changes in, SERs. The nature of these effects appears to differ between the two country groups.I am grateful to an anonymous referee and to Subrata Ghatak for helpful comments on an earlier draft of this paper, and to Massimo Suardi for research assistance