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.      

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.     

Capital depreciation and the underdetermination of rate of return: A unifying perspective

This paper shows that the notion of rate of return is best understood through the lens of the average-internal-rate-of-return (AIRR) model, first introduced in Magni (2010a). It is an NPV-consistent approach based on a coherent definition of rate of return and on the notion of Chisini mean, it is capable of solving the conundrums originated by the rate-of-return notion and represents a unifying theoretical paradigm under which every existing measure of wealth creation can be subsumed. We show that a rate of return is underdetermined by the project’s cash-flow stream; in particular, a unique return function (not a unique rate of return) exists for every project which maps depreciation classes into rates of return. The various shapes a rate of return can take on (internal rate of return, average accounting rate of return, modified internal rate of return, etc.) derive from the (implicit or explicit) selection of different depreciation patterns. To single out the appropriate rate of return for a project, auxiliary assumptions are needed regarding the project’s capital depreciation. This involves value judgment. On one side, this finding opens terrain for a capital valuation theory yet to be developed; on the other side, it triggers the creation of a toolkit of domain-specific and purpose-specific metrics that can be used, jointly or in isolation, for analyzing the economic profitability of a given project. We also show that the AIRR perspective has a high explanatory power that enables connecting seemingly unrelated notions and linking various disciplines such as economics, finance, and accounting. Some guidelines for practitioners are also provided.     

Analysis of divisional profitability using the residual income profile: a note on cash flows and rates of growth

The present note expands on the (Frigo and Ciecka (1995)) residual income profile in order to analyze the relationship between residual income (RI), return on investment (ROI) and cash flows. The results indicate that in addition to the question of whether RI and ROI are useful in divisional performance evaluation, both measures also have an important role to play as a means of approximating actual cash flow. © 1998 John Wiley & Sons, Ltd.     

An Average-Based Accounting Approach to Capital Asset Investments: The Case of Project Finance

Literature and textbooks on capital budgeting endorse net present value (NPV) and generally treat accounting rates of return as not being reliable tools. This paper shows that accounting numbers can be reconciled with NPV and fruitfully employed in real-life applications. Focusing on project finance transactions, an average return on investment (AROI) is drawn from the pro forma financial statements, obtained as the ratio of aggregate income to aggregate book value. We show that such a metric correctly captures a project's economic profitability, as long as it is compared with a comprehensive weighted average cost of capital (WACC) that includes a correction factor that takes account of the capital foregone by the investors. In contrast to the internal rate of return, AROI is unique, and we provide an explicit functional relation that links it to the NPV. The approach holds for levered and unlevered projects, constant and non-constant leverage ratios, and constant and non-constant WACCs.     

Measure of privacy in randomized response model

Randomized response (say, RR) techniques on survey are used for collecting data on sensitive issues while trying to protect the respondents’ privacy. The degree of confidentiality will clearly determine whether or not respondents choose to cooperate. There have been many proposals for privacy measures with very different implications for an optimal model design. These derived measures of protection privacy involve both conditional probabilities of being perceived as belonging to sensitive group, denoted as P(A|yes) and P(A|no). In this paper, we introduce an alternative criterion to measure privacy protection and reconsider and compare some RR models in the light of the efficiency/protection privacy. This measure is known to the respondents before they agree to use the RR model. This measure is helpful for choosing an optimal RR model in practice.     

Contracting with Private Knowledge of Production Capacity

We analyze the design of procurement contracts when the supplier is privately informed about both his innate production capacity (K) and his innate unit cost of production. We identify conditions under which the supplier will strategically employ an inefficient production technology to expand output above K. We also show that when the buyer employs the simple fixed‐price cost‐reimbursement (FPCR) contracts in the setting examined by Rogerson (2003), the supplier has no incentive to exaggerate K. Furthermore, the buyer can secure with FPCR contracts at least 75% of the surplus she secures with fully optimal contracts.     

Lattice structure of knowledge and agreeing to disagree

This paper is a contribution to the study of the underlying mathematical structure of common-knowledge, which gives the well-known result of Aumann about the impossibility of ‘agreeing to disagree’. We present the Bayesian subjective probability model with player's belief: i.e. a triple (? %plane1D;4AF;, μ), in which i is a player. ? is a lattice in the field of sets of a state space Ω, %plane1D;4AF;, is a correspondence assigning to each state ω a filter %plane1D;4AF;(ω) in ?, and μ is a common-prior. For this model, we impose none of the important restrictions on the information structure in the Aumann-Bacharach model: axiom of knowledge K₁. axiom of transparency K₂ and axiom of wisdom K₃. We can extend both the disagreement theorem of Aumann and the agreement theorem of Geanacoplos and Polemarchakis under the assumption that each ? is an Artinian lattice.     

Arbitrage pricing theory and risk-neutral measures

Abstract We consider a market with countably many risky assets and finite factor structure, as in the “arbitrage pricing theory” of Ross (1976). We prove necessary and sufficient conditions in terms of parameters for the existence of an equivalent risk-neutral measure, i.e., a measure under which each asset return has zero expected value. We relate these conditions to a certain absence of arbitrage property of the model. Mathematics Subject Classification (2000): 91B24, 91B28 Journal of Economic Literature Classification: G10, G12     

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 .     

Evaluation of inequivalent projections of Hadamard matrices of order 24

Screening designs are useful for situations where a large number of factors (q) is examined but only few (k) of these are expected to be important. It is of practical interest for a given k to know all the inequivalent projections of the design into the k dimensions. In this paper we give all the (combinatorially) inequivalent projections of inequivalent Hadamard matrices of order 24 into k=3,4 and 5 dimensions, as well as their frequencies. Then, we sort these projections according to their generalized resolution, generalized aberration and centered L₂-discrepancy measure of uniformity. Then, we study the hidden projection properties of these designs as they are introduced by Wang and Wu (1995). The hidden projection property suggests that complex aliasing allows some interactions to be estimated without making additional runs.     

Scharfe untere und obere Schranken für die Wahrscheinlichkeit der Realisation vonk untern Ereignissen

Summary LetA ₁,...,A _n be events in a probability space (,A,W). We denote byL _k the event, that at leastk events among then eventsA ₁,...A _n occur, and byK _k the event, that exactlyk events occur. If only the inequalities _i W(A _i ) _i ,i=1,...,n, are known, we calculate sharp lower and upper bounds forW(L _k ) andW(K _k ). These bounds only depend onn, k and _i , _i ,i=1,...,n. They are relevant, when treating combined tests or confidence procedures.     

A characterization for solutions of stochastic discrete time optimization models

A notion of finitely optimal plan for intertemporal optimization problems as a necessary condition for optimality is introduced. Under interiority of a feasible plan and differentiability of the return function, such a plan satisfies the stochastic analogue of deterministicEuler-Lagrange conditions, which become also sufficient conditions under concavity of the return function. Then, under more general assumptions, a sufficient criterion of optimality based on competitive plans supported by price systems and transversality conditions is discussed. Differently from the current literature, no restrictive hypotheses on the probability measure of the random shocks are assumed.

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Sommario Nella prima parte del lavoro si studiano le soluzioni dettefinitamente ottime che costituiscono una condizione necessaria per l'ottimalità di un piano ammissibile . Sotto ipotesi di interiorità del piano ammissibile e di differenziabilità della funzione obiettivo uniperiodaleF, vengono formulate le condizioni necessarie diEuler-Lagrange in ambito stocastico; aggiungendo l'ulteriore ipotesi di concavità perF, tali condizioni diventano anche sufficienti per l'ottimalità finita.La teoria viene poi estesa al caso sopradifferenziabile ottenendo una generalizzazione delle condizioni di Euler-Lagrange stocastiche che consente di individuare i piani finitamente ottimi mediante la nozione dicompetitività, ovvero supportabilità di un sistema di prezzi aleatori. Con una condizione di trasversalità all'infinito, la competitività diventa condizione sufficiente anche per l'ottimalità propria (cioè non solo finita).La peculiarità della presente trattazione è che tali condizioni sono formulate sotto ipotesi molto generali per quanto riguarda il processo degli shocks esogeni.

     

Restoring Credible Economic Governance to the Eurozone

This paper discusses new institutional rules for a stable and lasting European Monetary Union (EMU). We propose a return to a strict and sustainable economic governance framework, which is mainly driven by market forces within a smart rule‐based environment. Our recommendations are: (a) ex ante conditionalities with a tough monitoring process to avoid moral hazard in the future; (b) further enhancement of the Stability and Growth Pact; and (c) ultimo ratio punishment to be able to respond to the unique constellation of fiscal–monetary interaction and new rescue facilities. Countries violating fiscal rules for more than four years in a row will thus lose their fiscal sovereignty or will have to resign from the Eurozone. After fulfilling the ex ante conditionalities as well as all required criteria, the country either will recover its fiscal sovereignty, or, in case of exclusion, will be given the option to rejoin EMU under certain conditions.     

Pairwise Multiple Comparisons under Violation of the Independence Assumption

Several multiple comparison procedures (MCPs)were compared for their rates of Type I error and fortheir ability to detect true pairwise differencesamong means when independence of observationsassumption were not satisfied. Monte Carlo resultsshowed that, if independence is not met, none of theprocedures maintain controlled at the chosennominal level, neither using error rate per comparisonor the error rate experimentwise. However, once thedependence of the data was corrected the Type I errorrate was maintained at the same level as when thecorrelation was zero in all the procedures, except forthe Fisher's (1935) least significant differenceprocedure (LSD) and Hayter's (1986) two-stagemodified LSD procedure (FH). At the sametime, conform the correlation increased by a smallamount the power rates also, specially, when the powerwas examined using per-pair power.     

Rates of return to public agricultural research in 48 US states

The internal rate of return to public investment in agricultural R&D is estimated for each of the continental US states. Theoretically, our contribution provides a way of obtaining the returns to a local public good using Rothbart’s concept of virtual prices. Empirically, a stochastic cost function that includes own knowledge capital stock as well as spillover capital stock variables is estimated. Stochastic spatial dependency among states generated by knowledge spillovers is used to define the ‘appropriate’ jurisdictions. We estimate an average own-state rate of 17% and a social rate of 29% that compare well to the 9 and 12% average returns of the S&P500 and NASDAQ composite indexes during the same period.     

A dynamic measure of inaccuracy between two past lifetime distributions

In the present communication we introduce a dynamic measure of inaccuracy between two past lifetime distributions over the interval (0, t). Based on proportional reversed hazard rate model (PRHRM), a characterization problem for this dynamic inaccuracy measure has been studied. An upper bound to the dynamic measure of inaccuracy H*(f, g; t) has also been derived.     

TypeA operating characteristic functions of defects sampling plans

Investigations on acceptance sampling have attached rather few attention to defects inspection. As to sampling models and the corresponding operating characteristic (OC) function of single defects sampling plans, generally only typeB (process model) has been considered: sampling from a production process where the OC is conceived as a function of sample sizen, acceptance numberc, and process average number of defects per itemλ. For modern quality control, both the steadily increasing complexity of products and the need for differentiated cost calculation involve a clear demand for defects sampling in its practically most relevant form: lot-by-lot single sampling plans, where the OC (typeA, lot OC) is considered as a function of lot sizeN, sample sizen, acceptance numberc, number of defects in the lotK. The present paper develops two lot OC functions from suitable assumptions on the arrangements of the total numberK of defects over theN elements of the lot. Limiting theorems on these OC functions are used to justify to some extent the customary assumption that the Poisson process OC can serve as an approximation for typeA. The dependence of the OC functions on sample sizen, acceptance numberc, total number of defects in the lotK is described by simple formulae.     

Robust trend inference with series variance estimator and testing-optimal smoothing parameter

The paper develops a novel testing procedure for hypotheses on deterministic trends in a multivariate trend stationary model. The trends are estimated by the OLS estimator and the long run variance (LRV) matrix is estimated by a series type estimator with carefully selected basis functions. Regardless of whether the number of basis functions K is fixed or grows with the sample size, the Wald statistic converges to a standard distribution. It is shown that critical values from the fixed-K asymptotics are second-order correct under the large-K asymptotics. A new practical approach is proposed to select K that addresses the central concern of hypothesis testing: the selected smoothing parameter is testing-optimal in that it minimizes the type II error while controlling for the type I error. Simulations indicate that the new test is as accurate in size as the nonstandard test of Vogelsang and Franses (2005) and as powerful as the corresponding Wald test based on the large-K asymptotics. The new test therefore combines the advantages of the nonstandard test and the standard Wald test while avoiding their main disadvantages (power loss and size distortion, respectively).     

On the upper and lower semicontinuity of the Aumann integral

Let (T,τ,μ) be a finite measure space, X be a Banach space, P be a metric space and let L₁(μ,X) denote the space of equivalence classes of X-valued Bochner integrable functions on (T,τ,μ). We show that if φ:T×P→2^X is a set-valued function such that for each fixed pεP, φ(·,p) has a measurable graph and for each fixed tεT, φ(t,·) is either upper or lower semicontinuous then the Aumann integral of φ, i.e.,∫_Tφ(t,p)dμ(t)= {∫_Tx(t)dμ(t):xεS_φ(p)}, where S_φ(p)= {yεL₁(μ,X):y(t)εφ(t,p)μ−a.e.}, is either upper or lower semicontinuous in the variable p as well. Our results generalize those of Aumann (1965, 1976) who has considered the above problem for X=Rⁿ, and they have useful applications in general equilibrium and game theory.