Games of chance with multiple objectives

We consider games of chance between two players: Player M can win only by amassing point totals in several categories before player N scores a prescribed total number n of points. Let M have k objectives, with m _i points required in category i and probability q _i of scoring a point in that category. We resolve certain special cases: (a) For all m _i equal, the probabilities of M winning are ordered by majorization of the vectors (q ₁,...,q _k ). (b) For all q _i equal, the probabilities of M winning are ordered by majorization of the vectors (m ₁,...,m _k ). (c) For all m _i equal and all q _i equal, the probability of M winning approaches 0 as n → ∞ or as k → ∞. The results, which follow from inequalities of majorization and Schur convexity, are in accord with intuition.      

A view on Bhattacharyya bounds for inverse Gaussian distributions

Shanbhag (J Appl Probab 9:580–587, 1972; Theory Probab Appl 24:430–433, 1979) showed that the diagonality of the Bhattacharyya matrix characterizes the set of Normal, Poisson, Binomial, negative Binomial, Gamma or Meixner hypergeometric distributions. In this note, using Shanbhag (J Appl Probab 9:580–587, 1972; Theory Probab Appl 24:430–433, 1979) and Pommeret (J Multivar Anal 63:105–118, 1997) techniques, we evaluated the general form of the 5 × 5 Bhattacharyya matrix in the natural exponential family satisfying f(x|q)=\fracexp{xg(q)}b(g(q))y(x){f(x|\theta)=\frac{\exp\{xg(\theta)\}}{\beta(g(\theta))}\psi(x)} with cubic variance function (NEF-CVF) of θ. We see that the matrix is not diagonal like distribution with quadratic variance function and has off-diagonal elements. In addition, we calculate the 5 × 5 Bhattacharyya matrix for inverse Gaussian distribution and evaluated different Bhattacharyya bounds for the variance of estimator of the failure rate, coefficient of variation, mode and moment generating function due to inverse Gaussian distribution.     

Bayesian robust designs for linear models with possible bias and correlated errors

Consider the design problem for the approximately linear model with serially correlated errors. The correlated structure is the qth degree moving average process, MA(q), especially for q = 1, 2. The optimal design is derived by using Bayesian approach. The Bayesian designs derived with various priors are compared with the classical designs with respect to some specific correlated structures. The results show that any prior knowledge about the sign of the MA(q) process parameters leads to designs that are considerately more efficient than the classical ones based on homoscedastic assumptions.     

Optimal design of nonlinear experiments with parameter constraints

The nonlinear regression model with N observations y _i=η(x _i,θ) +ε_i, and with the parameter θ subject to q nonlinear constraints C _j (θ)=0; j=1, …,q, is considered. As an example, the spline regression with unknown nodes is taken. Expressions for the variances (variance matrices) of the LSE are discussed. Because of the complexity of these expressions, and the singularity of the variance matrix of the LSE for θ, the optimality criteria and their properties, in particular the convexity and the equivalence theorem are considered from different aspects. Also the possibility of restriction to designs with limited values of measures of nonlinearity is mentioned. Research supported by the VEGA-grant of the Slovak grant agency No. 1/7295/20.     

Response of double-auction markets to instantaneous Selling�CBuying signals with stochastic Bid�CAsk spread

Statistical properties of order-driven double-auction markets with Bid–Ask spread are investigated through the dynamical quantities such as response function. We first attempt to utilize the so-called Madhavan–Richardson–Roomans model (MRR for short) to simulate the stochastic process of the price-change in empirical data sets (say, EUR/JPY or USD/JPY exchange rates) in which the Bid–Ask spread fluctuates in time. We find that the MRR theory apparently fails to simulate so much as the qualitative behaviour (‘non-monotonic’ behaviour) of the response function R(l) (l denotes the difference of times at which the response function is evaluated) calculated from the data. Especially, we confirm that the stochastic nature of the Bid–Ask spread causes apparent deviations from a linear relationship between the R(l) and the auto-correlation function C(l), namely, R(l) μ -C(l){R(l) \propto -C(l)}. To make the microscopic model of double-auction markets having stochastic Bid–Ask spread, we use the minority game with a finite market history length and find numerically that appropriate extension of the game shows quite similar behaviour of the response function to the empirical evidence. We also reveal that the minority game modeling with the adaptive (‘annealed’) look-up table reproduces the non-linear relationship R(l) μ -f(C(l)){R(l) \propto -f(C(l))} (f(x) stands for a non-linear function leading to ‘λ-shapes’) more effectively than the fixed (‘quenched’) look-up table does.     

Estimation of quantities determined as implicit functions of unknown parameters

Consider a family of distribution functions ${\{F(x, \theta),\,\theta \in \Theta\}}Consider a family of distribution functions {F(x, q), q ? Q}{\{F(x, \theta),\,\theta \in \Theta\}} . Suppose that there exists an estimator of the unknown parameter vector θ based on given data set. Then it is readily to obtain an estimator of any quantity given as an explicit function g(θ). Particularly, it is the case when the maximum likelihood estimator of θ is available. However, often some quantities of interest can not be expressed as an explicit function, rather it is determined as an implicit function of θ. The present article studies this problem. Sufficient conditions are given for deriving estimators of these quantities. The results are then applied to estimate change point of failure rate function, and change point of mean residual life function.     

Locally minimax test of independence in elliptically symmetrical distributions with additional observations

Summary LetX=(X _ij )=(X ₁, ...,X _n )’,X’ _i =(X _i1, ...,X _ip )’,i=1,2, ...,n be a matrix having a multivariate elliptical distribution depending on a convex functionq with parameters, 0,σ. Let ϱ²=ϱ ₂ ^-2 be the squared multiple correlation coefficient between the first and the remainingp ₂+p ₃=p−1 components of eachX _i . We have considered here the problem of testingH ₀:ϱ²=0 against the alternativesH ₁:ϱ ₁ ^-2 =0, ϱ ₂ ^-2 >0 on the basis ofX andn ₁ additional observationsY ₁ (n ₁×1) on the first component,n ₂ observationsY ₂(n ₂×p ₂) on the followingp ₂ components andn ₃ additional observationsY ₃(n ₃×p ₃) on the lastp ₃ components and we have derived here the locally minimax test ofH ₀ againstH ₁ when ϱ ₂ ^-2 →0 for a givenq. This test, in general, depends on the choice ofq of the familyQ of elliptically symmetrical distributions and it is not optimality robust forQ.     

18-run nonisomorphic three level orthogonal arrays

In this paper we construct all possible orthogonal arrays OA(18,q, 3,2) with 18 runs and 3 ≤ q ≤ 7 columns and present those that are nonisomorphic. A discussion on the novelty and the superiority of many of the designs found in terms of isomorphism and generalized minimum aberration has been made.      

Multivariate likelihood ratio orderings between spacings of heterogeneous exponential random variables

Let X ₁, X ₂, ..., X _n be independent exponential random variables such that X _i has failure rate λ for i = 1, ..., p and X _j has failure rate λ^* for j = p + 1, ..., n, where p ≥ 1 and q = n − p ≥ 1. Denote by D _i:n (p,q) = X _i:n −X _i-1:n the ith spacing of the order statistics X _1:n ≤ X _2:n ≤ ... ≤ X _n:n , i = 1, ..., n, where X _0:n ≡ 0. The purpose of this paper is to investigate multivariate likelihood ratio orderings between spacings D _i:n (p,q), generalizing univariate comparison results in Wen et al.(J Multivariate Anal 98:743–756, 2007). We also point out that such multivariate likelihood ratio orderings do not hold for order statistics instead of spacings. Supported by National Natural Science Foundation of China, the Program for New Century Excellent Talents in University (No.: NCET-04-0569), and by the Knowledge Innovation Program of the Chinese Academy of Sciences (No.: KJCX3-SYW-S02).     

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     

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 the strong uniform consistency of the mode estimator for censored time series

Let (T _n ) _n≥1 be a sequence random variables (rv) of interest distributed as T. In censorship models the rv T is subject to random censoring by another rv C. Let θ be the mode of T. In this paper we define a new smooth kernel estimator [^(q)]_n{\hat{\theta}_n} of θ and establish its almost sure convergence under an α-mixing condition.     

CEO ownership and firm value

This paper examines the hypothesis that the amount of CEO ownership has a dominating effect on the value of the firm. Using a diverse sample of firms, firm value as measured by Tobin's q is found to be a nonmonotonic function of CEO ownership. Specifically, Tobin's q rises when the CEO owns between 0 and 15% and declines as CEO ownership increases to 50%. Beyond 50%, the value starts to rise. Firm value also is found not to be a function of management ownership when CEO ownership is separated out, indicating that CEO ownership does have a dominating effect on firm value. Copyright © 1999 John Wiley & Sons, Ltd.     

On characterizing discrete distributions via conditional expectations of weak record values

Let (W _n ,n ≥ 0) denote the sequence of weak records from a distribution with support S = { α₀,α₁,...,α _N }. In this paper, we consider regression functions of the form ψ _n (x) = E(h(W _n ) |W _n+1 = x), where h(·) is some strictly increasing function. We show that a single function ψ _n (·) determines F uniquely up to F(α₀). Then we derive an inversion formula which enables us to obtain F from knowledge of ψ _n (·), ψ _n-1(·), h(·) and F(α₀).     

Stochastic monotonicity of the MLEs of parameters in exponential simple step-stress models under Type-I and Type-II censoring

In two recent papers by Balakrishnan et al. (J Qual Technol 39:35–47, 2007; Ann Inst Stat Math 61:251–274, 2009), the maximum likelihood estimators [^(q)]₁{\hat{\theta}_{1}} and [^(q)]₂{\hat{\theta}_{2}} of the parameters θ ₁ and θ ₂ have been derived in the framework of exponential simple step-stress models under Type-II and Type-I censoring, respectively. Here, we prove that these estimators are stochastically monotone with respect to θ ₁ and θ ₂, respectively, which has been conjectured in these papers and then utilized to develop exact conditional inference for the parameters θ ₁ and θ ₂. For proving these results, we have established a multivariate stochastic ordering of a particular family of trinomial distributions under truncation, which is also of independent interest.     

A geometric property of indifference curves in the (mean-standard deviation) plane

We prove that the slope of the indifference curve through point (m σ) of the (meanstandard deviation) plane must be smaller than the ratio (1+m)/σ. We show that the indifference curves corresponding to a quadratic utility function do actually satisfy that inequality; and prove, by the way, that these curves are not quarters of circles (as it is usually asserted), but smaller parts of them.

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Riassunto Si prova che, nel piano (media-scarto quadratico medio), l'inclinazione della curva d'indifferenza per il punto (m, σ) deve essere minore del rapporto (1+m)/gs. Si mostra che le curve d'indifferenza corrispondenti ad una funzione d'utilià quadratica soddisfano di fatto tale limitazione; e si prova, a questo proposito, che tali curve non sono — come generalmente asserito — quarti di circonferenze, ma archi più piccoli delle medesime.

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The second part of this article was stimulated by an observation of E. Castagnoli. The author gratefully acknowledges this to him.     

Management ownership and corporate value

Existing literature on the relation between management ownership and firm value has provided competing hypotheses and conflicting evidence. Using samples of Fortune 500-sized firms in 1976, 1980 and 1984, we find that corporate value measured by Tobin's q is a function of management ownership. Specifically, the q rises when management ownership is between 0% and 5-7%, and falls as the ownership increases to 10-12%. Beyond this range, we find that the q continues to fall in the 1976 sample, and starts to rise in the 1980 and 1984 samples. The evidence supports the hypothesis that there is a nonmonotonic relation between management ownership and corporate value.     

Minimax estimation under convex loss when the parameter interval is bounded

In the present paper families of truncated distributions with a Lebesgue density forx=(x ₁,...,x _n ) ε ℝ ⁿ are considered, wheref ₀:ℝ → (0, ∞) is a known continuous function andC _n (ϑ) denotes a normalization constant. The unknown truncation parameterϑ which is assumed to belong to a bounded parameter intervalΘ=[0,d] is to be estimated under a convex loss function. It is studied whether a two point prior and a corresponding Bayes estimator form a saddle point when the parameter interval is sufficiently small.     

Minimax invariant estimation of a continuous distribution function under entropy loss

For the invariant decision problem of estimating a continuous distribution function F with two entropy loss functions, it is proved that the best invariant estimators d ₀ exist and are the same as the best invariant estimator of a continuous distribution function under the squared error loss function L (F, d)=∫|F (t) −d (t) |² dF (t). They are minimax for any sample size n≥1.     

Conditions for the optimality of exponential smoothing forecast procedures

Exponential smoothing procedures, in particular those recommended byBrown [1962] are used extensively in many areas of economics, business and engineering. It is shown in this paper that:

1. Brown's forecasting procedures are optimal in terms of achieving minimum mean square error forecasts only if the underlying stochastic process is included in a limited subclass of ARIMA (p, d, q) processes. Hence, it is shown what assumptions are made when using these procedures.
2. The implication of point (i) is that the users ofBrown's procedures tacitly assume that the stochastic processes which occur in the real world are from the particular restricted subclass of ARIMA (p, d, q) processes. No reason can be found why these particular models should occur more frequently than others.
3. It is further shown that even if a stochastic process which would lead toBrown's model occurred, the actual methods used for making the forecasts are clumsy and much simpler procedures can be employed.