Revelations of a gambler

Knowing that a decision maker maximizes expected utility with respect to some (unknown) utility U and some (unknown) probability P, what can one tell about P by observing his decisions? We discuss this revealed preference question primarily in the simple case of a two-element (H and T) state space, and show that the possible revelations of P_T/P_H are precisely those of the form P_T/P_Hε∪^K_k=1 (γ_k,δ_k), for some algebraic numbers γ_k,δ_k.     

Correcting for truncation bias caused by a latent truncation variable

We discuss estimation of the model Y_i=X_ib_Y+e_Yi, T_i=X_ib_T+ e_Ti, when data on the continuous dependent variable Y and on the independent variables X are observed iff the ‘truncation variable’ T>0 and when T is latent. This case is distinct from both (i) the‘censored sample’ case, in which Y data are available iff T>0, T is latent and X data are available for all observations, and (ii) the ‘observed truncation variable’ case, in which both Y and X are observed iff T>0 and in which the actual value of T is observed whenever T>0. We derive a maximum-likelihood procedure for estimating this model and discuss identification and estimation.     

Third-order optimum properties of estimator-sequences

SequencesT ⁽ⁿ⁾ ,n∈N, are considered, whereT ⁽ⁿ⁾ estimates a vector parameter ?∈R ^p from an i.i.d. sample of sizen, and such sequences are compared on the basis of their risks ∫L(n ^1/2(T ⁿ (x)?θ))P _θ ⁿ (dx) relative to loss functionsL:R ^p →R. A characterization is given for sequencesT ^*(n) ,n∈N, which generate an essentially complete class in the following sense: For any sequenceT ⁽ⁿ⁾ ,n∈N, there exist functions Φ _n ,n∈N, such that forn→∞ we have $$\begin{gathered} \smallint L (n^{1/2} (T^{*(n)} + n^{ - 1} \Phi _n (T^{*(n)} ) - \theta )) dP_\theta ^n \leqslant \hfill \\ \leqslant \smallint L (n^{1/2} (T^{(n)} - \theta )) dP_\theta ^n + o (n^{ - 1} ), \hfill \\ \end{gathered} $$ for every ? and everyL satisfying certain conditions. If the estimator-sequences are compared by their risks ∫W(T ⁽ⁿ⁾ d P _θ ⁿ ,θ) with respect to loss functionsW:R ^p ×Θ→R then a similar result on asymptotically complete classes is valid. The results are obtained under the assumption thatT ^*(n) ,n∈N, andT ⁽ⁿ⁾ ,n∈N, admit stochastic expansions which are sufficiently regular, that the loss functionsL andW are sufficiently smooth and bounded by polynomials, and that the estimator-sequences have asymptotically bounded moments; the latter condition is not needed for bounded functionsL andW.     

Sulle operazioni associative tra variabili casuali

Si mostra che, sotto condizioni di regolarità, seo è un’operazione associativa tra variabili casuali reali e indipendenti, è definibile una trasformata integrale ξ delle loro funzioni di ripartizione con la proprietà: ξx ₀ Y (t)=ξx(t)·ξ _y (t). Si indicano alcune proprietà di tale trasformata e si tratta della possibilità di estendere a un’operazione associativa risultati noti per l’addizione tra variabili casuali. In particolare ci si occupa dell’« infinita divisibilità » fornendo condizioni perché una variabile casualeX ammetta la rappresentazioneX=X _1O X _2O … _o X _n per ognin naturale con leX _i indipendenti e identicamente ditribuite.     

On the maximum number of balancing subsets

Let a₁,…,a_n be a finite sequence of n real numbers. A subset {i₁,…,i_r} of the set {1,…,n } is called a balancing subset for a₁,…,a_n if a_i1 + … + a_ir = 0. We discuss in this paper the maximum number of balancing subsets for given n.     

Utility functions for Debreu's ‘excess demands’

Given an arbitrary function x: $\text{R}$^l→$\text{R}$^l satisfying Walras law and homogeneity, Debreu decomposed x into the sum of l ‘individually rational’ functions x(p)=Σ^l_k=1[uvbar|x]^k(p). Here we find explicit utility functions u^k, constructed on the basis of a simple geometric intuition, which give rise to Debreu's excess demands [uvbar|x]^k(p).     

Cointegration and distance between differenced processes

In this note, it is argued that cointegration augments the distance between the differenced series. If two series, x _t and y _t , are integrated of order one and cointegrated and v _t and w _t are integrated of order one but not cointegrated then, under certain conditions, the distance between ??x _t and ??y _t is more than the distance between ??v _t and ??w _t .     

A numerical representation of preferences with intransitive indifference

It is shown that, under certain conditions on a preference relation on a subset X of R^m, there exist real-valued functions u and v on X such that x is preferred to y if and only if u(x)>v(y). This generalises the familiar representation of preferences by a utility function in the case where the preference and indifference relations are transitive. The continuity of the functions u and v is also discussed.     

A theory of coalition formation in committees

Let G = (N,W) be a strong weighted majority game and let A be a set of alternatives. Denote by L the set of linear orders on A. A social choice function F:L^N → A is a representation of G if the simple game G¹(F) associated with F equals G. A coalition S is determining in G if it satisfies the following condition. Let F be a representation of G and let R^N ? L^N. Then, if a simple majority of the members of S consider an alternative x to be their best choice, then S can ‘enforce’ x to be a Nash equilibrium payoff in the resulting non-cooperative voting game g(F,R^N). In this paper we generalize the above notion of a determining coalition to committees (i.e., proper and monotonic simple games), and give a complete characterization of the set of determining coalitions of a committee. Furthermore, we discuss our notion of a determining coalition in the light of some real-life data on formation of coalitions in town councils in Israel.     

Estimation in the first-order moving average model through the finite autoregressive approximation: Some asymptotic results

To estimate α in the model y_t = u_t+αu_t?1, we consider a proposal by Durbin (Biometrika, 1969). It consists in fitting an autoregression of order k to the data, and deriving from there an estimate α^. The probability limit and the variance of the limiting normal distribution of α^ are presented and discussed in detail, when the sample size T → ∞, but k remains fixed. The differences between the resulting values and those corresponding to the maximum likelihood estimator are exponentially decreasing functions of k. Several modifications of the estimator are discussed and found consistent, but asymptotically inefficient.     

Distributions determined by conditioning on a single order statistic

LetX ₁,X ₂, …,X _n(n ? 2) be a random sample on a random variablex with a continuous distribution functionF which is strictly increasing over (a, b), ?∞ ?a <b ? ∞, the support ofF andX _1:n ?X _2:n ? … ?X _n:n the corresponding order statistics. Letg be a nonconstant continuous function over (a, b) with finiteg(a ⁺) andE {g(X)}. Then for some positive integers, 1 <s ?n $$E\left\{ {\frac{1}{{s - 1}}\sum\limits_{i - 1}^{s - 1} {g(X_{i:n} )|X_{s:n} } = x} \right\} = 1/2(g(x) + g(a^ + )), \forall x \in (a,b)$$ iffg is bounded, monotonic and \(F(x) = \frac{{g(x) - g(a^ + )}}{{g(b^ - ) - g(a^ + )}},\forall x \in (a,b)\) . This leads to characterization of several distribution functions. A general form of this result is also stated.     

Nonparametric identification of a binary random factor in cross section data

Suppose V and U are two independent mean zero random variables, where V has an asymmetric distribution with two mass points and U has some zero odd moments (having a symmetric distribution suffices). We show that the distributions of V and U are nonparametrically identified just from observing the sum V+U, and provide a pointwise rate root n estimator. This can permit point identification of average treatment effects when the econometrician does not observe who was treated. We extend our results to include covariates X, showing that we can nonparametrically identify and estimate cross section regression models of the form Y=g(X,D^∗)+U, where D^∗ is an unobserved binary regressor.     

Some results on the existence of utility functions on path connected spaces

This paper concerns the existence of utility representations for preferences defined on a path connected space X. This includes any convex set. A classical result of Eilenberg (1941) proves the existence of utility representations when the consumption set is connected and separable. In an infinite dimensional space the above result may not be useful, because we lack, in general, the separability of the space. The non-separable spaces L^∞ and ca(K) are typical examples in mathematical economics. In this paper we show that a continuous preference relation ≽, on X has a continuous utility representation if and only if it is countably bounded, i.e., there is some countable subset F of X such that for all x in X there exist y and z in F with y≽x≽z. An easy corollary states that any continuous preference which has a best and a worst point has a continuous representation. We also obtain a convex continuous preference on a Banach lattice that has not a utility representation, because it is not countably bounded.     

A note on compact differentiability and theδ-method

Letη _n ,n ∈ ?, be arbitrary functions defined on a probability space (ω,A,P) with values in a normed vector spaceB ₁ ,μ ∈ B ₁ andξ ₀ a separable random element inB ₁ such thatξ _n :=√n(η _n ?μ) converges weakly toξ ₀ in the sense of Hoffmann-Jørgensen. Then with (B ₂, ∥·∥₂) being another normed vector space andφ:B ₁→B ₂ compactly differentiable atμ with derivateD _μ, the random variable $\parallel \sqrt n (\phi (n_n ) - \phi (\mu )) - D_\mu (\sqrt n (n_n - \mu ))\parallel 2*$ converges to 0P-stochastically where “*” denotes the measurable cover. We show that the classicalδ — method extends to the non-measurable case where in the proof we shall not make use of any representation theorems but only of a slight refinement of the usual characterisation of compact differentiability, due to the fact that we will not assume {ξ _n :n ∈ ?} being tight.     

Estimation in a disequilibrium model and the value of information

Suits' model of the watermelon market is reformulated as a disequilibrium model and the likelihood function is derived under various assumptions concerning the amount of available information. Such models are characterized by a min condition as in y_t = min(D_t, S_t) for an ordinary demand-supply model, where the observed quantity y_t is the smaller demand and supply. Varying amounts of information may be available depending on which variables are observed and on whether prior information is available for separating the sample into subsets for which D_t < S_t and conversely. The likelihood function corresponding to the least amount of prior information is shown to be unbounded without a certain variance ratio restriction. Computations are successfully carried out both with the original model and Monte Carlo experiments and the effect of different amounts of information on MSE's is analyzed.     

Existence of a convex extension of a preference relation

In this paper we say that a preference (i.e. irreflexive) relationP isregular (or aweak order) if both it and its non-comparability relation are transitive; we also say that a preference relationP ^* is aconvex extension of another preference relationP ifP?P ^* holds andP ^* is regular and convex-valued. We prove that a convex extension ofP exists if and only if every non-empty and finite set of alternativesA is not included in the convex hull of ∪ _x∈A P(x).     

Improved stein-rule estimator for regression problems

Stein-Rule estimator for regression problems has been studied by several authors including Sclove (1968) and others listed in Vinod's (1978) survey. Ullah and Ullah (1978) provide the expressions for the mean squared error (MSE) of a double k-class (KK) estimator with parameters k₁ and k₂. When k₂=1 the Stein-Rule estimator becomes a special case of KK and an optimal choice of k₁ is known. This paper explores optimal theoretical choice of k₁ and k₂. We note that negative choices of k₂ are permissible, and that thereis a large range of choices for K₁ and k₂ where the MSE of the Stein-Rule estimator can be reduced for regression problems based on multicollinear data. A simulation experiment is included.     

A new non-linear AR(1) time series model having approximate beta marginals

We consider the mixed AR(1) time series model $$X_t=\left\{\begin{array}{ll}\alpha X_{t-1}+ \xi_t \quad {\rm w.p.} \qquad \frac{\alpha^p}{\alpha^p-\beta ^p},\\ \beta X_{t-1} + \xi_{t} \quad {\rm w.p.} \quad -\frac{\beta^p}{\alpha^p-\beta ^p} \end{array}\right.$$ for ?1 < β ^p ≤ 0 ≤ α ^p  < 1 and α ^p ? β ^p  > 0 when X _t has the two-parameter beta distribution B₂(p, q) with parameters q > 1 and ${p \in \mathcal P(u,v)}$ , where $$\mathcal P(u,v) = \left\{u/v : u < v,\,u,v\,{\rm odd\,positive\,integers} \right\}.$$ Special attention is given to the case p = 1. Using Laplace transform and suitable approximation procedures, we prove that the distribution of innovation sequence for p = 1 can be approximated by the uniform discrete distribution and that for ${p \in \mathcal P(u,v)}$ can be approximated by a continuous distribution. We also consider estimation issues of the model.     

Practical models in credibility theory,including parameter estimation

In credibility theory, an unobservable random vector Y is approximated by a random vector ? in a pre-assigned set $\text{A}$ of admitted estimators for Y. The credibility approximation ? for Y is best in the sense that it is the vector V ? $\text{A}$ minimizing the distance between V and Y. The credibility estimator depends on observable random variables but also on unknown structure parameters. In the practical models, the latter can be estimated from realizations of the observable random variables.     

Inequalities for the Gini coefficient of composite populations

Unlike other popular measures of income inequality, the Gini coefficient is not decomposable, i.e., the coefficient G($\text{X}$) of a composite population $\text{X}\text{=}\text{X}{}_1\text{∪…∪}\text{X}{}_{\mathrm{r}}$ cannot be computed in terms of the sizes, mean incomes and Gini coefficients of the components $\text{X}$_i. In this paper upper and lower bounds (best possible for r=2) for G($\text{X}$) in terms of these data are given. For example, $\text{G(}\text{X}{}_1\text{∪…∪}\text{X}{}_{\mathrm{r}}\text{)≧Σα}{}_{\mathrm{i}}\text{G(}\text{X}{}_{\mathrm{i}}\text{)}$, where α_i is the proportion of the pop ulation in $\text{X}$_i. One of the tools used, which may be of interest for other applications, is a lower bound for ∫^∞₀f(x)g(x)dx (converse to Cauchy's inequality) for monotone decreasing functions f and g.