Median unbiased estimates for M.L.R.-families

Summary Lehmann [p. 83] has shown that some families of probability measures with monotone likelihood ratios (m.l.r.) admit median unbiased estimates which are optimum in the sense that among all median unbiased estimates they minimize the expected loss for any loss function which assumes its minimal value zero for the “true” parameter value and is nondecreasing as the parameter moves away from the true value in either direction. This very strong optimum property was proved under the assumption that all probability measures of the m.l.r.-family have continuous distribution functions, that they are mutually absolutely continuous and that each element of the support is the median of somep-measure of the family. This result does therefore not cover important cases such as the binomial families or thePoisson family. The purpose of the present paper is to show the existence ofrandomized median unbiased estimates with the same optimum property for m.l.r.-families which are closed and connected with respect to the strong topology. Such families are always dominated. We do, however, neither assume that thep-measures are mutually absolutely continuous nor that the distribution functions are continuous. We remark that the use of randomized estimates is indispensable here because nonrandomized median unbiased estimates do not always exist in the general case.     

Minimum kolmogorov distance estimates of parameters and parametrized distributions

Minimum Kolmogorov distance estimates of arbitrary parameters are considered. They are shown to be strongly consistent if the parameter space metric is topologically weaker than the metric induced by the Kolmogorov distance of distributions from the statistical model. If the parameter space metric can be locally uniformly upper-bounded by the induced metric then these estimates are shown to be consistent of ordern ^−1/2. Similar results are proved for minimum Kolmogorov distance estimates of densities from parametrized families where the consistency is considered in theL ₁-norm. The presented conditions for the existence, consistency, and consistency of ordern ^−1/2 are much weaker than those established in the literature for estimates with similar properties. It is shown that these assumptions are satisfied e.g. by all location and scale models with parent distributions different from Dirac, and by all standard exponential models. Supported by the scientific exchange program between the Hungarian Academy of Sciences and the Royal Belgian Academy of Sciences, and by GACR grant 201/93/0232.     

Note on minimum contrast estimates forMarkov processes

Summary In the present paper it is shown that the concept of minimum contrast estimates (m.c.e.) considered inPfanzagl [1969a] for independent and identically distributed (iid) observations can be modified to cover stationary discrete timeMarkov processes admitting a unique stationary distribution which dominates the transition probabilities (Condition (S)). Sufficient conditions on the measurability and strong consistency of m.c.e. stated inPfanzagl [1969a] for the idd case are reformulated to give sufficient conditions for the existence of measurable m.c.e. and their strong consistency for such processes (section 1). The proofs of the main theorems are only sketched, because they are nearly the same as those given inPfanzagl [1969a] for the iid case.The concept of m.c.e. covers maximum likelihood estimates (m.l.e.) as a special case; therefore an application of the results to m.l.e. yields sufficient conditions for the existence of measurable m.l.e. and their strong consistency if the parameter space is compact metrizable or locally compact with countable base (Section 2). These conditions are weaker than the usual regularity conditions (see for exampleBillingsley [1961b] and the references cited there) and under the assumption that the transition probabilities as well as the stationary distribution are absolutely continuous with respect to a -finite measure they can be expressed in terms of the corresponding equivalence classes of transition densities. This seems to be more transparent than the conditions given byRoussas [1965]. In Section 3 asymptotic normality of m.c.e. is proved under conditions which correspond to those used inRoussas [1968] for the case of m.l.e.     

Characterization of some types of completeness resp. Total completeness and their conservation under direct products

Some notions ofL _p (μ)-completeness resp. totally L _p (μ)-completeness (1≦p≦∞) are characterized for families of probability distributions dominated by aσ-finite measureμ and their conservation with respect to direct products is proved. Furthermore, it is shown that totallyL _∞(μ)-completeness does not implyL ₁(μ)-completeness and that there are families of probability distributions in the i.i.d. case induced by the order statistic, which are L₁(μ)-complete but not totallyL _∞(μ)-complete.     

Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof

In this paper, we present a new unified approach and an elementary proof of a very general theorem on the existence of a semicontinuous or continuous utility function representing a preference relation. A simple and interesting new proof of the famous Debreu Gap Lemma is given. In addition, we prove a new Gap Lemma for the rational numbers and derive some consequences. We also prove a theorem which characterizes the existence of upper semicontinuous utility functions on a preordered topological space which need not be second countable. This is a generalization of the classical theorem of Rader which only gives sufficient conditions for the existence of an upper semicontinuous utility function for second countable topological spaces.     

Eine Bemerkung zur Existenz erwartungstreuer Schätzer

Summary A theorem ofBickel andLehmann concerning the existence of unbiased estimates for parameters on the convex family of all absolutely continuous probability distributions is generalized to arbitrary convex families of absolutely continuous distributions. In contrast to the result ofBickel andLehmann, however, we give necessary and sufficient conditions for the existence of only approximately unbiased estimates. An example, which is borrowed from density estimation, shows that in general the term approximately cannot be ommitted.     

Necessary and sufficient conditions for consistency of generalizedM-estimates

GeneralizedM-estimates (minimum contrast estimates) and their asymptotically equivalent approximate versions are considered. A relatively simple condition is found which is equivalent with consistency of all approximateM-estimates under wide assumptions about the model. This condition is applied in several directions. (i) A more easily verifiable condition equivalent with consistency of all approximateM-estimates is derived and illustrated on models with stationary and ergodic observations. (ii) A condition sufficient for inconsistency of all approximateM-estimates is obtained and illustrated on models with i.i.d. observations. (iii) A simple necessary and sufficient condition for consistency of all approximateM-estimates in linear regression with i.i.d. errors is found. This condition is weaker than sufficient conditions for consistency ofM-estimators known from the literature. A linear regression example is presented where theM-estimate is consistent and an approximateM-estimate is incosistent.Supported by CSAS grant N. 17503.     

Generalized weak transfer continuity and the Nash equilibrium

This paper introduces the notion of generalized weak transfer continuity and establishes that a bounded, compact locally convex metric quasiconcave and generalized weak transfer continuous game has a Nash equilibrium. Our equilibrium existence result neither implies nor is implied by the existing results in the literature such as those in [Carmona, G., 2011. Understanding some recent existence results for discontinuous games. Economic Theory 48, 31–45], [Prokopovych, P., 2011. On equilibrium existence in payoff secure games. Economic Theory 48, 5–16], [Carmona, G., 2009. An existence result for discontinuous games. Journal of Economic Theory 144, 1333–1340], and [Reny, P.J., 1999. On the existence of pure and mixed strategy Nash equilibria in discontinuous games, Econometrica 67, 1029–1056].     

Interim correlated rationalizability in infinite games

In a Bayesian game, assume that the type space is a complete, separable metric space, the action space is a compact metric space, and the payoff functions are continuous. We show that the iterative and fixed-point definitions of interim correlated rationalizability (ICR) coincide, and ICR is non-empty-valued and upper hemicontinuous. This extends the finite-game results of Dekel et al. (2007), who introduced ICR. Our result applies, for instance, to discounted infinite-horizon dynamic games.     

Utility representation theorems for Debreu separable preorders

We prove the existence of arbitrary (resp., semicontinuous, continuous) utility representations for arbitrary (resp., semicontinuous, continuous) preorders satisfying some weakened Debreu order separability conditions. In this way we widely generalize a classical result for total preorders that essentially is due to Debreu.     

On selecting the best ofk lognormal distributions

Summary Fork lognormal populations, which differ only in one certain parameter Ϙ, the problem of finding the population with the largest value ofϑ is considered. For two-parameter lognormal families, several natural choices ofϑ are treated, where the problem can be solved, through logarithmic transformation of the observations, within the framework of estimating parameters ink, possibly restricted, normal populations. For three-parameter lognormal families, this standard approach of selecting in terms of natural estimators fails to work ifϑ is the “guaranteed lifetime”. For this case, a selection procedure is derived which is based on anL-statistic which has the smallest asymptotic variance. Of importance here is that it is location equivariant, whereas it does not matter what it actually estimates. Comparisons are made with other suitable selection rules, through the asymptotic relative efficiencies, as well as in an example of intermediate sample sizes. It is shown that only in the latter, the selection rule, which is based on the sample minima, compares favorably. The research of this author was supported by the Office of Naval Research Contract N00014-88-K-0170 and NSF Grant Number DMS-8606964 at Purdue University. Reproduction in whole or in part is permitted for any purpose of the United States Government. The research of this author was supported by the Air Force Office of Scientific Research Grant 85-0347 at the University of Illinois at Chicago.     

The joint continuity of the expected payoff functions

Glicksberg [Glicksberg, I.L., 1952. A further generalization of the Kakutani fixed point theorem, with applications to Nash equilibrium points. In: Proceedings of the American Mathematical Society 3, pp. 170–174] generalized the Kakutani fixed point theorem to the setting of locally convex spaces and used it to prove that every k-person strategic game with action sets convex compact subsets of locally convex spaces and continuous payoff functions has a Nash equilibrium. He subsequently used this result to establish the following fundamental theorem of game theory: Every k-person strategic game with action sets metrizable compact topological spaces and continuous payoff functions has a mixed strategies equilibrium. However, in his proof of the latter result, Glicksberg did not show that the expected payoff functions were jointly continuous, something that was required for the existence of a mixed strategies equilibrium.     

Nonparametric inference on multivariate versions of Blomqvist’s beta and related measures of tail dependence

We consider nonparametric estimation of multivariate versions of Blomqvist’s beta, also known as the medial correlation coefficient. For a two-dimensional population, the sample version of Blomqvist’s beta describes the proportion of data which fall into the first or third quadrant of a two-way contingency table with cutting points being the sample medians. Asymptotic normality and strong consistency of the estimators are established by means of the empirical copula process, imposing weak conditions on the copula. Though the asymptotic variance takes a complicated form, we are able to derive explicit formulas for large families of copulas. For the copulas of elliptically contoured distributions we obtain a variance stabilizing transformation which is similar to Fisher’s z-transformation. This allows for an explicit construction of asymptotic confidence bands used for hypothesis testing and eases the analysis of asymptotic efficiency. The computational complexity of estimating Blomqvist’s beta corresponds to the sample size n, which is lower than the complexity of most competing dependence measures.      

Hypothesis testing and parameter estimation based onM-statistics ink samples with unequal variances

Summary Ink samples with unequal variances,M-tests for homogeneity ofk location parameters are proposed. The asymptoticχ ²-distributions of the test statistics and the robustness of the tests are investigated. NextM-estimators (ME’s) of parameters are discussed. Furthermore positive-part shrinkage versions (PSME’s) of theM-estimators for the location parameters are considered along with modified James-Stein estimation rule. In asymptotic distributional risks based on a special feasible loss, it is shown that the PSME’s dominate the ME’s, and preliminary test and shrinkageM-versions fork≧4.     

Strong consistency of the MLE under random censoring

LetX ₁, ...,X _n be an i.i.d. sample from some parametric family {_θ :θ (Θ} of densities. In the random censorship model one observesZ _i =min (X _i ,Y _i ) andδ _i =1_{ x _i ≤Y _i}, whereY _i is a censoring variable being independent ofX _i . In this paper we investigate the strong consistency ofθ _n maximizing the modified likelihood function based on (Z _i ,δ _i , 1≤i≤n. The main result constitutes an extension of Wald’s theorem for complete data to censored data. Work partially supported by the “Deutsche Forschungsgemeinschaft”.     

A note on reliability properties of k-record statistics

LetY _k,n denote the nth (upper) k-record value of an infinite sequence of independent, identically distributed random variables with common continuous distribution function F. We show that if the nth k-record valueY _k,n has an increasing failure rate (IFR), thenY _l,n (l<k) andY _k+1,n+1(n≤k+1) also have IFR distributions. On the other hand, ifY _k,n has a decreasing failure rate (DFR), thenY _l,n (1>k) has also a DFR distribution. We also present some results concerning log convexity and log concavity ofY _k,n .     

Kernel estimation of a smooth distribution function based on censored data

The problem of estimating a smooth distribution functionF at a pointτ based on randomly right censored data is treated under certain smoothness conditions onF. The asymptotic performance of a certain class of kernel estimators is compared to that of the Kaplan-Meier estimator ofF(τ). It is shown that the relative deficiency of the Kaplan-Meier estimator ofF(τ) with respect to the appropriately chosen kernel type estimator tends to infinity as the sample sizen increases to infinity. Strong uniform consistency and the weak convergence of the normalized process are also proved. Research Surported in part by NIH grant 1R01GM28405.     

Approximation of contractible valued correspondences by functions

Let X and Y be absolute neighborhood retracts (this is a large class of spaces) with X compact, and let F:X→Y be an upper hemicontinuous correspondence whose values are compact and contractible. It is shown that any neighborhood of the graph of F contains the graph of a continuous function f:X→Y. The relevance of this result to fixed point theory is indicated. It is also shown that if X is ‘locally infinite’, then F can be approximated in the stronger sense of the graph of f being close to the graph of F and every point in the graph of F being close to some point in the graph of f. A conjectured generalization of the main result is stated.     

The weighted empirical process of row independent random variables with arbitrary distribution functions

Abstract  The relative compactness and weak convergence of reduced empirical, quantile and weighted empirical processes are demonstrated under mild conditions. For example, the reduced empirical proces of arbitrary independent iv'S is always relatively compact in any ‖ /q ‖ metric with q square integrable; and weak convergence takes place if and only if the covariance function converges. V an Z uijlen's representation of the general empirical process is a fundamental tool. This paper both unifies and extends the previous treatment of these processes - processes that have application to rank statistics and linear combinations of order statistics. Most proofs are contained in Section 3. A brief introduction to weak convergence is presented in the appendix for readers lacking this background. Applications are indicated in Section 4.     

Semicontinuous extension of a partial order

This paper shows that it is always possible to determine an extension of a strict partial order to a strict weak order that preserves simultaneously some order denseness and topological properties; as a corollary, sufficient conditions for the existence of an upper semicontinuous utility on a strictly partially ordered space are derived; a direct proof of a complementary result is also given.