A Comparison of Methods for Estimating the Extremal Index

The extremal index, (01), is the key parameter when extending discussions of the limiting behavior of the extreme values from independent and identically distributed sequences to stationary sequences. As measures the limiting dependence of exceedances over a threshold u, as u tends to the upper endpoint of the distribution, it may not always be informative about the extremal dependence at levels of practical interest. Therefore we also consider a threshold-based extremal index, (u). We compare the performance of a range of different estimators for and (u) covering processes with < 1 and = 1. We find that the established methods for estimating actually estimate (u), so perform well only when (u) . For Markov processes, we introduce an estimator which is as good as the established methods when (u) but provides an improvement when (u) < = 1. We illustrate our methods using simulated data and daily rainfall measurements.     

On the speed of convergence in the central limit theorem of log-likelihood ratio processes

Let , the parameter space, be an open subset ofR ^k ,k1. For each , let the r.v.'sX _n ,n=1, 2,... be defined on the probability space (X, P ) and take values in (S,S,L) whereS is a Borel subset of a Euclidean space andL is the -field of Borel subsets ofS. ForhR _k and a sequence of p.d. normalizing matrices _n = _n ^{k × k} (₀ set _n ^* = ^* = ₀ + _n h, where ₀ is the true value of , such that ^*, . Let _n (^*, _*)( be the log-likelihood ratio of the probability measure with respect to the probability measure , whereP _n is the restriction ofP over _n = (X ₁,X ₂,...,X _n . In this paper we, under a very general dependence setup obtain a rate of convergence of the normalized log-likelihood ratio statistic to Standard Normal Variable. Two examples are taken into account.     

A graded scale of parametric families of distributions,and parameter estimates based on the sample mean

Summary Denote by _k a class of familiesP={P} of distributions on the line R¹ depending on a general scalar parameter , being an interval of R¹, and such that the moments µ₁()=xdP ,...,µ_2k ()=x ^2k dP are finite, ₁ (), ..., _k (), _k+1 () ..., _k () exist and are continuous, with ₁ () 0, and _j +1 ()= ₁ () _j () +[₂() -₁()²] _j ()/ ₁ (), J=2, ..., k. Let ₁=¯x=x ₁ + ... +x _n/n, ₂=x ₁ ² + ... +x _n ²/n, ..., _k =(x ₁ ^k + ... +x _n ^k/n denote the sample moments constructed for a sample x₁, ..., x_n from a population with distribution Pg. We prove that the estimator of the parameter by the method of moments determined from the equation ₁= ₁() and depending on the observations x₁, ..., x_n only via the sample mean ¯x is asymptotically admissible (and optimal) in the class _k of the estimators determined by the estimator equations of the form ₀ () + ₁ () ₁ + ... + _k () _k =0 if and only ifP _k .The asymptotic admissibility (respectively, optimality) means that the variance of the limit, as n (normal) distribution of an estimator normalized in a standard way is less than the same characteristic for any estimator in the class under consideration for at least one 9 (respectively, for every ).The scales arise of classes _{1 2}... of parametric families and of classes _{1 2} ... of estimators related so that the asymptotic admissibility of an estimator by the method of moments in the class _k is equivalent to the membership of the familyP in the class _k .The intersection consists only of the families of distributions with densities of the form h(x) exp {C₀() + C₁() x } when for the latter the problem of moments is definite, that is, there is no other family with the same moments ₁ (), ₂ (), ...Such scales in the problem of estimating the location parameter were predicted by Linnik about 20 years ago and were constructed by the author in [1] (see also [2, 3]) in exact, not asymptotic, formulation.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, pp. 41–47, 1981.     

Parametric stochastic convexity and concavity of stochastic processes

A collection of random variables {X(), } is said to be parametrically stochastically increasing and convex (concave) in if X() is stochastically increasing in , and if for any increasing convex (concave) function , E(X()) is increasing and convex (concave) in whenever these expectations exist. In this paper a notion of directional convexity (concavity) is introduced and its stochastic analog is studied. Using the notion of stochastic directional convexity (concavity), a sufficient condition, on the transition matrix of a discrete time Markov process {X _n(), n=0,1,2,...}, which implies the stochastic monotonicity and convexity of {X _n(), }, for any n, is found. Through uniformization these kinds of results extend to the continuous time case. Some illustrative applications in queueing theory, reliability theory and branching processes are given.Supported by the Air Force Office of Scientific Research, U.S.A.F., under Grant AFOSR-84-0205. Reproduction in whole or in part is permitted for any purpose by the United States Government.     

Efficient Estimation of the Canonical Dependence Function

The canonical dependence function (z), z [0,1], is introduced and studied in detail for distributions, which belong to the -neighborhood of a bivariate generalized Pareto distribution. We establish local asymptotic normality (LAN) of the loglikelihood function of a 2×2 table sorting of n i.i.d. observations and derive efficient estimators of (z) from the Hájek-LeCam Convolution Theorem. These results extend results by Falk and Reiss (2003) for the canonical dependence parameter (1/2) to arbitrary z (0,1).     

On dense sequences of polynomials in several variables

In the present paper a form of generalization of Gelfond's lemma on dense sequences of polynomials is proposed. For a set of complex numbers ₁, ..., _s we define the coefficients_gk( ₁, ..., _s ) (0ks) and give the relations between them and the transcendental degrees or the transcendence types of the field © ( ₁, ..., _s ) or its subfields.This work was completed at the Dept. of Math., Univ. of Southern Mississippi, USA in 1987.     

Lévy homeomorphic parametrization and exponential families

Summary Let P={P : } be an exponential family of probability distributions with the canonical parameter and consider the one to one mapping : P . It is shown that, under mild regularity assumptions, and ^–1 are continuous with respect to the Lévy metric in P and Euclidean metric in .     

Recurrence vs. transience for spatially inhomogeneous birth and death in a random environment

Summary We consider birth and death in a random environment, having probabilitiesp(,x) andq(,x) of a step to the right or left, respectively, from statex in environment. Under several sets of conditions involving the existence of limp(,x) and limq(,x) asx, we give conditions for certain extinction or escape to when 0 is absorbing, corresponding to recurrence or transience when 0 is reflecting. The environmental sequence is assumed to be stationary ergodic, and for some results stronger assumptions are required.     

Exact Solution of the Ising Model on the Cayley Tree with Competing Ternary and Binary Interactions

The exact solution is found for the problem of phase transitions in the Ising model with competing ternary and binary interactions. For the pair of parameters =(J) and ₁=₁(J ₁) in the plane (₁,), we find two critical curves such that a phase transition occurs for all pairs (₁,) lying between the curves.     

Asymptotic Formulae and Divisor Problems

We investigate the asymptotic behaviour of the summatory functions of _z(n, ), _k(n, ) z ⁽ⁿ⁾ and _k(n, ) z ⁽ⁿ⁾.     

On the Embedding of an Affine Space into a Projective Space

Let k, K be fields, and assume that |k| 4 and n, m 2, or |k| = 3 and n 3, m 2. Then, for any embedding of AG(n, k) into PG(m, K), there exists an isomorphism from k into K and an (n+1) × (m+1) matrix B with entries in K such that can be expressed as (x₁,x₂,...,x_n) = [(1,x₁ ,x₂ ,...,x_n )B], where the right-hand side is the equivalence class of (1,x₁ ,x₂ ,...,x_n )B. Moreover, in this expression, is uniquely determined, and B is uniquely determined up to a multiplication of element of K*. Let l 1, and suppose that there exists an embedding of AG(m+l, k) into PG(m, K) which has the above expression. If we put r = dim _k K, then we have r 3 and m > 2 l-1)/(r-2). Conversely, there exists an embedding of AG(l+m, k) into PG(m, K) with the above expression if K is a cyclic extension of k with dim _k K=r 3, and if m 2l/(r-2) with m even or if m 2l/(r-2) +1 with m odd.     

Admissibility of unbiased tests for a composite hypothesis with a restricted alternative

This paper discusses -admissiblility and d-admissiblity which are important concepts in studying the performance of statistical tests for composite hypotheses. A sufficient condition for -admissibility is presented. When =1/m, the Nomakuchi-Sakata test, which is uniformly more powerful than the likelihood ratio test for hypotheses min (₁, ₁) = 0 versus min (₁, ₁) > 0, is generalized for a class of distributions in an exponential family, and its unbiasedness and -admissibility are shown. Finally, the case of 1/m is discussed in brief.     

On-line k-Truck Problem and Its Competitive Algorithms

In this paper, based on the Position Maintaining Strategy (PMS for short), on-line scheduling of k-truck problem, which is a generalization of the famous k-server problem, is originally presented by our team. We proposed several competitive algorithms applicable under different conditions for solving the on-line k-truck problem. First, a competitive algorithm with competitive ratio 2k+1/ is given for any 1. Following that, if (c+1)/(c-1) holds, then there must exist a (2k-1)-competitive algorithm for k-truck problem, where c is the competitive ratio of the on-line algorithm about the relevant k-server problem. And then a greedy algorithm with competitive ratio 1+/, where lambda is a parameter related to the structure property of a given graph, is given. Finally, competitive algorithms with ratios 1+1/ are given for two special families of graphs.     

The Busy Period of the M/GI/∞ Queue

An equation for the distribution Z() of the duration T of the busy period in a stationary M/GI/ service system is constructed from first principles. Two scenarios are examined, being distinguished by the half-plane Re()>₀ for some ₀0 in which the generic service time random variable S, always assumed to have a finite mean E(S), has an analytic Laplace–Stieltjes transform E(e^–S ). If ₀<0 then E(e^–T ) is analytic in a half-plane (₁,), where ₀₁<0 and ₁ is determined by the distribution of S; then for any 0<s<|₁|.When ₀=0, E(e^–T ) is analytic in (0,), and now more is known about T. Inequalities on the tail () are used to show that for any 1, E(T ) is finite if and only if E(S ) is finite. It follows that the point process consisting of the starting epochs of busy periods is long range dependent if and only if E(S ²)=, in which case it has Hurst index equal to [frac12](3–), where is the moment index of S.If also the tail (x)=Pr{Sx} of the service time distribution satisfies the subexponential density condition ₀ ^x (x–u) (u)du/ (x)2E(S) as x, then (x)/ (x)e^E(S), where is the arrival rate.     

On the factor-thickness of regular graphs

For eachr-regular graphG, define a binary sequence(G) = ( ₁, ₂,..., _r-1) by _k = 0 ifG has ak-factor, and _k = 1 otherwise. A binary sequence = ( _i |i = 1, 2,...,r – 1) is said to be realizable if there exists anr-regular graphG such that(G) = . In this paper we characterize all binary sequences which are realizable.     

Inner functions and related spaces of pseudocontinuable functions

Let be an inner function, let C, ¦¦=1. Then the harmonic function [(+)]/(–)] is the Poisson integral of a singular measure D. N. Clark's known theorem enables us to identify in a natural manner the space H² H² with the space L² ( ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 7–33, 1989.     

A New Bound for the Fejér-Jackson Sum

We establish a new positive functional estimate for the Fejér-Jackson sum Sn() = k=1n k-1 sin k. This result enables us to give a simple proof of a result of Askey and Steinig on a monotonic sine sum associated with the sum Sn().     

Estimation of Spectral Densities with Multiplicative Parameter

A two-step estimation procedure is presented for spectral densities of the form f()=g() with and being unknown parameters. The classes of random fields for which the procedure is applicable are defined by restrictions on spectral densities of second and higher orders. The procedure suggests a minimum contrast estimator for the parameter which is then used to construct the estimator for . The delta method provides the asymptotic normality of our estimator for the parameter .     

On θ-Principal Extension and H-Closedness

A type of extensions called the -extention of topological spaces and their -equivalence and -trace systems are introduced, which ultimately characterize H-closed -extensions of a Hausdorff topological space. Also, the notion of -principal extensions is defined. A typical -principal extension consisting of certain grills on a Hausdorff space is constructed, and finally, some characterizations of H-closedness of a Hausdorff space are obtained.AMS Subject Classification (2000) 54D30 54D99     

Bayesian estimation in the symmetric location problem

Summary Let X _i =+ _i for i=1, ..., n, where the _i's are i.i.d. F and F is symmetric about 0. F is assumed unknown or only partially known, and the problem is to estimate . Priors are put on the pair (F,). The priors on F are obtained from Doksum's neutral to the right priors, and include symmetrized Dirichlet priors. The marginal posterior distribution of given X ₁, ..., X _nis computed and its general properties studied. It is found that for certain classes of distributions of the _i's, the posterior distribution of is for all large n a point mass at the true value of . If the distribution of the _i's is not exactly symmetric, the Bayes estimates can behave very poorly.