Asymptotic normality of a quadratic measure of orthogonal series type density estimate

LetX ₁,...,X _n be i.i.d. random variable with a common densityf. Let be an estimate off(x) based on a complete orthonormal basis {φ _k :k≧0} ofL ₂[a, b]. A Martingale central limit theorem is used to show that , where and .     

Asymptotic bounds for estimators without limit distribution

Let be a general family of probability measures,κ : a functional, and the optimal limit distribution for regular estimator sequences of κ. On intervals symmetric about 0, the concentration of this optimal limit distribution can be surpassed by the asymptotic concentration of an arbitrary estimator sequence only forP in a “small” subset of . For asymptotically median unbiased estimator sequences the same is true for arbitrary intervals containing 0. The emphasis of the paper is on “pointwise” conditions for , as opposed to conditions on shrinking neighbourhoods, and on “general” rather than parametric families.     

Nonparametric estimation of Matusita's measure of affinity between absolutely continuous distributions

LetF andG be two distribution functions defined on the same probability space which are absolutely continuous with respect to the Lebesgue measure with probability densitiesf andg, respectively. Matusita [3] defines a measure of the closeness, affinity, betweenF andG as: . Based on two independent samples fromF andG we propose to estimate ρ by , where and are taken to be the kernel estimates off(x) andg(x), respectively, as given by Parzen [5]. In this note sufficient conditions are given such that (i) asx→∞ and (ii) with probability one, asn→∞. Research supported in part by the National Research Council of Canada and by McMaster University Science and Engineering Research Board. The author is presently with the Department of Mathematical Sciences, Memphis State University, Memphis, Tennessee 38152.     

Asymptotic expansion of the log-likelihood function based on stopping times defined on a Markov process

Consider the parameter space Θ which is an open subset of ℝ ^k ,k≧1, and for each θ∈Θ, let the r.v.′sY _n ,n=0, 1, ... be defined on the probability space (X,A,P _θ) and take values in a Borel setS of a Euclidean space. It is assumed that the process {Y _n },n≧0, is Markovian satisfying certain suitable regularity conditions. For eachn≧1, let υ _n be a stopping time defined on this process and have some desirable properties. For 0 < τ _n → ∞ asn→∞, set h _n →h ∈R ^k , and consider the log-likelihood function of the probability measure with respect to the probability measure . Here is the restriction ofP _θ to the σ-field induced by the r.v.′sY ₀,Y ₁, ..., . The main purpose of this paper is to obtain an asymptotic expansion of in the probability sense. The asymptotic distribution of , as well as that of another r.v. closely related to it, is obtained under both and . This research was supported by the National Science Foundation, Grant MCS77-09574. Research supported by the National Science Foundation, Grant MCS76-11620.     

Primitive systems of elements in the variety\mathfrak{A}\mathfrak{N}_2 and some locally finite varieties of groupsand some locally finite varieties of groups

Part of any basis of a relatively free group in the variety is called a primitive system of elements. We provide a criterion of being primitive for , where is a variety of Abelian groups satisfying x^m=1, and a variety generated by a finite group. Let be a variety of nilpotent groups of class ≤c. It is proved that, for the group , the property of being primitive for an element g is stronger than the condition of being unimodular on a vector composed of values of Fox derivatives in the ring . The group is not residually finite whenever a system of elements is primitive. Supported by RFFR grant No. 96-01-01948. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 687–699, November–December, 1998.     

What we know about the second adjunction mapping

Let be an algebraic submanifold of complex projective space ℙ ^N . Assume thatn:=dim and let denote the restriction of the hyperplane section bundle to . The meromorphic map associated to fork≥1 ties together the pluricanonical maps of the surface sections of . Known results show that the behavior of is far better than one would expect from experience with the pluricanonical mappings of algebraic surfaces. In this article we discuss the known results on the structure of the mappings and describe the open problems. Conferenza tenuta il 5 giugno 1997     

A local duality theorem for categories of modules

Let Λ be an associative ring with identity and let be the category of left unitary Λ-modules. A subcateqory of the category is said to be small if the pairwise nonisomorphic objects of form a set. The main result of this paper consists of the fact that for every small full subcategory , there exists a ring Γ such that is dual to a small full subcategory of the category . Some applications of this result are indicated. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 66–73.     

A question in the theory of saturated formations of finite soluble groups

This paper examines the following question. If and are saturated formations then is defined to be the class of all soluble groups whose belong to . In general is a formation, but need not be a saturated formation. Here the smallest saturated formation containing is studied.     

Maximal deviation theory of some estimators of prior distribution functions

Summary Let be a sequence of independent identically distributed random variables withθ ₁∼G and the conditional distribution ofx ₁ givenθ ₁=θ given by . HereG is unknown andF _θ(·) is known. This paper provides estimators ofG based onx ₁, …,x _n such that the random variable sup has an asymptotic distribution asn→∞ under certain on conditionsG and for certain choices ofF _θ. A simulation model has been discussed involving the uniform distribution on (0, θ) forF _θ and an exponential distribution forG. Research supported by the National Science Foundation under Grant #MCS77-26809.     

Nonparametric estimation of an affinity measure between two absolutely continuous distributions with hypotheses testing applications

LetF andG denote two distribution functions defined on the same probability space and are absolutely continuous with respect to the Lebesgue measure with probability density functionsf andg, respectively. A measure of the closeness betweenF andG is defined by: . Based on two independent samples it is proposed to estimate λ by , whereF _n (x) andG _n (x) are the empirical distribution functions ofF(x) andG(x) respectively and and are taken to be the so-called kernel estimates off(x) andg(x) respectively, as defined by Parzen [16]. Large sample theory of is presented and a two sample goodness-of-fit test is presented based on . Also discussed are estimates of certain modifications of λ which allow us to propose some test statistics for the one sample case, i.e., wheng(x)=f ₀ (x), withf ₀ (x) completely known and for testing symmetry, i.e., testingH ₀:f(x)=f(−x).     

Measure-valued almost periodic functions

We consider Stepanov almost periodic functions μ ∈ ranging in the metric space of Borel probability measures on a complete separable metric space is equipped with the Prokhorov metric). The main result is as follows: a function , belongs to if and only if for each bounded continuous function , the function is Stepanov almost periodic (of order 1) and

On the representation of differentiations by Hamiltonians,operating from an algebra of local observable spin systems

Let and be algebras of local and quasilocal observable spin systems corresponding to the group Zr, be a differentiation invariant with respect to displacements. The question of representation of D in the form of formal Hamiltonian formed by the displacements of an elementx ε is considered. It is shown that such a representation exists if the condition holds, where means an element obtained from the elements [T_kX,a] by some r-multiple process of summation. Translated from Matematicheskii Zametki, Vol. 21, No. 1, pp. 93–98, January, 1977.     

Elementary properties of free extensions

For any partial groupoid , let Fr be the free extension of to a total groupoid. We show that implies and that the theory of Fr is uniformly recursive in the theory of . These results fail if “groupoid” is replaced by “semigroup”, “commutative semigroup”, “group”, “abelian group”, “semilattice”, “K-lattice” for any nontrivial varietyK of lattices, or “Boolean algebra”. Research supported in part by NSF Grant MCS78-01867. We thank the referee for his valuable comments. Presented by B. Jónsson.     

Baer cotorsion Pairs

LetR be a unital associative ring and two classes of leftR-modules. In [St3] the notion of a ( ) pair was introduced. In analogy to classical cotorsion pairs, a pair (V,W) of subclasses is called a ( ) pair if it is maximal with respect to the classes and the condition Ext _R ¹ (V, W)=0 for all . In this paper we study pairs whereR = ℤ and is the class of all torsion-free abelian groups andT is the class of all torsion abelian groups. A complete characterization is obtained assumingV=L. For example, it is shown that every pair is singly cognerated underV=L. The author was supported by a DFG grant.     

Integral-valued rational functions on valued fields

Let K be a field and a non-trivial valuation ring of K withm as its maximal ideal. Denote by and the rings of polynomials f∈K[X] and rational functions f∈K(X) resp. such that . We prove that for one variable X we have if and only if the completion of (K, ) is locally compact or algebraically closed. In the second case—i.e. if K is dense in the algebraic closure of (K, )—we even get for any number of variables X=(X₁,...,X_n). This work contains parts of the second author's thesis [Ri] written under the supervision of the first author.     

Construction of representations of the Poincaré Lie algebra by extension of direct integrals of representations ofSL(2,C)

Summary Continuing the program initiated in the first paper of this series, representations of the Lie algebra of the Poincaré group are constructed by forming suitable direct integrals of representations of SL(2, C) and extending them to by means of four additional momentum operators whose matrix elements are derived in a basis adapted to the direct integral structure. The representations obtained here include Wigner’s representations with positive mass and spin and Wigner’s representations with zero mass, spin and helicity ± . Lavoro eseguito con contributo del C.N.R. nell’ambito dell’attività di ricerca del Gruppo Nazionale per la Fisica Matematica e per le applicazioni della Matematica alla Fisica ed all’Ingegneria. Entrata in Redazione il 12 marzo 1972.     

A minimax regret estimator of a normal mean after preliminary test

Summary This paper considers the problem of estimating a normal mean from the point of view of the estimation after preliminary test of significance. But our point of view is different from the usual one. The difference is interpretation about a null hypothesis. Let denote the sample mean based on a sample of sizen from a normal population with unknown mean μ and known varianceσ ². We consider the estimator that assumes the value when and the value when where ω is a real number such that 0≤ω≤1 andC is some positive constant. We prove the existence of ω, satisfying the minimax regret criterion and make a numerical comparison among estimators by using the mean square error as a criterion of goodness of estimators.     

Interval estimation of the critical value in a general linear model

Summary This paper concerns interval estimation of the critical value θ which satisfies under the general linear model,Y _i =μ(x _i )+ε _i (i=1,2,···), where for and the functional forms off _j ^′ s are known. From an asymptotic expansion it is shown that, under reasonable conditions, the limiting distribution of is normal. Thus in the large-sample case a confidence interval for θ can be obtained. Such a result is useful when one is interested in carrying out a retrospective analysis rather than designing the experiment (as in the Kiefer-Wolfowitz procedure). In Section 3 a sequential procedure is considered for confidence intervals with fixed width 2d. It is shown that, for a given stopping variableN, is also asymptotically normal asd→0. Thus the coverage probability converges to 1−α (preassigned) asd→0. An example of application in estimating the phase parameter in circadian rhythms is given for the purpose of illustration. Research partially supported by the NSF Grant DMS-8502346.     

On the word problem and the conjugacy problem for groups of the formF/V(R)

LetF be a free group with at most countable system of free generators, letR be its normal subgroup recursively enumerable with respect to , and let be a variety of groups that differs from and for which the corresponding verbal subgroupV of the free group of countable rank is recursive. It is proved that the word problem inF/V(R) is solvable if and only if this problem is solvable inF/R, and if , then there exists anR such, that the conjugacy problem inF/R is solvable, but this problem is unsolvable inF/V(R) for any Abelian variety (all algorithmic problems are regarded with respect to the images of under the corresponding natural epimorphisms). Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 3–9, January, 1997. Translated by M. I. Anokhin