Empirical Bayes Test for the Parameter of the Truncated-Type Distribution Families

The empirical Bayes test (EBT) is proposed for testing H₀ : ₀ H₁ : > ₀ in the truncated-type distribution families. It is found that the EBT proposed is obtained asymptotically optimal and its convergence rate is also obtained.AMS Subject Classification (2000) 62J10, 62G99     

Solution of the nonlinear functional equations representing the roll gap relationships in a cold mill

In the development of a roll force model for cold rolling, techniques were developed for solving the system equations which are of general interest. This paper gives a brief introduction to the physical model but concentrates on the solution of the model equations and the simulation. An unusual feature of the model was that the calculated profiles had to satisfy a number of boundary conditions at different points throughout the roll arc. A new method was developed for calculating these profiles and for determining the gradient functions which satisfied the boundary constraints.Nomenclature p() pressure at roll angle - h() gauge - a() roll radius - y() yield stress - g _i () gradient function on iterationi - e() gauge error - (, ) transition function - H(–) Heaviside unit step function at = - (–) unit impulse function at = - H(, ₁, ₂) defined asH( – ₁) –H( – ₂) - angular position from the roll center line - _T angular limits of roll arc represented - _n angular position of the neutral angle - _i angular position ofith strip elastic-plastic boundary - _pi pressure change at the boundaryi - _i , _i , _i constants defined in Appendix A - k ₁,k ₂ elastic region constants - k total number of strip boundaries (elastic-plastic and entry and exit points) - R undeformed work roll radius - R _s roll separation—distance between roll centers - h ₀₁ unstrained gauge in an elastic region - h _in gauge of the strip at the entry to the roll gap - J gauge error cost function - <x, y> inner product ofx andy - x norm ofx - L ₂[0, _T ] the space of Lebesgue square-integrable functions defined on the interval [0, _T ] - JUVY denotes (Dx)() =dx()/d The author would like to acknowledge the help given by Dr. G. F. Bryant, Director, and Mr. M. A. Fuller, Senior Research Engineer, the Industrial Automation Group, Imperial College of Science and Technology, London. He is also grateful to M. J. G. Henderson of the University of Birmingham for his advice and encouragement during the project. He would like to thank the Directors of GEC Electrical Projects Limited for allowing him to undertake the work and also Mr. J. McTaggart and Mr. C. McKenzie (GEC), Professor H. A. Prime of the University of Birmingham, and Dr. G. F. Bryant for arranging the project.     

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.     

Minimality of root vectors of operator functions analytic in an angle

We study the minimality of elementsx _h,j,k of canonical systems of root vectors. These systems correspond to the characteristic numbers _k of operator functionsL() analytic in an angle; we assume that operators act in a Hilbert space . In particular, we consider the case whereL()=I+T()c, >0,I is an identity operator,C is a completely continuous operator, (I- C)^–1c for ¦arg¦, 0<<, the operator functionT() is analytic, and T()c for ¦arg¦<. It is proved that, in this case, there exists >0 such that the system of vectorsC ^v x _h,j,k is minimal in for arbitrary positive <1+, provided that ¦_k¦>.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 545–566, May, 1994.This research was partially supported by the Ukrainian State Committee of Science and Technology.     

Criteria of algebraic independence of complex numbers over a field of finite transcendence type (II)

In the present paper we establish a criterion of algebraic independence of complex numbers ₁, ..., _n over a field of finite transcendence type using a sequence of nonzero polynomials in several variables with integral coefficients, which satisfy simultaneously certain upper and lower estimates in different orders of magnitude at the point ( ₁, ..., _q , ₁, ..., _n ), where { ₁, ..., _q } is a transcendence basis of over .The Project Supported by the National Natural Science Foundation of China     

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.     

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.     

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.     

Two consistency results concerning thin-tall Boolean algebras

In this paper we show that the following is relatively consistent withZFC +CH: There is no superatomic Boolean algebra of height ₂+1 and width, and there is no superatomic Boolean algebraA with for 0<<₁ and Presented by J. Mycielski.     

Growth along a ray of a subharmonic function,having a mass distributed on the negative axis

Let U be a subharmonic function in C with a Riesz mass , distributed on the negative semiaxis without some neighborhood of zero, let and be its order and lower order, and let B(r, U) be the maximum of U(z) for ¦z¦=r. Estimates are obtained for the measure of sets of those values of r 0 for which certain inequalities hold. The following result is typical. LetE = {r:u(re ^l)–cosB<(r,U) > 0}. If < < 1, ¦¦=., then the lower logarithmic density of the set E is at least 1 – /. If < > 1,¦¦ ., then the upper logarithmic density of the set E is at least 1 – /.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 31–38, 1988.     

On regular embedding integrally in R3 of metrics of class C4 of negative curvature

On the x₀y plane let there be specified a complete metric of negative curvature K by means of the line element ds²=dx²+B²(x, y) dy², and, in the strip _a={0xa, -4-bounded function B>0,K-²<0 ( and are constants). Then, the metric in strip _a is embedded in R³ by means of a surface of class C³.Translated from Matematicheskie Zametki, Vol. 14, No. 2, pp. 261–266, August, 1973.     

A new stochastic approximation procedure using quantile curves

Summary Suppose that for each real number x we can sample at will from a population having unknown distribution function F _x (·), that p is a fixed number in (0,1), that a is a fixed real number, and that there is a unique unknown value of x (call it ) such that is the p-th quantile of F .We propose scheme for sequentially approximating , present a theorem giving conditions under which these approximations converge to with probability one, discuss the conditions of the theorem, give some examples, and mention some variations.The research of this author was supported by National Science Foundation Grant No. MCS 77-00841The research of this author was supported by the Research Foundation of the State University of New York     

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⁾.     

A decomposition for the exponential dispersion model generated by the invariant measure on the hyperboloid

For = ₀, ₁, ₂) andx=(x₀, x₁, x₂) in R³, define [,x] = ₀ x ₀ – ₁ x ₁ – ₂ x ₂,C = {x³:x ₀ > 0 and [x, x]>0},R(x)=([x, x]) ^1/2 forx inC andH ₁={xC: x₀>0,R(x)=1}. Define the measure onH ₁ such that if is inC and =R(), then exp (–[,x])(dx = ( exp )^–1. Therefore, is invariant under the action ofSO (1, 2), the connected component ofO(1, 2) containing the identity. We first prove that there exists a positive measure in ³ such that its Laplace transform is ( exp )^– if and only if >1. Finally, for 1 and inC, denotingP(,)(dx) = ( exp )^– exp (–[,x])(dx, we show that ifY ₀,...,Y _n aren+1 independent variables with densityP(,),j=0,...,n and ifS _k =X ₀ + ... +X _k andQ _k =R(S _k) –R(S _k–1) –R(Y _k),k=1,...,n, then then+1 statisticsD _n = [/,S _k ] –R _{k – 1} ),Q ₁,...,Q _n are independent random variables with the exponential () or gamma (1,1/) distribution.This research has been partially funded by NSERC Grant A8947.     

Minimum Divergence Estimators Based on Grouped Data

The paper considers statistical models with real-valued observations i.i.d. by F(x, ₀) from a family of distribution functions (F(x, ); ), R ^s , s 1. For random quantizations defined by sample quantiles (F _n ^–1 (₁),, F _n ^–1 ( _m–1)) of arbitrary fixed orders 0 < ₁ < _m-1 < 1, there are studied estimators _,n of ₀ which minimize -divergences of the theoretical and empirical probabilities. Under an appropriate regularity, all these estimators are shown to be as efficient (first order, in the sense of Rao) as the MLE in the model quantified nonrandomly by (F ^–1 (₁,₀),, F ^–1 ( _m–1, ₀)). Moreover, the Fisher information matrix I _m (₀, ) of the latter model with the equidistant orders = ( _j = j/m : 1 j m – 1) arbitrarily closely approximates the Fisher information J(₀) of the original model when m is appropriately large. Thus the random binning by a large number of quantiles of equidistant orders leads to appropriate estimates of the above considered type.     

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.     

The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms

Given a vector of real numbers=(₁,... _d ) ^d , the Jacobi-Perron algorithm and related algorithms, such as Brun's algorithm and Selmer's algorithm, produce a sequence of (d+1)×(d+1) convergent matrices {C⁽ⁿ⁾():n1} whose rows provide Diophantine approximations to . Such algorithms are specified by two mapsT:[0, 1] ^d [0, 1] ^d and A:[0,1] ^d GL(d+1,), which compute convergent matrices C⁽ⁿ⁾())...A(T())A(). The quality of the Diophantine approximations these algorithms find can be measured in two ways. The best approximation exponent is the upper bound of those values of for which there is some row of the convergent matrices such that for infinitely many values ofn that row of C⁽ⁿ⁾() has . The uniform approximation exponent is the upper bound of those values of such that for all sufficiently large values ofn and all rows of C⁽ⁿ⁾⁽⁾ one has . The paper applies Oseledec's multiplicative ergodic theorem to show that for a large class of such algorithms and take constant values and on a set of Lebesgue measure one. It establishes the formula where are the two largest Lyapunov exponents attached by Oseledec's multiplicative ergodic theorem to the skew-product (T, A,d), whered is aT-invariant measure, absolutely continuous with respect to Lebesgue measure. We conjecture that holds for a large class of such algorithms. These results apply to thed-dimensional Jacobi-Perron algorithm and Selmer's algorithm. We show that; experimental evidence of Baldwin (1992) indicates (nonrigorously) that. We conjecture that holds for alld2.     

Diffusion Equations and Geometric Inequalities

Let =(₀, ₁) be a fixed vector in R ² with strictly positive components and suppose ₀, ₁ > 0. Set = _{0 0} + _{1 1} and, if x ₀, x ₁ R ⁿ , set x = ₀ x ₀ + ₁ x ₁. Moreover, for any j {0, 1, }, let c _j : R ⁿ R be a continuous, bounded function and denote by p _j , c _j (t, x, y) the fundamental solution of the diffusion equation

Powers of Sign Portraits of Real Matrices

The sign portrait S of a real n× n matrix is a matrix over the semiring with elements 0, 1, -1, and , where symbolizes indeterminateness. It is proved that if k is the least positive integer such that all the entries of S ^k are equal to , then k 2n ² – 3n + 2, and this bound is sharp. Bibliography: 6 titles.     

The order of approximation to functions of the Zα. Class by means of positive linear operators

Let C_n (, ) be the upper bound for deviations of periodic functions which form the Zygmund class Z,0 0<<2 from a class of positive linear operators. A study is made of the conditions under which there exists a limit nC_n(, )=C(, ). An explicit expression is given for the functions C(,).Translated from Matematicheskie Zametki, Vol. 4, No. 2, pp. 201–210, August, 1968.