Entropy loss and risk of improved estimators for the generalized variance and precision

Let the distributions of X(p×r) and S(p×p) be N(, I _r) and W _p(n, ) respectively and let them be independent. The risk of the improved estimator for || or {ei329-1} based on X and S under entropy loss (=d/|| –log(d/||)–1 or d||–log(d||)–1) is evaluated in terms of incomplete beta function of matrix argument and its derivative. Numerical comparison for the reduction of risk over the best affine equivariant estimator is given.Dedicated to Professor Yukihiro Kodama on his 60th birthday.     

Percentage Points of the Largest Among Student's T Random Variable

Let us consider k( 2) independent random variables U₁, . . . ,U_k where U_i is distributed as the Student's t random variable with a degree of freedom m_i, i=1, . . . ,k. Here, m₁, . . . ,m_k are arbitrary positive integers. We denote m=(m₁, . . . ,m_k) and U_k:k=max {U₁, . . . ,U_k}, the largest Student's t random variable. Having fixed 0< <1, let a a(k,) and h_m h_m (k,) be two positive numbers for which we can claim that (i) ^k(a)–^k(–a)=1–, and (ii) P{–h_m U_k:k h_m}=1–. Then, we proceed to derive a Cornish–Fisher expansion (Theorem 3.1) of the percentage point h_m. This expansion involves a as well as expressions such as _i=1 ^k m_i ^–1, _i=1 ^km_i ^–2, and _i=1 ^k m_i ^–3. The corresponding approximation of h_m is shown to be remarkably accurate even when k or m₁, . . . ,m_k are not very large.     

Spreads admitting net generating regulizations

We say a spread S carries a regulization , if is a collection of reguli contained in S and if each element of S, except at most two lines, is contained either in exactly one regulus of or in all reguli of . Replacement of each regulus of by its complementary regulus (exceptional lines remain unchanged) yields the complementary congruence S^c of S with respect to . If S^c is a hyperbolic or parabolic or elliptic linear congruence of lines, then is called a net generating, in particular, a hyperbolic or parabolic or elliptic regulization, respectively. For hyperbolic and parabolic regulizations we also give other geometric characterizations.     

On an optimum test of the equality of two covariance matrices

Let X: p × 1, Y: p × 1 be independently and normally distributed p-vectors with unknown means ₁, ₂ and unknown covariance matrices ₁, ₂ (>0) respectively. We shall show that Pillai's test, which is locally best invariant, is locally minimax for testing H ₀: ₁=₂ against the alternative H ₁: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaaeiDaiaabkhacaqGOaWaaabmaeaadaaeqaqaaiabgkHiTiaadMea% caGGPaGaaiiiaiabg2da9iaacccacqaHdpWCcaGGGaGaeyOpa4Jaai% iiaiaaicdaaSqaaiaaigdaaeqaniabggHiLdaaleaacaqGYaaabaGa% aeylaiaabgdaa0GaeyyeIuoaaaa!4E3F!\[{\rm{tr(}}\sum\nolimits_{\rm{2}}^{{\rm{ - 1}}} {\sum\nolimits_1 { - I) = \sigma > 0} }\]as 0. However this test is not of type D among G-invariant tests.Research supported by the Canadian N.S.E.R.C. Grant.     

The uniform convergence of the sum of independent random variables

LetX _i (i=1, 2, ...) be the independent random variables on the probability space (, ,P). In this paper we will show that the necessary and sufficient condition of the uniform convergence of X _i is the convergence of m _i ⁽¹⁾ and m _i ⁽⁰⁾ , wherem _i ⁽¹⁾ , m _i ⁽⁰⁾ denote the 1-quantile and 0-quantile ofX _i respectively.     

Manifolds of infinite topological type with integrable geodesic flows

Let (M,k) be a complete surface of constant negative curvature (resp. an -geometric 3-manifold). This paper constructs a complete riemannian 8-manifold (resp. 9-manifold) (,h) such that is homotopy equivalent to M, the geodesic flow of h is completely integrable and there is a riemannian embedding (M,k)(,h). This embeds the geodesic flow of (M,k) as a subsystem of an integrable geodesic flow. Amongst the manifolds is an 8-dimensional manifold whose fundamental group is the free group on countably many generators.Thanks to Keith Burns and Leo Jonker for comments. Research partially supported by the Natural Sciences and Engineering Research Council of Canada.Mathematics Subject Classification (2000): 58F17, 53D25, 37D40     

Estimation of the Multivariate Normal Precision Matrix under the Entropy Loss

Let X ₁, , X _n (n > p) be a random sample from multivariate normal distribution N _p (, ), where R ^p and is a positive definite matrix, both and being unknown. We consider the problem of estimating the precision matrix ^–1. In this paper it is shown that for the entropy loss, the best lower-triangular affine equivariant minimax estimator of ^–1 is inadmissible and an improved estimator is explicitly constructed. Note that our improved estimator is obtained from the class of lower-triangular scale equivariant estimators.     

Vector Valued Differentiation Theorems for Multiparameter Additive Processes in Lp Spaces

Let X be a Banach space and (,,µ) be a -finite measure space. We consider a strongly continuous d-dimensional semigroup T={T(u):u=(u₁,..., u_d, u_i >0, 1 i d} of linear contractions on L_p((,,µ); X), with 1 p<. In this paper differentiation theorems are proved for d-dimensional bounded processes in L_p((,,µ); X) which are additive with respect to T. In the theorems below we assume that each T(u) possesses a contraction majorant P(u) defined on L_p((,,µ); R), that is, P(u) is a positive linear contraction on L_p((,,µ); R) such that T(u)f(w) P(u)f(·)() almost everywhere on for all f L_p((,,µ); X).     

The exact density function of the ratio of two dependent linear combinations of chi-square variables

A computable expression is derived for the raw moments of the random variableZ=N/D whereN= ₁ ⁿ m _iX_i+ _n ^+1s m _iX_i,D= _n ^+1s l _iX_i+ _s ^+1r n _iX_i, and theX _i's are independently distributed central chi-square variables. The first four moments are required for approximating the distribution ofZ by means of Pearson curves. The exact density function ofZ is obtained in terms of sums of generalized hypergeometric functions by taking the inverse Mellin transform of theh-th moment of the ratioN/D whereh is a complex number. The casen=1,s=2 andr=3 is discussed in detail and a general technique which applies to any ratio having the structure ofZ is also described. A theoretical example shows that the inverse Mellin transform technique yields the exact density function of a ratio whose density can be obtained by means of the transformation of variables technique. In the second example, the exact density function of a ratio of dependent quardratic forms is evaluated at various points and then compared with simulated values.     

Existence and Construction of Simple B-Splines of Class C k on a Four-Directional Mesh of the Plane

In this paper we present a study of spaces of splines in C ^k (R ²) with supports the square ₁ and the lozenge ₁ formed respectively by four and eight triangles of the uniform four directional mesh of the plane. Such splines are called ₁ and ₁-splines. We first compute the dimension of the space of ₁-splines. Then we prove the existence of a unique ₁-spline of minimal degree for any fixed k0. By using this last result, we also prove the existence of a unique ₁-spline of minimal degree. Finally, we describe algorithms allowing to compute the Bernstein–Bézier coefficients of ₁-spline and ₁-spline of minimal degree.     

Combinatorial principles concerning approximations of functions

Two infinite combinatorial principlesP( _n ) andT( _n ) concerning the existence of approximations of functions are studied.T( _n ) is shown to be equivalent toI _n andP( _n ) is shown to be incomparable withB _n+1 . Finally Pudláks principle, which is a finite miniaturization of bothT andP, is studied and its instances are related to instances of other known combinatorial statements.This paper was completed in September 1985 when the second author was a guest of the Mathematical Institute of the Czechoslovak Academy of Science in Prague.     

A strong law for weighted sums of i.i.d. random variables

A strong law is proved for weighted sumsS _n=a _in X _i whereX _i are i.i.d. and {a _in} is an array of constants. When sup(n ^–1|a _in | ^q )^1/q <, 1<q andX _i are mean zero, we showE|X| ^p <,p ^l+q ^–1=1 impliesS _n /n 0. Whenq= this reduces to a result of Choi and Sung who showed that when the {a _in} are uniformly bounded,EX=0 andE|X|< impliesS _n /n 0. The result is also true whenq=1 under the additional assumption that lim sup |a _in |n ^–1 logn=0. Extensions to more general normalizing sequences are also given. In particular we show that when the {a _in} are uniformly bounded,E|X|^1/< impliesS _n /n 0 for >1, but this is not true in general for 1/2<<1, even when theX _i are symmetric. In that case the additional assumption that (x ^1/ log^1/–1 x)P(|X|x)0 asx provides necessary and sufficient conditions for this to hold for all (fixed) uniformly bounded arrays {a _in}.     

Attainable order of rational approximations to the exponential function with only real poles

Rational approximations of the form _i=0 ^m a _i q ⁱ / _i=1 ⁿ (1+ _i q) to exp(–q),qC, are studied with respect to order and error constant. It is shown that the maximum obtainable order ism+1 and that the approximation of orderm+1 with least absolute value of the error constant has ₁=₂=...= _n . As an application it is shown that the order of av-stage semi-implicit Runge-Kutta method cannot exceedv+1.     

Co-monotone allocations,Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion

For every integrable allocation (X ₁,X ₂, ...,X _n ) of a random endowmentY= _i ^=1/n X _i amongn agents, there is another allocation (X ₁*,X ₂*, ...,X _n *) such that for every 1in,X _i * is a nondecreasing function ofY (or, (X ₁*,X ₂*, ...,X _n *) areco-monotone) andX _i * dominatesX _i by Second Degree Dominance.If (X ₁*,X ₂*, ...,X _n *) is a co-monotone allocation ofY= _i ^=1/n X _i *, then for every 1in, Y is more dispersed thanX _i * in the sense of the Bickel and Lehmann stochastic order.To illustrate the potential use of this concept in economics, consider insurance markets. It follows that unless the uninsured position is Bickel and Lehmann more dispersed than the insured position, the existing contract can be improved so as to raise the expected utility of both parties, regardless of their (concave) utility functions.     

Weighted Polynomial Approximation for Weights with Logarithmic Singularities in the Extremal Measure

Let w : [0, ) be a weight function on a set R. We assume that the associated extremal measure has density function v(t) with finitely many singularities of logarithmic type. We show that any continuous function f on which vanishes outside the set where v is positive or has a logarithmic singularity, is the uniform limit on of a sequence of weighted polynomials of the form wⁿ P_n, where P_n is of degree n. This extends previous results for continuous densities to densities having logarithmic singularities.     

Achtdimensionale lokalkompakte Translationsebenen mit großen kompakten Kollineationsgruppen

This paper is part of a program aiming at the classification of all higher-dimensional locally compact translation planes whose collineation groups have large dimension. In the present paper we determine all eight-dimensional locally compact translation planes which admit acompact collineation group of dimension at least 5 acting almost effectively on the translation axis. In fact, is isomorphic either to Spin₄ or toSO ₄(). The case Spin₄() has already been treated elsewhere ([6]). Here, the planes with SO ₄() are explicitly determined and studied in detail.     

The Automorphic Conjugacy Problem for Subgroups of Fundamental Groups of Compact Surfaces

Let be a compact connected surface with basepoint x and H ₁ and H ₂ be two finitely generated subgroups of ₁(, x) on finite sets of generators. It is proved that there exists an algorithm which decides whether there is an automorphism for which (H ₁) = H ₂, and if so, it finds such.     

ESTIMATION OF SCATTER MATRIX BASED ON i．i．d． SAMPLE FROM ELLIPTICAL DISTRIBUTIONS

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Spectral Properties of Hamiltonians of Charged Systems in a Homogeneous Magnetic Field: II. The Structure of the Pure Point Spectrum

We study the pure point spectrum of the energy operator H(P ) of a many-particle charged quantum system in a homogeneous magnetic field based on the results in our previous work under fixation of the sum P of the pseudomomentum components of the system. We prove that the discrete spectrum H(P ) of a short-range system is infinite under some conditions (which, for example, hold for a system of two oppositely charged particles) even in the case of a finitely supported potential. For a long-range system of the type of a (+)-ion of an atom (including the ion), the discrete spectrum is infinite.     

Developments on MTP2 properties of absolute value multinormal variables with nonzero means

LetX=(X ₁,...,X _n ) have a multivariate normal distribution with mean and covariance matrix. In the case=0, Karlin and Rinott^[6] obtained a necessary and sufficient condition on for |X|=(|X ₁|,...|X _n|) to be MTP₂. In this paper we consider the case0, and give some conditions under which |X| is MTP₂. A necessary and sufficient condition is given for |X| to be TP₂ whenn=2 and0. Some results about the TP₂ stochastic ordering are also given. The results are applied to obtain positive dependence and associated inequalities for multinormal and related distributions.This research is supported by the National Natural Science Foundation of China, the Doctoral Program Foundation of Institute of High Education and a grant of the Chinese Academy of Sciences.