Lift zonoid and barycentric representation on a Banach space with a cylinder measure

We show that the lift zonoid concept for a probability measure on $ {{\mathbb{R}}^d} $ , introduced in [G.A. Koshevoy and K. Mosler, Zonoid trimming for multivariate distributions, Ann. Stat., 25(5):1998–2017, 1997], naturally leads to a oneto-one representation of any interior point of the convex hull of the support of a continuous measure as the barycenter w.r.t. this measure of either a half-space or the whole space. We prove an infinite-dimensional generalization of this representation, which is based on the extension of the concept of lift zonoid for a cylindrical probability measure.     

Regular Score Tests of Independence in Multivariate Extreme Values

The score tests of independence in multivariate extreme values derived by Tawn (Tawn, J.A., “Bivariate extreme value theory: models and estimation,” Biometrika 75, 397–415, 1988) and Ledford and Tawn (Ledford, A.W. and Tawn, J.A., “Statistics for near independence in multivariate extreme values,” Biometrika 83, 169–187, 1996) have non-regular properties that arise due to violations of the usual regularity conditions of maximum likelihood. Two distinct types of regularity violation are encountered in each of their likelihood frameworks: independence within the underlying model corresponding to a boundary point of the parameter space and the score function having an infinite second moment. For applications, the second form of regularity violation has the more important consequences, as it results in score statistics with non-standard normalisation and poor rates of convergence. The corresponding tests are difficult to use in practical situations because their asymptotic properties are unrepresentative of their behaviour for the sample sizes typical of applications, and extensive simulations may be needed in order to evaluate adequately their null distribution. Overcoming this difficulty is the primary focus of this paper. We propose a modification to the likelihood based approaches used by Tawn (Tawn, J.A., “Bivariate extreme value theory: models and estimation,” Biometrika 75, 397–415, 1988) and Ledford and Tawn (Ledford, A.W. and Tawn, J.A., “Statistics for near independence in multivariate extreme values,” Biometrika 83, 169–187, 1996) that provides asymptotically normal score tests of independence with regular normalisation and rapid convergence. The resulting tests are straightforward to implement and are beneficial in practical situations with realistic amounts of data. AMS 2000 Subject Classification Primary—60G70 Secondary—62H15     

On Fiber Coefficients of Fiber Cones †

Let (R,𝔪) be a Cohen–Macaulay local ring of dimension d > 0, I an 𝔪-primary ideal of R, and K an ideal containing I. When depth G(I) ≥ d ? 1, depth F_K(I) ≥ d ? 2, and r(I|K) < ∞, we calculate the fiber coefficients f_i(I). Under the above assumptions on depth G(I) and r(I|K), we give an upper bound for f₁(I), and also provide a characterization, in terms of f₁(I), of the condition depth F_K(I) ≥ d ? 2.     

Projection based scatter depth functions and associated scatter estimators

In this article, we study a class of projection based scatter depth functions proposed by Zuo [Y. Zuo, Robust location and scatter estimators in multivariate analysis, The Frontiers in Statistics, Imperial College Press, 2005. Invited book chapter to honor Peter Bickel on his 65th Birthday]. In order to use the depth function effectively, some favorable properties are suggested for the common scatter depth functions. We show that the proposed scatter depth totally satisfies these desirable properties and its sample version possess strong and uniform consistency. Under some regularity conditions, the limiting distribution of the empirical process of the scatter depth function is derived. We also found that the aforementioned depth functions assess the bounded influence functions.A maximum depth based affine equivariant scatter estimator is induced. The limiting distributions as well as the strong and consistency of the sample scatter estimators are established. The finite sample performance of the related scatter estimator shows that it has a very high breakdown point and good efficiency.     

Fixed points in intuitionistic fuzzy metric spaces

The purpose of this paper, using the idea of intuitionistic fuzzy set due to Atanassov [Atanassov K. Intuitionistic fuzzy sets. Fuzzy Sets Syst 1986;20:87–96], we define the notion of intuitionistic fuzzy metric spaces due to Kramosil and Michalek [Kramosil O, Michalek J. Fuzzy metric and statistical metric spaces. Kybernetika 1975;11:326–34]. Further the well-known fixed point theorems of Banach and Edelstein are extended to intuitionistic fuzzy metric spaces with the help of Grabiec [Grabiec M. Fixed points in fuzzy metric spaces. Fuzzy Sets Syst 1988;27:385–9].     

Exact sampling with highly uniform point sets

In 1996, Propp and Wilson came up with a remarkably clever method for generating exact samples from the stationary distribution of a Markov chain [J.G. Propp, D.B. Wilson, Exact sampling with coupled Markov chains and applications to statistical mechanics, Random Structures and Algorithms 9 (1–2) (1996) 223–252]. Their method, called “perfect sampling” or “exact sampling” avoids the inherent bias of samples that are generated by running the chain for a large but fixed number of steps. It does so by using a strategy called “coupling from the past”. Although the sampling mechanism used in their method is typically driven by independent random points, more structured sampling can also be used. Recently, Craiu and Meng [R.V. Craiu, X.-L. Meng, Antithetic coupling for perfect sampling, in: E.I. George (Ed.), Bayesian Methods with Applications to Science, Policy, and Official Statistics (Selected Papers from ISBA 2000), 2000, pp. 99–108; R.V. Craiu, X.-L. Meng, Multi-process parallel antithetic coupling for forward and backward Markov Chain Monte Carlo, Annals of Statistics 33 (2005) 661–697] suggested using different forms of antithetic coupling for that purpose. In this paper, we consider the use of highly uniform point sets to drive the exact sampling in Propp and Wilson’s method, and illustrate the effectiveness of the proposed method with a few numerical examples.     

Identifying Cyclic Patterns of Interaction to Study Individual and Collective Learning

Our analysis of a college level mathematics course for prospective secondary mathematics teachers revealed that each student developed, at least to some degree, a conceptual orientation for teaching mathematics (A. G. Thompson, Philipp, Thompson, & Boyd, 1994). This initial finding led to a more in-depth question: If we assume an emergent perspective (Cobb & Bauersfeld, 1995; Cobb & Yackel, 1996) in which the values, practices, and social motivations of the classroom are believed to play critical roles in students' conceptual development, what social aspects emerged that supported these individual constructions? To address this question, we documented the emergence of a collective conceptual orientation and then used this construct to explore the reflexivity between its emergence and individual students' development of conceptual orientations.     

Weighted-mean regions of a probability distribution

In this paper we investigate a new class of central regions for probability distributions on R^d, called weighted-mean regions. Their restrictions to an empirical distribution are the weighted-mean trimmed regions investigated by Dyckerhoff and Mosler (2011) for d-variate data. Furthermore a new class of stochastic orderings of variability, the weighted-mean orderings, is introduced.     

Polynomially bounded operators and Ext groups

In this paper, we consider the Ext functor in the category
of Hilbert modules over the disk algebra. We characterize the group
as a quotient of operators and explicitly calculate
, where is a weighted Hardy space. We then use our results to give a simple proof of a result due to Bourgain.

     

On the λY calculus

The λY calculus is the simply typed λ calculus augmented with the fixed point operators. We show three results about λY: (a) the word problem is undecidable, (b) weak normalisability is decidable, and (c) higher type fixed point operators are not definable from fixed point operators at smaller types.     

Guest Commentary

Abstract

Clustering and Classification edited by P. Arabie, L. J. Hubert, and G. De Soete, River Edge, New Jersey: World Scientific, 1996, 490 pp., $109 hardcover, $61 paperback. Reviewed by Jon R. Kettenring

Statistics as Principled Argument by Robert P. Abelson, Hillsdale, New Jersey: Lawrence Erlbaum Associates, 1995, 221 pp. Reviewed by Edward J. Shoben     

Statistical inference with minimumd-risk

Translated from Issledovaniya po Prikladnoi Matematike, No. 11, Part 2, pp. 25–39, 1984. Presented at the seminar of the Department of Probability Theory and Mathematical Statistics, NIIMM, May 21, 1981.     

On the jordan form of the tensor product over fields of prime characteristic

Let A, B denote the companion matrices of the polynomials x^m,xⁿ over a field F of prime order p and let λ,μ be non-zero elements of an extension field K of F. The Jordan form of the tensor product (λI + A)⊗(μI + B) of invertible Jordan matrices over K is determined via an equivalent study of the nilpotent tranformation S of m × n matrices X over F where(X)S = A ^TX + XB. Using module-theoretic concepts a Jordan basis for S is specified recursively in terms of the representations of m and n in the scale of p, and reduction formulae for the elementary divisors of S are established.     

A bi-Bayesian procedure with constraints on prior refusal probability

Translated from Issledovaniya po Prikladnoi Matematike, No. 11, Part 2, pp. 39–43, 1984. Presented at the seminar of the Department of Probability Theory and Mathematical Statistics, NIIMM, October 15, 1981.     

Karp complexity and classes with the independence property

A class K of structures is controlled if for all cardinals λ, the relation of L_∞,λ-equivalence partitions K into a set of equivalence classes (as opposed to a proper class). We prove that no pseudo-elementary class with the independence property is controlled. By contrast, there is a pseudo-elementary class with the strict order property that is controlled (see Arch. Math. Logic 40 (2001) 69–88).     

Music by Design: A Sesquicentennial Statistics Songbook

Abstract

Fox, J. (1997), Applied Regression Analysis, Linear Models, and Related Methods, Thousand Oaks, CA: Sage Publications.

French, S., and Smith, J. Q. (eds.) (1997), The Practice of Bayesian Analysis, New York: Wiley.

Hettmansperger, T. P., and McKean, J. W. (1998), Robust Nonparametric Statistical Methods, London: Arnold.

Jacoby, W. G. (1998), Statistical Graphics For Visualizing Multivariate Data, Thousand Oaks, CA: Sage Publications.

Johnson, N. L., and Balakrishnan, N. (1997), Advances in the Theory and Practice of Statistics: A Volume in Honor of Samuel Kotz, New York: Wiley.

Korner, T. W. (1995), The Pleasures of Counting, Cambridge, U.K.: Cambridge University Press.

McLachlan, G. J., and Krishman, T. (1996), The EM Algorithm and Extensions, New York: Wiley.

Ruelle, D. (1991), Chance and Chaos, Princeton, NJ: Princeton University Press.

Ryan, T. P. (1997), Modern Regression Methods, New York: Wiley.

Bernstein, D. (1997), Statistical Applications of Baseball, Alexandria, VA: SPSTATE.     

Non-commutative disc algebras and their representations

It is shown that the smallest closed subalgebra

generated by any sequence of isometries on a Hilbert space such that is completely isometrically isomorphic to the non-commutative ``disc' algebra introduced in Math. Scand. 68 (1991), 292--304. We also prove that for the Banach algebras and are not isomorphic. In particular, we give an example of two non-isomorphic Banach algebras which are completely isometrically embedded in each other. The completely bounded (contractive) representations of the ``disc' algebras on a Hilbert space are characterized. In particular, we prove that a sequence of operators is simultaneously similar to a contractive sequence (i.e., ) if and only if it is completely polynomially bounded. The first cohomology group of with coefficients in is calculated, showing, in particular, that the disc algebras are not amenable. Similar results are proved for the non-commutative Hardy algebras introduced in Math. Scand. 68 (1991), 292--304. The right joint spectrum of the left creation operators on the full Fock space is also determined.

     

Convergence rate of the best-r-point-average estimator for the maximizer of a nonparametric regression function

The best-r-point-average (BRPA) estimator of the maximizer of a regression function, proposed in Changchien (in: M.T. Chao, P.E. Cheng (Eds.), Proceedings of the 1990 Taipei Symposium in Statistics, June 28–30, 1990, pp. 63–78) has certain merits over the estimators derived through the estimation of the regression function. Some of the properties of the BRPA estimator have been studied in Chen et al. (J. Multivariate Anal. 57 (1996) 191) and Bai and Huang (Sankhya: Indian J. Statist. Ser. A. 61 (Pt. 2) (1999) 208–217). In this article, we further study the properties of the BRPA estimator and give its convergence rate under some quite general conditions. Simulation results are presented for the illustration of the convergence rate. Some comparisons with existing estimators such as the Müller estimator are provided.     

Schaefer type theorem and periodic solutions of evolution equations

Two fixed point theorems for the sum of contractive and compact operators are obtained in this paper, which generalize and improve the corresponding results in [H. Schaefer, Über die methode der a priori-Schranken, Math. Ann. 129 (1955) 415-416; T.A. Burton, Integral equations, implicit functions and fixed points, Proc. Amer. Math. Soc. 124 (1996) 2383-2390; V.I. Istrǎtescu, Fixed Point Theory, an Introduction, Reidel, Dordrecht, 1981; T.A. Burton, K. Colleen, A fixed point theorem of Krasnoselskii-Schaefer type, Math. Nachr. 189 (1998) 23-31; D.R. Smart, Fixed Point Theorems, Cambridge Univ. Press, Cambridge, 1980]. As the applications for the results, we obtain the existence of periodic solutions for some evolution equations with delay, which extend the corresponding results in [T.A. Burton, B. Zhang, Periodic solutions of abstract differential equations with infinite delay, J. Differential Equations 90 (1991) 357-396].