Bahadur efficiency and robustness of studentized score tests

We derive the exact Bahadur slopes of studentized score tests for a simple null hypothesis in a one-parameter family of distributions. The Student's t-test is included as a special case for which a recent result of Rukhin (1993, Sankhy Ser. A, 55, 159–163) was improved upon. It is shown that locally optimal Bahadur efficiency for one-sample location models with a known or estimated scale parameter is attained within the class of studentized score tests. The studentized test has an asymptotic null distribution free of the scale parameter, and the optimality of likelihood scores does not depend on the existence of a moment generating function. We also consider the influence function and breakdown point of such tests as part of our robustness investigation. The influence of any studentized score test is bounded from above, indicating certain degree of robustness of validity, but a bounded score function is needed to cap the influence from below and to ensure a high power breakdown point. We find that the standard Huber-type score tests are not only locally minimax in Bahadur efficiency, but also very competitive in global efficiency at a variety of location models.     

On Appell Sets and the Fueter-Sce Mapping

It is proved in Clifford algebras generated by an odd number of basis vectors e ₁, ... , e _n , that the recently discussed Appell polynomials in Clifford algebras are the Fueter-Sce extension of the complex monomials z ^k . Furthermore, it is shown, for which complex functions the Fueter-Sce extension and the extension method using Appell polynomials coincide.      

Algebras with a compatible uniformity

Given a variety of algebras V, we study categories of algebras in V with a compatible structure of uniform space. The lattice of compatible uniformities of an algebra, Unif A, can be considered a generalization of the lattice of congruences Con A. Mal'cev properties of V influence the structure of Unif A, much as they do that of Con A. The category V[CHUnif] of complete, Hausdor. such algebras in the variety V is particularly interesting; it has a factorization system , and V embeds into V[CHUnif] in such a way that is the subcategory of onto and the subcategory of one-one homomorphisms. Received February 17, 2000; accepted in final form April 1, 2001.     

Some autoregressive moving average processes with generalized Poisson marginal distributions

Some simple models are introduced which may be used for modelling or generating sequences of dependent discrete random variables with generalized Poisson marginal distribution. Our approach for building these models is similar to that of the Poisson ARMA processes considered by Al-Osh and Alzaid (1987,J. Time Ser. Anal.,8, 261–275; 1988,Statist. Hefte,29, 281–300) and McKenzie (1988,Adv. in Appl. Probab.,20, 822–835). The models have the same autocorrelation structure as their counterparts of standard ARMA models. Various properties, such as joint distribution, time reversibility and regression behavior, for each model are investigated.     

Influence Diagnostics in the Common Canonical Variates Model

As a generalization of the canonical correlation analysis to k random vectors, the common canonical variates model was recently proposed based on the assumption that the canonical variates have the same coefficients in all k sets of variables, and is applicable to many cases. In this article, we apply the local influence method in this model to study the impact of minor perturbations of data. The method is non-standard because of the restrictions imposed on the coefficients. Besides investigating the joint local influence of the observations, we also obtain the elliptical norm of the empirical influence function as a special case of local influence diagnostics. Based on the proposed diagnostics, we find that the results of common canonical variates analysis for the female water striders data set is largely affected by omitting just one single observation.     

Parafermion vertex operator algebras

This paper reviews some recent results on the parafermion vertex operator algebra associated to the integrable highest weight module L(k, 0) of positive integer level k for any affine Kac-Moody Lie algebra ĝ, where g is a finite dimensional simple Lie algebra. In particular, the generators and the C ₂-cofiniteness of the parafermion vertex operator algebras are discussed. A proof of the well-known fact that the parafermion vertex operator algebra can be realized as the commutant of a lattice vertex operator algebra in L(k, 0) is also given.     

The Catline for Deep Regression

Motivated by the notion of regression depth (Rousseeuw and Hubert, 1996) we introduce thecatline, a new method for simple linear regression. At any bivariate data setZ_n={(x_i, y_i);i=1, …, n} its regression depth is at leastn/3. This lower bound is attained for data lying on a convex or concave curve, whereas for perfectly linear data the catline attains a depth ofn. We construct anO(n log n) algorithm for the catline, so it can be computed fast in practice. The catline is Fisher-consistent at any linear modely=βx+α+ein which the error distribution satisfies med(e | x)=0, which encompasses skewed and/or heteroscedastic errors. The breakdown value of the catline is 1/3, and its influence function is bounded. At the bivariate gaussian distribution its asymptotic relative efficiency compared to theL¹line is 79.3% for the slope, and 88.9% for the intercept. The finite-sample relative efficiencies are in close agreement with these values. This combination of properties makes the catline an attractive fitting method.     

Elliptic Curve Cryptosystems in the Presence of Permanent and Transient Faults

Elliptic curve cryptosystems in the presence of faults were studied by Biehl et al., Advances in Cryptology CRYPTO 2000, Springer Verlag (2000) pp. 131–146. The first fault model they consider requires that the input point P in the computation of dP is chosen by the adversary. Their second and third fault models only require the knowledge of P. But these two latter models are less practical in the sense that they assume that only a few bits of error are inserted (typically exactly one bit is supposed to be disturbed) either into P just prior to the point multiplication or during the course of the computation in a chosen location.This paper relaxes these assumptions and shows how random (and thus unknown) errors in either coordinates of point P, in the elliptic curve parameters or in the field representation enable the (partial) recovery of multiplier d. Then, from multiple point multiplications, we explain how this can be turned into a total key recovery. Simple precautions to prevent the leakage of secrets are also discussed.communication by : P. WildThe work described in this paper has been supported [in part] by the Commission of the European Communities through the IST Programme under Contract IST-1999-12324, http://www.cryptonessie.org/. The information in this document is provided as is, and no guarantee or warranty is given or implied that the information is fit for any particular purpose. The user thereof uses the information at his sole risk and liability. The views expressed are those of the authors and do not represent an official view/position of the NESSIE project (as a whole)     

ASSOCIATIONS AND VALUATIONS

Abstract

Right cones are semigroups for which the lattice of right ideals is a chain and a left cancellation law holds; valuation rings, the cones of ordered groups, and initial segments of ordinal numbers are examples. Two such cones are associated if they have isoniorphic lattices of right ideals so that ideals, prime ideals, and completely prime ideals correspond to each other. A list of problems is discussed. In Proposition 3.11 it is proved that the canonical mapping from a right invariant right chain domain R onto the associated right holoid can be extended to a valuation from the skew field Q(R) of quotients of R onto an ordered group if and only if Ja ? aJ for all a ∈ R and J = J(R), the Jacobson radical of R.     

Improved complexity results on solving real-number linear feasibility problems

We present complexity results on solving real-number standard linear programs LP(A,b,c), where the constraint matrix the right-hand-side vector and the objective coefficient vector are real. In particular, we present a two-layered interior-point method and show that LP(A,b,0), i.e., the linear feasibility problem A x = b and x ≥ 0, can be solved in in O(n ^2.5 c(A)) interior-point method iterations. Here 0 is the vector of all zeros and c(A) is the condition measure of matrix A defined in [25]. This complexity iteration bound is reduced by a factor n from that for general LP(A, b, c) in [25]. We also prove that the iteration bound will be further reduced to O(n ^1.5 c(A)) for LP(A, 0, 0), i.e., for the homogeneous linear feasibility problem. These results are surprising since the classical view has been that linear feasibility would be as hard as linear programming. This author was supported in part by NSF Grants DMS-9703490 and DMS-0306611     

Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix

The plain Newton-min algorithm to solve the linear complementarity problem (LCP for short) can be viewed as a semismooth Newton algorithm without globalization technique to solve the system of piecewise linear equations min(x, Mx + q) = 0, which is equivalent to the LCP. When M is an M-matrix of order n, the algorithm is known to converge in at most n iterations. We show in this paper that this result no longer holds when M is a P-matrix of order ≥ 3, since then the algorithm may cycle. P-matrices are interesting since they are those ensuring the existence and uniqueness of the solution to the LCP for an arbitrary q. Incidentally, convergence occurs for a P-matrix of order 1 or 2.     

Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials

This paper is devoted to the analysis of a one-dimensional model for phase transition phenomena in thermoviscoelastic materials. The corresponding parabolic-hyperbolic PDE system features a strongly nonlinear internal energy balance equation, governing the evolution of the absolute temperature ϑ, an evolution equation for the phase change parameter χ, including constraints on the phase variable, and a hyperbolic stress-strain relation for the displacement variable u. The main novelty of the model is that the equations for χ and u are coupled in such a way as to take into account the fact that the properties of the viscous and of the elastic parts influence the phase transition phenomenon in different ways. However, this brings about an elliptic degeneracy in the equation for u which needs to be carefully handled. First, we prove a global well-posedness result for the related initial-boundary value problem. Secondly, we address the long-time behavior of the solutions in a simplified situation. We prove that the ω-limit set of the solution trajectories is nonempty, connected and compact in a suitable topology, and that its elements solve the steady state system associated with the evolution problem. Dedicated to Jürgen Sprekels on the occasion of his 60th birthday     

An alternating iterative method and its application in statistical inference

This paper studies non-convex programming problems. It is known that, in statistical inference, many constrained estimation problems may be expressed as convex programming problems. However, in many practical problems, the objective functions are not convex. In this paper, we give a definition of a semi-convex objective function and discuss the corresponding non-convex programming problems. A two-step iterative algorithm called the alternating iterative method is proposed for finding solutions for such problems. The method is illustrated by three examples in constrained estimation problems given in Sasabuchi et al. （Biometrika, 72, 465472 （1983））, Shi N. Z. （J. Multivariate Anal., 50, 282-293 （1994）） and El Barmi H. and Dykstra R. （Ann. Statist., 26, 1878 1893 （1998））.     

First-Quantized Fermions in Compact Dimensions

We discuss a path integral representation for fermionic particles and strings in the spirit of V. Ya. Fainberg and the author (Nucl. Phys. B, 306, 659–676, (1998); Phys. Lett. B, 211, 81–85, (1988)). We concentrate on the problems arising when some target-space dimensions are compact. We consider the partition function for a fermionic particle at a finite temperature or in compact time in detail as an example. We demonstrate that a self-consistent definition of the path integral generally requires introducing nonvanishing background Wilson loops and that modulo some common problems for real fermions in the Grassmannian formulation, these loops can be interpreted as condensates of world-line fermions. Properties of the corresponding string-theory path integrals are also discussed.     

Null distribution of the sum of squared z-transforms in testing complete independence

Brien et al. (1984, Biometrika, 71, 545–554; 1988, Biometrika, 75, 469–476) have proposed, illustrated and discussed advantages of using Fisher's z-transforms for analyzing correlation structures of multinormal data. Chen and Mudholkar (1988, Austral. J. Statist., 31, 105–110) have studied the sum of squared z-transforms of sample correlations as a test statistic for complete independence. In this paper Brown's (1987, Ann. Probab., 15, 416–422) graph-theoretic characterization of the dependence structure of sample correlations is used to evaluate moments of the test statistic. These moments are then used to approximate its null distribution accurately over a broad range of parameters, including the case where the population dimension exceeds the sample size.     

On the admissibility of an estimator of a normal mean vector under a linex loss function

For ap-variate normal mean with known variances, the model proposed by Zellner (1986,J. Amer. Statist. Assoc.,81, 446–451) is discussed in a slightly different framework. A generalized Bayes estimate is derived from a three-stage Bayes point of view under the asymmetric loss function, and the admissibility of such estimators is proved.     

Regularity properties of H-graphs

It is proved that if H(u) is non-decreasing and if , then if u (x) describes a graph over a disk B _R (0), with (upward oriented) mean curvature H(u), there is a bound on the gradient that depends only on R, on u (0), and on the particular function H (u). As a consequence a form of Harnack's inequality is obtained, in which no positivity hypothesis appears. The results are qualitatively best possible, in the senses a) that they are false if H is constant, and b) the dependences indicated are essential.?The demonstrations are based on an existence theorem for a nonlinear boundary problem with singular data, which is of independent interest. Received: April 22, 1996     

Comparison of ‘Tail Method’ Exact Confidence Limits for the Difference of Binomial Probabilities

Suppose that Y ₁ and Y ₂ are independent and have Binomial(n ₁,p ₁) and Binomial (n ₂,p ₂) distributions respectively. Also suppose that θ=p ₁−p ₂ is the parameter of interest. We consider the problem of finding an exact confidence limit (either upper or lower) for θ. The solution to this problem is very important for statistical practice in the health and life sciences. The ‘tail method’ provides a solution to this problem. This method finds the exact confidence limit by exact inversion of a hypothesis test based on a specified test statistic. Buehler (J. Am. Stat. Assoc. 52, 482–493, 1957) described, for the first time, a finite-sample optimality property of this confidence limit. Consequently, this confidence limit is sometimes called a Buehler confidence limit. An early tail method confidence limit for θ was described by Santner and Snell (J. Am. Stat. Assoc. 75, 386–394, 1980) who used the maximum likelihood estimator of θ as the test statistic. This confidence limit is known to be very inefficient (see e.g. Cytel Software, StatXact, version 6, vol. 2, 2004). The efficiency of the confidence limit resulting from the tail method depends greatly on the test statistic on which it is based. We use the results of Kabaila (Stat. Probab. Lett. 52, 145–154, 2001) and Kabaila and Lloyd (Aust. New Zealand J. Stat. 46, 463–469, 2004, J. Stat. Plan. Inference 136, 3145–3155, 2006) to provide a detailed explanation for the dependence of this efficiency on the test statistic. We consider test statistics that are estimators, Z-statistics and approximate upper confidence limits. This explanation is used to find the situations in which the tail method exact confidence limits based on test statistics that are estimators or Z-statistics are least efficient.     

On equalities for BLUEs under misspecified Gauss-Markov models

This paper studies relationships between the best linear unbiased estimators （BLUEs） of an estimable parametric functions Kβunder the Gauss-Markov model {y, Xβ, σ^2]E} and its misspecified model {y, X0β,σ^2∑0}. In addition, relationships between BLUEs under a restricted Gauss Markov model and its misspecified model are also investigated.     

Semi-empirical likelihood ratio confidence intervals for the difference of two sample means

We all know that we can use the likelihood ratio statistic to test hypotheses and construct confidence intervals in full parametric models. Recently, Owen (1988,Biometrika,75, 237–249; 1990,Ann. Statist.,18, 90–120) has introduced the empirical likelihood method in nonparametric models. In this paper, we combine these two likelihoods together and use the likelihood ratio to construct confidence intervals in a semiparametric problem, in which one model is parametric, and the other is nonparametric. A version of Wilks's theorem is developed.