Second-Order Properties of a Two-Stage Fixed-Size Confidence Region for the Mean Vector of a Multivariate Normal Distribution

We consider the classical fixed-size confidence region estimation problem for the mean vectorμin theN_p(μ, Σ) population where Σ is unknown but positive definite. We writeλ₁for the largest characteristic root of Σ and assume thatλ₁is simple. Moreover, we suppose that, in many practical applications, we will often have available a numberλ*(>0) and that we can assumeλ₁>λ*. Given this addi- tional, and yet very minimal, knowledge regardingλ₁, the two-stage procedure of Chatterjee (Calcutta Statist. Assoc. Bull.8(1959a), 121–148;9(1959b), 20–28;11(1962), 144–159) is revised appropriately. The highlight in this paper involves the verification ofsecond-order propertiesassociated with such revised two-stage estimation techniques, along with the maintenance of the nominal confidence coefficient.     

Vertex Set Partitions Preserving Conservativeness

Let G be an undirected graph and ={X₁, …, X_n} be a partition of V(G). Denote by G/ the graph which has vertex set {X₁, …, X_n}, edge set E, and is obtained from G by identifying vertices in each class X_i of the partition . Given a conservative graph (G, w), we study vertex set partitions preserving conservativeness, i.e., those for which (G/ , w) is also a conservative graph. We characterize the conservative graphs (G/ , w), where is a terminal partition of V(G) (a partition preserving conservativeness which is not a refinement of any other partition of this kind). We prove that many conservative graphs admit terminal partitions with some additional properties. The results obtained are then used in new unified short proofs for a co-NP characterization of Seymour graphs by A. A. Ageev, A. V. Kostochka, and Z. Szigeti (1997, J. Graph Theory34, 357–364), a theorem of E. Korach and M. Penn (1992, Math. Programming55, 183–191), a theorem of E. Korach (1994, J. Combin. Theory Ser. B62, 1–10), and a theorem of A. V. Kostochka (1994, in “Discrete Analysis and Operations Research. Mathematics and its Applications (A. D. Korshunov, Ed.), Vol. 355, pp. 109–123, Kluwer Academic, Dordrecht).     

On the distribution of simple zeros of polynomials

Erd s and Turán discussed in (Ann. of Math. 41 (1940), 162–173; 51 (1950), 105–119) the distribution of the zeros of monic polynomials if their Chebyshev norm on [−1, 1] or on the unit disk is known. We sharpen this result to the case that all zeros of the polynomials are simple. As applications, estimates for the distribution of the zeros of orthogonal polynomials and the distribution of the alternation points in Chebyshev polynomial approximation are given. This last result sharpens a well-known error bound of Kadec (Amer. Math. Soc. Transl. 26 (1963), 231–234).     

Superapproximation and Commutator Properties of Discrete Orthogonal Projections for Continuous Splines

This paper builds upon the L_p-stability results for discrete orthogonal projections on the spaces S_h of continuous splines of order r obtained by R. D. Grigorieff and I. H. Sloan in (1998, Bull. Austral. Math. Soc.58, 307–332). Properties of such projections were proved with a minimum of assumptions on the mesh and on the quadrature rule defining the discrete inner product. The present results, which include superapproximation and commutator properties, are similar to those derived by I. H. Sloan and W. Wendland (1999, J. Approx. Theory97, 254–281) for smoothest splines on uniform meshes. They are expected to have applications (as in I. H. Sloan and W. Wendland, Numer. Math. (1999, 83, 497–533)) to qualocation methods for non-constant-coefficient boundary integral equations, as well as to the wide range of other numerical methods in which quadrature is used to evaluate L₂-inner products. As a first application, we consider the most basic variable-coefficient boundary integral equation, in which the constant-coefficient operator is the identity. The results are also extended to the case of periodic boundary conditions, in order to allow appplication to boundary integral equations on closed curves.     

M-Band Burt–Adelson Biorthogonal Wavelets

For every integer M>2 we introduce a new family of biorthogonal MRAs with dilation factor M, generated by symmetric scaling functions with small support. This construction generalizes Burt–Adelson biorthogonal 2-band wavelets. For M{3,4} we are able to find simple explicit expressions for two different families of wavelets associated with these MRAs: one with better localization and the other with interesting symmetry–antisymmetry properties. We study the regularity of our scaling functions by determining their Sobolev exponent, for every value of the parameter and every M. We also study the critical exponent when M=3.     

The estimation of the bispectral density function and the detection of periodicities in a signal

In a recent paper Subba Rao and Gabr (J. Time Ser. Anal. (1987), in press) considered the estimation of the spectrum and the inverse spectrum based on the method by Pisarenko (Geophys. J. Roy. Astronom. Soc. 28 (1972), 511–531). The asymptotic properties of these estimates were studied using the properties of Wishart matrices. In this paper we show how the method can be extended to the estimation of the bispectral density function, an important tool in the study of non-Gaussian time series. All these methods of estimation are illustrated with simulated examples. In the illustrations considered, the emphasis is on the detection of periodicities in the “signal” (possibly in the presence of noise). We also considered an example based on real data. These data arise in the study of the earth's magnetic reversals and the detection of periodicities.     

On a Small Sample Adjustment for the Profile Score Function in Semiparametric Smoothing Models

We consider the profile score function in models with smooth and parametric components. If local respectively weighted likelihood estimation is used for fitting the smooth component, the resulting profile likelihood estimate for the parametric component is asymptotically efficient as shown in T. A. Severini and W. H. Wong (1992, Ann. Statist.20, 1768–1802). However, as in solely parametric models the profile score function is not unbiased. We propose a small sample bias adjustment which results by extending the correction suggested in P. McCullagh and R. Tibshirani (1990, J. Roy. Statist. Soc. Ser. B52, 325–344) to the framework of semiparametric models.     

Cross-validation and the smoothing of orthogonal series density estimators

We describe a class of smoothed orthogonal series density estimates, including the classical sequential-series introduced by [6], Soviet Math. Dokl. 3 1559–1562) and [16], Ann. Math. Statist. 38 1261–1265), and [23], Ann. Statist 9 146–156) two-parameter smoothing. The Bowman-Rudemo method of least-squares cross-validation (1982, Manchester-Sheffield School of Probability and Statistics Research Report 84/AWB/1; 1984, Biometrika 71 353–360; [14], Scand. J. Statist. 9 65–78), is suggested as a practical way of choosing smoothing parameters automatically. Using techniques of [18], Ann. Statist. 12 1285–1297), that method is shown to perform asymptotically optimally in the case of cosine and Hermite series estimators. The same argument may be used for other types of series.     

Properties of convergence for -Bernstein polynomials

In this paper, we discuss properties of the ω,q-Bernstein polynomials introduced by S. Lewanowicz and P. Woźny in [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT 44 (1) (2004) 63–78], where fC[0,1], ω,q>0, ω≠1,q⁻¹,…,q⁻ⁿ⁺¹. When ω=0, we recover the q-Bernstein polynomials introduced by [G.M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511–518]; when q=1, we recover the classical Bernstein polynomials. We compute the second moment of , and demonstrate that if f is convex and ω,q(0,1) or (1,∞), then are monotonically decreasing in n for all x[0,1]. We prove that for ω(0,1), q_n(0,1], the sequence converges to f uniformly on [0,1] for each fC[0,1] if and only if lim_n→∞q_n=1. For fixed ω,q(0,1), we prove that the sequence converges for each fC[0,1] and obtain the estimates for the rate of convergence of by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions.     

Finite element analysis of an A– method for time-dependent Maxwell’s equations based on explicit-magnetic-field scheme

In this paper, error analysis of a finite element A– method for the time-dependent Maxwell’s equations is presented. An explicit-magnetic-field scheme is applied. Provided that the time-stepsize τ is sufficiently small, the proposed algorithm yields for finite time T an error of in the L²-norm for the electric field E and the magnetic field H, where h is the mesh size.     

r-k Class estimation in regression model with concomitant variables

We treat with the r-k class estimation in a regression model, which includes the ordinary least squares estimator, the ordinary ridge regression estimator and the principal component regression estimator as special cases of the r-k class estimator. Many papers compared total mean square error of these estimators. Sarkar (1989, Ann. Inst. Statist. Math., 41, 717–724) asserts that the results of this comparison are still valid in a misspecified linear model. We point out some confusions of Sarkar and show additional conditions under which his assertion holds.     

On the Asymptotics of Trimmed Best k-Nets

Trimmed best k-nets were introduced in J. A. Cuesta-Albertos, A. Gordaliza and C. Matrán (1998, Statist. Probab. Lett.36, 401–413) as a robustified L_∞-based quantization procedure. This paper focuses on the asymptotics of this procedure. Also, some possible applications are briefly sketched to motivate the interest of this technique. Consistency and weak limit law are obtained in the multivariate setting. Consistency holds for absolutely continuous distributions without the (artificial) requirement of a trimming level varying with the sample size as in J. A. Cuesta-Albertos, A. Gordaliza and C. Matrán (1998, Statist. Probab. Lett.36, 401–413). The weak convergence will be stated toward a non-normal limit law at a O_P(n^−1/3) rate of convergence. An algorithm for computing trimmed best k-nets is proposed. Also a procedure is given in order to choose an appropriate number of centers, k, for a given data set.     

Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions: II. The Heterogeneous Case

We consider the problem of discriminating between two independent multivariate normal populations, N_p(μ₁, Σ₁) and N_p(μ₂, Σ₂), having distinct mean vectors μ₁ and μ₂ and distinct covariance matrices Σ₁ and Σ₂. The parameters μ₁, μ₂, Σ₁, and Σ₂ are unknown and are estimated by means of independent random training samples from each population. We derive a stochastic representation for the exact distribution of the “plug-in” quadratic discriminant function for classifying a new observation between the two populations. The stochastic representation involves only the classical standard normal, chi-square, and F distributions and is easily implemented for simulation purposes. Using Monte Carlo simulation of the stochastic representation we provide applications to the estimation of misclassification probabilities for the well-known iris data studied by Fisher (Ann. Eugen.7 (1936), 179–188); a data set on corporate financial ratios provided by Johnson and Wichern (Applied Multivariate Statistical Analysis, 4th ed., Prentice–Hall, Englewood Cliffs, NJ, 1998); and a data set analyzed by Reaven and Miller (Diabetologia16 (1979), 17–24) in a classification of diabetic status.     

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Let (E,H,μ) be an abstract Wiener space and let D_V:=VD, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space . Given a bounded operator B on , coercive on the range , we consider the operators A:=V^*BV in H and in , as well as the realisations of the operators and in L^p(E,μ) and respectively, where 1<p<∞. Our main result asserts that the following four assertions are equivalent:

(1) with for ;

(2) admits a bounded H^∞-functional calculus on ;

(3) with for ;

(4) admits a bounded H^∞-functional calculus on .

Moreover, if these conditions are satisfied, then . The equivalence (1)–(4) is a non-symmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where , V=I, ). A one-sided version of (1)–(4), giving L^p-boundedness of the Riesz transform in terms of a square function estimate, is also obtained. As an application let −A generate an analytic C₀-contraction semigroup on a Hilbert space H and let −L be the L^p-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A. The boundedness of the Riesz transform is used to obtain an L^p-domain characterisation for the operator L.

Rank estimates in a class of semiparametric two-sample models

We consider a two-sample semiparametric model involving a real parameter and a nuisance parameter F which is a distribution function. This model includes the proportional hazard, proportional odds, linear transformation and Harrington-Fleming models (1982, Biometrika, 69, 533–546). We propose two types of estimates based on ranks. The first is a rank approximation to Huber's M-estimates (1981, Robust Statistics, Wiley) and the second is a Hodges-Lehmann type rank inversion estimate (1963, Ann. Math. Statist., 34, 598–611). We obtain asymptotic normality and efficiency results. The estimates are consistent and asymptotically normal generally but fully efficient only for special cases.Research partially supported by National Science Foundation Grant DMS-86-02083 and National Institute of General Medical Sciences Grant SSS-Y1RO1GM35416-01     

Convergence of Finite Element Approximations and Multilevel Linearization for Ginzburg–Landau Model of d-Wave Superconductors

In this paper, we consider the finite element approximations of a recently proposed Ginzburg–Landau-type model for d-wave superconductors. In contrast to the conventional Ginzburg–Landau model the scalar complex valued order-parameter is replaced by a multicomponent complex order-parameter and the free energy is modified according to the d-wave paring symmetry. Convergence and optimal error estimates and some superconvergent estimates for the derivatives are derived. Furthermore, we propose a multilevel linearization procedure to solve the nonlinear systems. It is proved that the optimal error estimates and superconvergence for the derivatives are preserved by the multilevel linearization algorithm.     

Characterizations of multivariate normality II. Through linear regressions

It is established that a vector (X′₁, X′₂, …, X′_k) has a multivariate normal distribution if (i) for each X_i the regression on the rest is linear, (ii) the conditional distribution of X₁ about the regression does not depend on the rest of the variables, and (iii) the conditional distribution of X₂ about the regression does not depend on the rest of the variables, provided that the regression coefficients satisfy some more conditions that those given by [4]J. Multivar. Anal. 6 81–94].     

Some properties of BLUE in a linear model and canonical correlations associated with linear transformations

Let (x, Xβ, V) be a linear model and let A′ = (A′₁, A′₂) be a p × p nonsingular matrix such that A₂X = 0, Rank A₂ = p − Rank X. We represent the BLUE and its covariance matrix in alternative forms under the conditions that the number of unit canonical correlations between y₁ ( = A₁x) and y₂ ( = A₂x) is zero. For the second problem, let x′ = (x′₁, x′₂) and let a g-inverse V⁻ of V be written as (V⁻)′ = (A′₁, A′₂). We investigate the reations (if any) between the nonzero canonical correlations {1 ₁ … ₁ > 0} due to y₁ ( = A₁x) and y₂ ( = A₂x), and the nonzero canonical correlations {1 λ₁ … λ_v+r > 0} due to x₁ and x₂. We answer some of the questions raised by Latour et al. (1987, in Proceedings, 2nd Int. Tampere Conf. Statist. (T. Pukkila and S. Puntanen, Eds.), Univ. of Tampere, Finland) in the case of the Moore-Penrose inverse V⁺ = (A′₁, A′₂) of V.     

MU-Estimation and Smoothing

In the M-estimation theory developed by Huber (1964, Ann. Math. Statist.43, 1449–1458), the parameter under estimation is the value of θ which minimizes the expectation of what is called a discrepancy measure (DM) δ(X, θ) which is a function of θ and the underlying random variable X. Such a setting does not cover the estimation of parameters such as the multivariate median defined by Oja (1983) and Liu (1990), as the value of θ which minimizes the expectation of a DM of the type δ(X₁, …, X_m, θ) where X₁, …, X_m are independent copies of the underlying random variable X. Arcones et al. (1994, Ann. Statist.22, 1460–1477) studied the estimation of such parameters. We call such an M-type MU-estimation (or μ-estimation for convenience). When a DM is not a differentiable function of θ, some complexities arise in studying the properties of estimators as well as in their computation. In such a case, we introduce a new method of smoothing the DM with a kernel function and using it in estimation. It is seen that smoothing allows us to develop an elegant approach to the study of asymptotic properties and possibly apply the Newton–Raphson procedure in the computation of estimators.     

On Thompson's Theorem

Let p be a prime divisor of the order of a finite group G. Thompson (1970, J. Algebra14, 129–134) has proved the following remarkable result: a finite group G is p-nilpotent if the degrees of all its nonlinear irreducible characters are divisible by p (in fact, in that case G is solvable). In this note, we prove that a group G, having only one nonlinear irreducible character of p′-degree is a cyclic extension of Thompson's group. This result is a consequence of the following theorem: A nonabelian simple group possesses two nonlinear irreducible characters χ₁ and χ₂ of distinct degrees such that p does not divide χ₁(1)χ₂(1) (here p is arbitrary but fixed). Our proof depends on the classification of finite simple groups. Some properties of solvable groups possessing exactly two nonlinear irreducible characters of p′-degree are proved. Some open questions are posed.