One-parameter Case Research Articles

The Calderón-Zygmund decomposition and interpolation on weighted Hardy spaces

We apply discrete Littlewood-Paley-Stein theory, developed by Han and Lu, to establish Calderon-Zygmund decompositions and interpolation theorems on weighted Hardy spaces Hwp for ω ∈ A∞ in both the one-parameter and two-parameter cases.

Quantum channel estimation and asymptotic bound

Quantum channel estimation in the one-parameter case is treated with the asymptotic setting. This problem sometimes has square speedup. A sufficient condition for this speedup is obtained. In this case, the existing results in Cramér-Rao approach seem to contradict the existing results in covariant approach. In order to resolve this problem, we introduce two kinds of asymptotic bounds.

Testing for Homogeneity in Meta-Analysis I. The One-Parameter Case: Standardized Mean Difference

Meta-analysis seeks to combine the results of several experiments in order to improve the accuracy of decisions. It is common to use a test for homogeneity to determine if the results of the several experiments are sufficiently similar to warrant their combination into an overall result. Cochran's Q statistic is frequently used for this homogeneity test. It is often assumed that Q follows a chi-square distribution under the null hypothesis of homogeneity, but it has long been known that this asymptotic distribution for Q is not accurate for moderate sample sizes. Here, we present an expansion for the mean of Q under the null hypothesis that is valid when the effect and the weight for each study depend on a single parameter, but for which neither normality nor independence of the effect and weight estimators is needed. This expansion represents an order O(1/n) correction to the usual chi-square moment in the one-parameter case. We apply the result to the homogeneity test for meta-analyses in which the effects are measured by the standardized mean difference (Cohen's d-statistic). In this situation, we recommend approximating the null distribution of Q by a chi-square distribution with fractional degrees of freedom that are estimated from the data using our expansion for the mean of Q. The resulting homogeneity test is substantially more accurate than the currently used test. We provide a program available at the Paper Information link at the Biometrics website http://www.biometrics.tibs.org for making the necessary calculations.

Comparison of three semiparametric methods for estimating dependence parameters in copula models

Three semiparametric methods for estimating dependence parameters in copula models are compared, namely maximum pseudo-likelihood estimation and the two method-of-moment approaches based on the inversion of Spearman’s rho and Kendall’s tau. For each of these three asymptotically normal estimators, an estimator of their asymptotic (co)variance is stated in three different situations, namely the bivariate one-parameter case, the multivariate one-parameter case and the multivariate multiparameter case. An extensive Monte Carlo study is carried out to compare the finite-sample performance of the three estimators under consideration in these three situations. In the one-parameter case, it involves up to six bivariate and four-variate copula families, and up to five levels of dependence. In the multiparameter case, attention is restricted to trivariate and four-variate normal and t copulas. The maximum pseudo-likelihood estimator appears as the best choice in terms of mean square error in all situations except for small and weakly dependent samples. It is followed by the method-of-moment estimator based on Kendall’s tau, which overall appears to be significantly better than its analogue based on Spearman’s rho. The simulation results are complemented by asymptotic relative efficiency calculations. The numerical computation of Spearman’s rho, Kendall’s tau and their derivatives in the case of copula families for which explicit expressions are not available is also investigated.

Comparison of Bartlett-type adjusted tests in the multiparameter case

Considering some Bartlett-type adjusted tests for a simple hypothesis about a multidimensional parameter, this paper clarifies similarities and dissimilarities with the one-parameter case developed in the 1990s, where a major emphasis is put on the issue posed by Rao and Mukerjee [C.R. Rao, R. Mukerjee, Comparison of Bartlett-type adjustments for the efficient score statistic, J. Statist. Plann. Inference 46 (1995) 137–146] on the power under a sequence of local alternatives. Not surprisingly, there is an infinite number of adjustments which extend Chandra–Mukerjee and Taniguchi approaches to the multiparameter case. Revisiting their ideas, this paper presents four specific cases (type K , K = 0 , 1 , 2 , 3 ) and gives a sufficient condition under which our generalized adjustment for each case is uniquely determined, where type 0 is a counterpart of Chandra and Mukerjee’s original proposal for Rao’s test statistic, whereas the latter three types are introduced as double adjustments related to the Cordeiro and Ferrari approach. If the adjustment of type 1 is made instead of type K , K = 0 , 2 , 3 , it is shown that Chandra and Mukerjee’s approach is equivalent to Taniguchi’s approach in terms of the third-order local power. The same is partially true for type 0, depending on the model under consideration. However, the adjustments of type K , K = 2 , 3 , reveal, in general, the non-equivalence of these two approaches in terms of the third-order local power.

The two-parameter Poisson–Dirichlet point process

The two-parameter Poisson--Dirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (that is, the random point process obtained by regarding the masses as points in the positive real line) is given in terms of the correlation functions. Using this, we apply the theory of point processes to reveal the mathematical structure of the two-parameter Poisson--Dirichlet distribution. Also, developing the Laplace transform approach due to Pitman and Yor, we are able to extend several results previously known for the one-parameter case. The Markov--Krein identity for the generalized Dirichlet process is discussed from the point of view of functional analysis based on the two-parameter Poisson--Dirichlet distribution.

Asymptotic posterior normality for multiparameter problems

For asymptotic posterior normality in the one-parameter cases, Weng [2003. On Stein's identity for posterior normality. Statist. Sinica 13, 495–506] proposed to use a version of Stein's Identity to write the posterior expectations for functions of a normalized quantity in a form that is more transparent and can be easily analyzed. In the present paper we extend this approach to the multi-parameter cases and compare our conditions with earlier work. Three examples are used to illustrate the application of this method.

Numerical calculation of the combinatorial entropy of partially ordered ice

Using a one-parameter case as an example, we demonstrate that multicanonical simulations allow for accurate estimates of the residual combinatorial entropy of partially ordered ice. For the considered case, corrections to an (approximate) analytical formula are found to be small, never exceeding 0.5%. The method allows one as well to calculate combinatorial entropies for other systems.

Vector-valued multiparameter singular integrals and pseudodifferential operators

We consider multiparameter singular integrals and pseudodifferential operators acting on mixed-norm Bochner spaces L p 1 , … , p N ( R n 1 × ⋯ × R n N ; X ) where X is a UMD Banach space satisfying Pisier's property ( α ) . These geometric conditions are shown to be necessary. We obtain a vector-valued version of a result by R. Fefferman and Stein, also providing a new, inductive proof of the original scalar-valued theorem. Then we extend a result of Bourgain on singular integrals in UMD spaces with an unconditional basis to a multiparameter situation. Finally we carry over a result of Yamazaki on pseudodifferential operators to the Bochner space setting, improving the known vector-valued results even in the one-parameter case.

Koszul algebras and the quantum MacMahon master theorem

We give a new proof of the quantum version of MacMahon's master theorem due to Garoufalidis, Lê and Zeilberger (in the one-parameter case) and to Konvalinka and Pak (in the multiparameter case) by deriving it from known facts about Koszul algebras.

Modified iterative aggregation algorithms

The iterative aggregation methods were developed in 1960s due to the necessity of practical solution of problems in mathematical economy; so they admit an economical interpretation. These methods are still insufficiently investigated from the mathematical point of view and not well-known. The development of the iterative aggregation methods, as well as the term, is connected with the works of L. M. Dudkin and E. B. Ershov (see [1], pp. 155–158). The following view on these methods is very significant ([1], p. 158): “Since the theory of the method is developed insufficiently and no convergence conditions are obtained for it, many numerical tests were performed; in many cases the method proved to be efficient”. An important result for an one-parameter case is established in [1]. As applied to a system of linear algebraic equations of the form x = Ax+ b, the convergence conditions for this method adduced in [1] (p. 156) require, in particular, that the elements aij (i = 1, . . . , n; j = 1, . . . ,m) of the matrix A be positive, the components bi of the vector of free terms be nonnegative, and the spectral radius of the matrix Ameets the inequality ρ(A) < 1. In this paper we study several modifications of the iterative aggregation methods for the linear equation

Accuracy of approximate progressively censored reliability sampling plans for exponential models

Reliability sampling plans provide an efficient method to determine the acceptability of a product based upon the lifelengths of some test units. Usually, they depend on the producer and consumer’s quality requirements and do not admit closed form solutions. Acceptance sampling plans for one- and two-parameter exponential lifetime models, derived by approximating the operating characteristic curve, are presented in this paper. The accuracy of these approximate plans, which are explicitly expressible and valid for failure and progressive censoring, is assessed. The approximation proposed in the one-parameter case is found to be practically exact. Explicit lower and upper bounds on the smallest sample size are given in the two-parameter case. Some additional advantages are also pointed out.

Foundation for Quantum Computing II

In this continuation of an earlier paper we develop further the theme of quantum logical specification and derive from it some apparently physically viable instantiations of potential quantum computing devices. Specifically, in the case of a one-parameter set of terms (or labels)—read as instants of time—we find, emerging quite naturally from the algebraic setup, the paradigm for a single qubit epitomized by the case of a two-state fermion interacting with an external single mode boson. This covers the cases: cavity QED, trapped ions, and, when the qubits are multiplexed appropriately, NMR based systems. (This case degenerates to one in which only bosons are relevant as in the case of pure bosonic harmonic oscillator models in the “dual rail” representation. Such models fly in the face of the logic itself, thus clearly revealing even at this level their well-known shortcomings as practical quantum computing devices. Here as elsewhere logical constraints apparently dominate physical ones.) In a final section we indicate briefly how this process exactly generalizes, in the case of a manifold of terms more general than the one-parameter case, to yield the notion of holonomic quantum computation. In the course of this investigation we find an interpretation of path integrals as limits of sequences of logical CUTS, thus establishing a link—though still tenuous—between ensembles of acts of quantum computation and Lagrangians.

Saddlepoint approximation in exponential models with boundary points

A saddlepoint approximation is developed for the likelihood ratio test statistic in non-regular exponential models. For the one-parameter case, it is proved that the saddlepoint correction has a relative error of order O ( n - 1 ) , assuming that the canonical statistic has four finite moments. The results are applied in reliability theory and survival analysis, testing for exponentiality. Note that we are using the saddlepoint method for statistics without moment generating function.

Some linear fractional stochastic equations

Using the multiple stochastic integrals, we prove an existence and uniqueness result for a linear stochastic equation driven by the fractional Brownian motion with any Hurst parameter. We study both the one- and two-parameter cases. When the drift is zero, we show that in the one-parameter case the solution is an exponential—thus positive—function while in the two-parameter setting the solution is negative on a non-negligible set.

Statistical inference for the Rayleigh model based on progressively type-II censored data

Maximum likelihood and Bayes estimates for one and two parameters and the reliability function of the Rayleigh distribution are obtained on the basis of progressively type-II censored samples. The inverted gamma conjugate prior density is assumed for the one-parameter case, whereas the joint prior density of the two-parameter case is composed of the inverted gamma and the uniform densities. A comparison between the obtained estimators is made through a Monte Carlo simulation study.

Pointwise convergence of ergodic averages for polynomial actions of $\mathbb{Z}^{d}$ by translations on a nilmanifold

Generalizing the one-parameter case, we prove that the orbit of a point on a compact nilmanifold X under a polynomial action of $\mathbb{Z}^{d}$ by translations on X is uniformly distributed on the union of several sub-nilmanifolds of X . As a corollary we obtain the pointwise ergodic theorem for polynomial actions of $\mathbb{Z}^{d}$ by translations on a nilmanifold.

The energy operator for a model with a multiparametric infinite statistics

In this paper we consider an energy operator (a free Hamiltonian), in the second-quantized approach, for the multiparameter quon algebras: aia†j − qija†jai = δij, i, j ∊ I with (qij)i,j∊I any Hermitian matrix of deformation parameters. We obtain an elegant formula for normally ordered (sometimes called Wick-ordered) series expansions of number operators (which determine a free Hamiltonian). As a main result (see theorem 1) we prove that the number operators are given, with respect to a basis formed by 'generalized Lie elements', by certain normally ordered quadratic expressions with coefficients given precisely by the entries of the inverses of Gram matrices of multiparticle weight spaces. (This settles a conjecture by Meljanac S and Perica A (1994 J. Phys. A: Math. Gen. 27 4737–44).) These Gram matrices are Hermitian generalizations of the Varchenko matrices, associated with a quantum (symmetric) bilinear form of diagonal arrangements of hyperplanes. The solution of the inversion problem of such matrices in Meljanac S and Svrtan D (1996 Math. Commun. 1 1–24 (theorem 2.2.17)), leads to an effective formula for the number operators studied in this paper. The one-parameter case, in the monomial basis, was studied by Zagier, Stanciu and Møller.

BEST TRACKING AND REGULATION PERFORMANCE UNDER CONTROL EFFORT CONSTRAINT: TWO-PARAMETER CONTROLLER CASE

This paper studies optimal tracking and regulation control problems by two-parameter controllers, in which objective functions of tracking error and regulated response, defined by integral square measures, are to be minimized jointly with the control effort, where the latter is measured by the plant input energy. The approach here is similar to our recent study for one-parameter controller case, and the problems are solved explicitly by deriving analytical expressions for the best achievable performance. Besides the plant non-minimum phase zeros, time delays, and unstable poles, the results reveal and quantify how the gain characteristics of the plant may all affect the performance. These effects are nonexistent when the control effort is not taken into account.

Weak Approximation of the Brownian Sheet from a Poisson Process in the Plane

We show an approximation in law of the Brownian sheet by processes constructed from the Poisson process in the plane. This result was inspired by a similar result of Stroock in the one-parameter case.