Asymptotic Linearity Research Articles

Asymptotic linearity of serial and nonserial multivariate signed rank statistics

Asymptotic linearity plays a key role in estimation and testing in the presence of nuisance parameters. This property is established, in the very general context of a multivariate general linear model with elliptical VARMA errors, for the serial and nonserial multivariate rank statistics considered in Hallin and Paindaveine (Ann. Statist. 30 (2002a) 1103; Bernoulli 8 (2002b) 787 Ann. Statist. 32 (2004), to appear) and Oja and Paindaveine (J. Statist. Plann. Inference (2004), to appear). These statistics, which are multivariate versions of classical signed rank statistics, involve (i) multivariate signs based either on (pseudo-)Mahalanobis residuals, or on a modified version ( absolute interdirections) of Randles's interdirections, and (ii) a concept of ranks based either on (pseudo-)Mahalanobis distances or on lift-interdirections.

Asymtotics of M-estmation in Non-linear Regression

Consider the standard non-linear regression model yi= g(xi, θ0)+ei, i = 1, ... ,n where g(x, θ) is a continuous function on a bounded closed region X × Θ, θ0 is the unknown parameter vector in Θ ⊂ Rp, {x1, x2, ... , xn} is a deterministic design of experiment and {e1, e2, ... , en} is a sequence of independent random variables. This paper establishes the existences of M-estimates and the asymptotic uniform linearity of M-scores in a family of non-linear regression models when the errors are independent and identically distributed. This result is then used to obtain the asymptotic distribution of a class of M-estimators for a large class of non-linear regression models. At the same time, we point out that Theorem 2 of Wang (1995) (J. of Multivariate Analysis, vol. 54, pp. 227–238, Corrigenda. vol. 55, p. 350) is not correct.

Affine-invariant aligned rank tests for the multivariate general linear model with VARMA errors

We develop optimal rank-based procedures for testing affine-invariant linear hypotheses on the parameters of a multivariate general linear model with elliptical VARMA errors. We propose a class of optimal procedures that are based either on residual (pseudo-)Mahalanobis signs and ranks, or on absolute interdirections and lift-interdirection ranks, i.e., on hyperplane-based signs and ranks. The Mahalanobis versions of these procedures are strictly affine-invariant, while the hyperplane-based ones are asymptotically affine-invariant. Both versions generalize the univariate signed rank procedures proposed by Hallin and Puri (J. Multivar. Anal. 50 (1994) 175), and are locally asymptotically most stringent under correctly specified radial densities. Their AREs with respect to Gaussian procedures are shown to be convex linear combinations of the AREs obtained in Hallin and Paindaveine (Ann. Statist. 30 (2002) 1103; Bernoulli 8 (2002) 787) for the pure location and purely serial models, respectively. The resulting test statistics are provided under closed form for several important particular cases, including multivariate Durbin–Watson tests, VARMA order identification tests, etc. The key technical result is a multivariate asymptotic linearity result proved in Hallin and Paindaveine (Asymptotic linearity of serial and nonserial multivariate signed rank statistics, submitted).

The Functional Delta Existence Theorem and the Reduction of a Proof of the Fukui Conjecture to That of the Special Functional Asymptotic Linearity Theorem

This article establishes a fundamental existence theorem, called the Functional Delta Existence Theorem (DET), which is significant for a new development in the repeat space theory (RST) and also for elucidating an empirical asymptotic principle from experimental chemistry. By using the Functional DET, we reduce a proof of the Fukui conjecture to that of a special and simpler version of the Asymptotic Linearity Theorem (ALT). This reduction provides a basis for the forthcoming series of articles entitled “Proof of the Fukui conjecture via resolution of singularities and related methods”. A proof of the Functional DET is given here in a unifying manner so that an investigative link is formed among: (i) fundamental methodology in the RST, which is referred to as the approach via the aspect of form and general topology, (ii) frontier electron theory of reactivity indices, and (iii) the Shingu–Fujimoto empirical asymptotic principle for long chain molecules.

New Proof of the Fukui Conjecture by the Functional Asymptotic Linearity Theorem

The present article provides a new proof of the Fukui conjecture concerning the additivity problem of the zero-point vibrational energies of hydrocarbons. This conjecture played a prominent role in the initial development of the repeat space theory (RST), and continues to be of vital significance in the recent development of the theory of the generalized repeat space X r (q,d). The new proof of the Fukui conjecture has been given here by establishing the functional version of the Asymptotic Linearity Theorem (ALT), the Functional ALT. This enhanced version of the ALT directly implies the validity of the Fukui conjecture; it easily unifies, in a broad perspective, a variety of additivity phenomena in physico-chemical network systems having many identical moieties, and efficiently solves some interpretational problems of the empirical additivity formulae from experimental chemistry. The proof of the functional version of the ALT is based on a new method transferable to the extended theoretical framework of the generalized repeat space X r (q,d).

On the asymptotic distribution of a multivariate GR-estimate for a VAR( p) time series

This paper introduces a new class of estimates for estimating the parameters of a vector autoregressive time series. The estimates minimize a sum of weighted pairwise Euclidean distances and extend the univariate GR-estimates of Terpstra et al. (Statist. Probab. Lett. 51 (2001) 165; Statist. Inference Stochastic Process. 4 (2001) 155) to the multivariate model. Asymptotic linearity properties are derived for the so called MGR-estimate. Based on these properties, the MGR-estimate is shown to be asymptotically normal at rate n 1/2.

Adaptive Linearizing Control of a Nonlinear Chemical Process Based on Reduced Design Model

In this paper, a control synthesis method requiring the least knowledge on a given process, such as the available outputs and a parametrized reduced design model, is presented. The reduced design model induced by the state transformation including independent output functions is described in the output variables as the transformed state and its minimal dimension is equal to the relative order of the system. It also has parametric uncertainties mainly resulting from the couplings with the disturbance model, which comprises the rest of the transformed system but is excluded from the design work. The method referred to as the ALIC (adaptive linearizing integral control) is a combination of the time-variant input-output linearization with generic model control-like parameter estimation for asymptotic linearization of the parametric design model. Thus, the method does not need the full state availability and the necessity of exact cancellations of nonlinear terms known as critical weak points of the state feedback linearization. Its applicability and effectiveness are demonstrated by simulation for typical control problems in the chemical processes. And, the simplicity and the clarity of the design procedure are elucidated.

A note on asymptotically linear elliptic problems in [formula omitted

for some V (x) ∈ C(R,R) and v∞ ∈ R. In case this equation is considered in a bounded domain ⊂ R (with, say, Dirichlet boundary condition) there is a large literature on existence and multiplicity results, with the case of resonance being of particular interest (see [1,4,5,7,9,12,15]). We recall that the problem is said to be at resonance if −λ ∈ σ(S), where σ(S) denotes the spectrum of S, the “asymptotic linearization” of the problem. In other words, S :D(S) ⊂ L2( )→ L2( ) is the operator given by Su(x)=−∆u(x)− V (x)u(x), D(S)=H 1 0 ( )∩H 2( ). (2)

GENERALIZED SIGNED-RANK ESTIMATORS FOR AUTOREGRESSION PARAMETERS

ABSTRACT A new class of weighted signed-rank-based estimates for estimating the parameter vector of an autoregressive time series is considered. The Wilcoxon signed-rank estimate and the GR-estimates of Terpstra et al. (GR-Estimates for an Autoregressive Time Series. Statistics and Probability Letters 2001, 51, 165–172; Generalized Rank Estimates for an Autoregressive Time Series: A U-Statistic Approach. Statistical Inference for Stochastic Processes 2001, 4, 155–179) can be viewed as special cases of the so-called GSR-estimates. Asymptotic linearity properties are derived for the GSR-estimates. Based on these properties, and a symmetry assumption, the GSR-estimates are shown to be asymptotically normal at rate n 1/2. The theory of U-Statistics along with a characterization of weak dependence that is inherent in stationary AR(p) models are the primary tools used to obtain the results. Tests of hypotheses as well as standard errors for confidence interval procedures can be based on such results. An efficiency study indicates that, for an appropriately chosen set of weights, the GSR-estimate is more efficient than the GR-estimate. Furthermore, the GSR-estimate has an added advantage in that an intercept term can be estimated simultaneously; unlike the GR-estimate. Two examples and a small simulation study are used to illustrate the computational and robust aspects of the GSR-estimates.

R-estimation in Autoregression with Square-Integrable Score Function

This paper develops an asymptotic theory for R-estimation based on a square-integrable, not necessarily bounded, score function in the pth order stationary autoregressive model. Asymptotic uniform linearity of a class of linear rank statistics is established and the asymptotic normality of the corresponding R-estimators is derived. This paper thus solves a long-standing problem in the development of the asymptotics for rank estimators under the autoregressive setup. The proofs use a combination of the approximation technique, the contiguity technique and the weak convergence technique of Hájek, Jurečková and Koul, respectively.

Theory & Methods: Weighted Wilcoxon Estimates for Autoregression

This paper explores the class of weighted Wilcoxon (WW) estimates in the context of autoregressive parameter estimation, giving special attention to three sub‐classes of so‐called WW‐estimates. When the weights are constant, the estimate is equivalent to using Jaeckel’s estimate with Wilcoxon scores. The paper presents asymptotic linearity properties for the three sub‐classes of WW‐estimates. These properties imply that the estimates are asymptotically normal at rate n½. Tests of hypotheses as well as standard errors for confidence interval procedures can be based on such results. Furthermore, the estimates can be computed with an L1 regression routine once the weights have been calculated. Examples and a Monte Carlo study over innovation and additive outlier models suggest that WW‐estimates can be both robust and highly efficient.

On a Class of Asymptotically Linear Elliptic Problems in [formula omitted

Under these conditions (P*) is an asymptotically linear problem. When this equation is considered in a bounded domain 0/R (with, say, the Dirichlet boundary condition) there is a large literature on existence and multiplicity results. Of particular interest is then the case of resonance, where &* # _(S) and S is the asymptotic linearization of the problem. In other words, S: D(S)/L (0) L (0) is the operator given by

Existence of guided cylindrical TM-modes in a homogeneous self-focusing dielectric

We establish the existence of nontrivial solutions for the nonlinear eigenvalue problem which describes self-trapped transverse magnetic field modes in a cylindrical optical fiber made from a self-focusing dielectric material. It amounts to finding solutions in the Sobolev space H01(0,∞) of a singular second order differential equation which is quasilinear and, in an appropriate sense, asymptotically linear. Solutions are critical points of an energy functional which has a mountain pass structure, although for all relevant parameter values the problem is in resonance at infinity. The quasilinearity complicates the proof of a Palais–Smale condition, and the asymptotic linearity means that the standard methods for showing that a P-S sequence is bounded do not apply. Both the linearization and the asymptotic linearization have only continuous spectrum. The eigenvalues determine the wavelengths of self-trapped modes and our results establish the existence of such modes for the largest possible range of wavelengths.

R-estimation for arma models

This paper is devoted to the It-estimation problem for the parameter of a stationary ARMA model. The asymptotic uniform linearity of a suitable vector of rank statistics leads to the asymptotic normality of v/w-consistent R-estimates resulting from the minimization of the norm of this vector. By using a discretized y^-consistent preliminary estimate, we construct a new class of one-step R-estimators. We compute the asymptotic relative efficiency of the proposed estimators with respect to the LS estimator. Efficiency properties are investigated via a Monte-Carlo study in the particular case of an AR(1) model.

Generalized Rank Estimates For An Autoregressive Time Series: A U-Statistic Approach

A class of weighted rank-based estimates for estimating the parameter vector of an autoregressive time series is considered. This class of estimates is similar to, and contains, the class proposed by Terpstra et al. [54]. Asymptotic linearity properties are derived for the so called GR-estimates. Based on these properties, the GR-estimates are shown to be asymptotically normal at rate n 1/2. The theory of U-statistics along with a characterization of weak dependence that is inherent in stationary AR(p) models are the primary tools used to obtain the results. The so called pair-wise slopes estimator, which is a special case of this class of estimates, is discussed in an AR(1) context.

GR-estimates for an autoregressive time series

A weighted rank-based (GR) estimate for estimating the parameter vector of an autoregressive time series is considered. When the weights are constant, the estimate is equivalent to using Jaeckel's estimate with Wilcoxon scores. Asymptotic linearity properties are derived for the GR-estimate. Based on these properties, the GR-estimate is shown to be asymptotically normal at rate n 1/2.

Application of convolution theorems in semiparametric models with non-i.i.d. data

A useful approach to asymptotic efficiency for estimators in semiparametric models is the study of lower bounds on asymptotic variances via convolution theorems. Such theorems are often applicable in models in which the classical assumptions of independence and identical distributions fail to hold, but to date, much of the research has focused on semiparametric models with independent and identically distributed (i.i.d.) data because tools are available in the i.i.d. setting for verifying pre-conditions of the convolution theorems. We develop tools for non-i.i.d. data that are similar in spirit to those for i.i.d. data and also analogous to the approaches used in parametric models with dependent data. This involves extending the notion of the tangent vector figuring so prominently in the i.i.d. theory and providing conditions for smoothness, or differentiability, of the parameter of interest as a function of the underlying probability measures. As a corollary to the differentiability result we obtain sufficient conditions for equivalence, in terms of asymptotic variance bounds, of two models. Regularity and asymptotic linearity of estimators are also discussed.

M-Estimators Converging to a Stable Limit

We discuss the asymptotic linearization of multivariate M-estimators, when the limit distribution is stable. We consider two different types of kernels: VC and bracketing. When applied to the case of normal limits, our work improves the known results to obtain the limit distribution of M-estimators. We give weak conditions for the asymptotic normality of M-estimators over differentiable kernels. To obtain these results, we present an inequality on empirical processes satisfying a bracketing condition with respect to a norm smaller than the L2 norm.

High-performance robust speed-flux tracking controller for induction motor

A new design strategy is proposed for the development of full-order indirect field-oriented induction motor controller. The proposed output feedback controller permits to achieve global exponential asymptotic speed and flux tracking in presence of unknown load torque. In addition, it ensures asymptotic decoupling of the speed and flux subsystems, asymptotic linearization of the speed subsystem and improved robustness with respect to rotor resistance variation, providing improved energy efficiency. The limited implementation complexity, joined to a simple procedure for performance specification and controller parameters' tuning, makes the proposed controller suitable for industrial application. Simulation results are presented, confirming excellent tracking quality and robustness. The proposed non-linear controller and the adopted design procedure are suitable for both further theoretical development (robust and adaptive control) and industrial applications where high dynamic performance is required. Copyright © 2000 John Wiley & Sons, Ltd.

Highly efficient weighted for autoregression wilcoxon estimes for autoregression

In this paper we explore the use of Schweppe-type weights for a class of weighted Wiicoxon estimates and apply the corresponding estimates to an autoregressive time series model This special class of estimates is essentially the autoregressive analog of the HBR-estimates proposed by Chang et al. (1999) in the linear regression context. Assuming a stationary finite second moment autoregressive model of order p, asymptotic linearity properties are derived for the HBR-estimate. Based on these properties, the HBR-estimate is shown to be asymptotically normal at rate nl/2. Tests of general linear hypotheses as well as standard errors for confidence interval procedures can be based on such results. In a linear regression setting, the HBR-estimate is highly efficient and inherits a totally bounded influence function and a 50percent breakdown point. Examples and a Monte Carlo study over innovated and additive outlier models indicate that these properties of the HBR-estimate are preserved in an autoregressive time series context, Thus, the HBR-estimate provides a highly efficient and robust alternative for autoregressive time series estimation.