Asymptotic Linearity Research Articles

Aligned Rank Tests for Linear Models with Autocorrelated Error Terms

Linear models in which the unobserved error constitutes a realization of some stationary ARMA process or, equivalently, ARMA processes with a linear regression trend, are considered under unspecified innovation densities. Due to serial dependence among the observations, the classical rank-based techniques, which have been developed for linear models with independent observations and unspecified error densities, do not apply; nor do the existing rank-based procedures for serial dependence problems, where the observations are assumed to be stationary in the mean (or the median). Moreover, all problems of practical interest (testing the significance of a subset of regression coefficients, identifying the orders p and q of the ARMA ( p, q) dependence, overall diagnostic checking of the model,...) involve nuisance parameters. Typically, one is interested either in the regression trend, and the serial dependence parameters are nuisance parameters; or the serial dependence structure is the main concern, and trend somehow has to be removed. A rank-based approach to such problems thus not only requires extending the classical Hájek-type theory of rank tests to serially dependent situations, it also requires a generalized theory of aligned rank tests. This is the purpose of the present paper. The key result is a local asymptotic normality (LAN) result involving a rank-measurable central sequence; depending on the model considered (with symmetric or totally unspecified innovation densities), the ranks to be used are either signed or unsigned. This LAN result, along with a particular local asymptotic linearity property, implies the local asymptotic sufficiency of (aligned) ranks for a broad class of testing problems-mainly, testing linear restrictions on the parameters of the model. Asymptotically invariant aligned rank tests which are locally asymptotically most stringent similarly are derived. Unlike former results on aligned rank tests for linear models with independent observations, the present ones are entirely obtained from the LAN structure of the underlying problem, and provide a remarkable illustration of the power and appropriateness of LeCam′s approach in nonparametric inference. They generalize and reinforce most previous results on rank-based tests for general linear models, as well as the few available ones on rank-based inference for time series analysis. The resulting testing procedures are invariant or asymptotically invariant, hence (asymptotically) distribution-free. They also are as powerful as (often, strictly more powerful than) their classical, correlogram-based counterparts. In addition, they are considerably more robust, and remain fully reliable in the presence of outliers, heavy-tailed distributions, atypical startup behavior, etc.

Nonparametric Spectral Density Estimation Using Penalized Whittle Likelihood

Abstract The penalized likelihood approach is not well developed in time series analysis, even though it has been applied successfully in a number of nonparametric function estimation problems. Chow and Grenander proposed a penalized likelihood-type approach to the nonparametric estimation of the spectral density of Gaussian processes. In this article this estimator is extended to more general stationary processes, its practical implementation is developed in some detail, and some asymptotic rates of convergence are established. Its performance is also compared to more widely used alternatives in the field. A computational algorithm involving an iterative least squares, initialized by the log-periodogram, is first developed. Then, motivated by an asymptotic linearization, an estimator of the integrated squared error between the estimated and true log-spectral densities is proposed. From this, a data-dependent procedure for selection of the amount of smoothing is constructed. The methodology is illustrated...

Nonparametric Spectral Density Estimation Using Penalized Whittle Likelihood

Abstract The penalized likelihood approach is not well developed in time series analysis, even though it has been applied successfully in a number of nonparametric function estimation problems. Chow and Grenander proposed a penalized likelihood-type approach to the nonparametric estimation of the spectral density of Gaussian processes. In this article this estimator is extended to more general stationary processes, its practical implementation is developed in some detail, and some asymptotic rates of convergence are established. Its performance is also compared to more widely used alternatives in the field. A computational algorithm involving an iterative least squares, initialized by the log-periodogram, is first developed. Then, motivated by an asymptotic linearization, an estimator of the integrated squared error between the estimated and true log-spectral densities is proposed. From this, a data-dependent procedure for selection of the amount of smoothing is constructed. The methodology is illustrated with some real and simulated data sets. A simulation study with autoregressive and moving average processes is conducted to provide quantification of the performance characteristics of the approach relative to some well-established methods in the literature, including the smoothed log-periodogram, the logarithm of the smoothed periodogram, and ARMA spectral density estimators. Empirical rates of convergence of the estimator are evaluated and shown to be well predicted by an asymptotic analysis.

Weak Convergence of Randomly Weighted Dependent Residual Empiricals with Applications to Autoregression

This paper establishes the uniform closeness of a randomly weighted residual empirical process to its natural estimator via weak convergence techniques. The weights need not be independent, bounded or even square integrable. This result is used to yield the asymptotic uniform linearity of a class of rank statistics in $p$th-order autoregression models. The latter result, in turn, yields the asymptotic distributions of a class of robust and Jaeckel-type rank estimators. The main result is also used to obtain the asymptotic distributions of the least absolute deviation and certain other robust minimum distance estimators of the autoregression parameter vector.

Rank-Based Estimates in the Linear Model with High Breakdown Point

Abstract An estimator in the linear model is defined by minimizing an objective function, the derivative of which is a signed rank statistic. The scores are generated from a function h+ : (0, 1) → [0, ∞), which is not necessarily nondecreasing, as is usually assumed. It is shown that this estimator can be chosen with a maximal breakdown point of. 5. Moreover, strong consistency and asymptotic normality (with convergence rate n −1/2 of the proposed estimator are proved under various regularity conditions. Because the objective function generally is not convex in the regression parameters, the usual proofs of asymptotic normality do not carry over. Instead the proof is based on an asymptotic linearity result, similar to that obtained by Huber for M estimates, and some moment estimates for signed rank statistics. Numerical examples illustrate the behavior of the estimator.

Asymptotic linearization of the Fisher equation for a class of initial conditions

We present a novel method of linearizing the Fisher equation asymptotically in time, i.e. we use a nonlinear transformation to convert the initial value problem for the Fisher equation into an initial value problem for the diffusion equation for a class of initial conditions. The analytic result we obtain is in excellent agreement with numerical simulations. Our method easily generalizes to other reaction-diffusion equations also.

Does asymptotic linearity of the regression extend to stable domains of attraction?

C. D. Hardin, Jr., G. Samorodnitsky, and M. S. Taqqu (1991, Ann. Appl. Probab. 1 582–612) have shown that the regression E [ Y | X = x ] is typically asymptotically linear when ( X , Y ) is an α -stable random vector with α < 2. We provide necessary and sufficient conditions for asymptotic linearity of E [ Y | X + δ = z ], where ( X , Y ) is an α -stable random vector and δ is a random variable, independent of ( X , Y ), such that X + δ is in the domain of normal attraction of X . Asymptotic linearity does not always hold even when E [ Y | X = x ] is linear. For some distributions of δ , the asymptotic rate of E [ Y | X + δ = z ] fluctuates.

ONE-STEP-M-ESTIMATORS IN CONDITIONALLY CONTAMINATED LINEAR MODELS

The notion of one-step-M-estimators is generalized for estimating general linear aspects of homoscedastic or heteroscedastic conditionally contaminated linear models with general parametrization and general underlying design. For such contaminated lineal: models it is shown that the generalized one-step-M-estimators at random and deterministic designs with finite support are asymptotically normally distributed with maximum asymptotic bias and asymptotic covariance matrix only depending on the score function of the one-step-M-estimator and the true variances of the model. This is obtained by deriving their asymptotic linearity under relative simple assumptions which are easy to verify and which admit a wide scope of applications. An example demonstrates the applicability of the results.

The use and interpretation of rank-based residuals

The diagnostic properties of residuals based on generalized rank-based (GR) estimates are investigated. These are bounded-influence estimates which are resistant to the effects of outliers in either the y- or x-space. Using first order approximations based on asymptotic linearity results, the interpretation of residuals based on GR-fits is discussed for true and misspecified models. A measure of departure from othogonality of both R- and GR-residuals and their corresponding fitted values is also proposed. This measure and the interpretation of GR-residuals prove insightful in an understanding of GR-residual plots. Proper standardization of GR-residuals is also discussed. This is effective in declaration of potential outliers. A small Monte Carlo study along with examples verify the asymptotic theory .

Asymptotics of R-, MD- and LAD-estimators in linear regression models with long range dependent errors

This paper establishes the uniform closeness of a weighted residual empirical process to its natural estimate in the linear regression setting when the errors are Gaussian, or a function of Gaussian random variables, that are strictly stationary and long range dependent. This result is used to yield the asymptotic uniform linearity of a class of rank statistics in linear regression models with long range dependent errors. The latter result, in turn, yields the asymptotic distribution of the Jaeckel (1972) rank estimators. The paper also studies the least absolute deviation and a class of certain minimum distance estimators of regression parameters and the kernel type density estimators of the marginal error density when the errors are long range dependent.

Bifurcation points of Hammerstein equations

We apply global bifurcation theorems to systems of nonlinear integral equations of Hammerstein type involving a scalar parameter. To this end, we give sufficient conditions for the continuous dependence, compactness, Frechet differentiability, and asymptotic linearity of the corresponding operators, which are more general than in the classical setting. These properties are ensured only after passing to some equivalent operator equation which typically contains fractional powers of the linear part. Finally, we show that the abstract hypotheses on the operators correspond to natural hypotheses on the kernel function and the nonlinearity in the Hammerstein equation under consideration.

A Large Sample Study of Rank Estimation for Censored Regression Data

Large sample approximations are developed to establish asymptotic linearity of the commonly used linear rank estimating functions, defined as stochastic integrals of counting processes over the whole line, for censored regression data. These approximations lead to asymptotic normality of the resulting rank estimators defined as solutions of the linear rank estimating equations. A second kind of approximations is also developed to show that the estimating functions can be uniformly approximated by certain more manageable nonrandom functions, resulting in a simple condition that guarantees consistency of the rank estimators. This condition is verified for the two-sample problem, thereby extending earlier results by Louis and Wei and Gail, as well as in the case when the underlying error distribution has increasing failure rate, which includes most parametric regression models in survival analysis. Techniques to handle the delicate tail fluctuations are provided and discussed in detail.

Practical version of the Asymptotic Linearity Theorem with applications to the additivity problems of thermodynamic quantities

By using fundamental notions and theorems of the Lesbesgue integral, a practical version of the Asymptotic Linearity Theorem has been derived. From this new version of the theorem, it easily follows that the theorem is also applicable to the additivity problems of thermodynamic quantities of molecules having many identical moieties, as well as to those of the zero-point vibrational energy of hydrocarbons and the total pi electron energies of altemant hydrocarbons.

Aspects of form and general topology: Alpha Space Asymptotic Linearity Theorem and the spectral symmetry of alternants

A theoretical device and a technique have been developed by which one can study, in a unifying manner, additive properties of hydrocarbons and the symmetric property of the spectrum of alternants. By using functionals introduced in the approach via the aspects of form and general topology, a special version of the Asymptotic Linearity Theorem (for the study of additivity problems of the zero-point vibrational energy of hydrocarbons and the total pi-electron energy of alternants) has been obtained in parallel with a derivation of the spectral symmetry of alternants.

The Asymptotic Linearity Theorem for the study of additivity problems of the zero-point vibrational energy of hydrocarbons and the total pi-electron energy of alternant hydrocarbons

By using the approach via the aspect of form and general topology, as well as basic notions of abstract algebra, a theoretical framework has been developed which elucidates the mechanism of the additivity relationships between structure and properties in molecules having many identical moieties. The main theorem, the Asymptotic Linearity Theorem (ALT), together with an auxiliary theorem, the a Independence Theorem, implies that the zero-point vibrational energy (or total pi-electron energy for the case of alternant hydrocarbons)En of a linearly extended system B-An-B′ having n repeating identical moieties has the asymptotic expansionEn = αn +β +o(1) as n → ∞, where α ∈ ℝ is independent of the choice of the end moieties B and B′. The theorem being formulated in a general context, the actual implication of the ALT is much broader than the above two applications would indicate.

Asymptotic linearization of uncertain nonlinear systems by means of continuous control

AbstractIn this paper a new method for the solution of the asymptotic linearization of uncertain nonlinear systems by means of a continuous control law is presented. The main feature of this approach consists of a continuous first‐order estimator and control laws with piecewise continuous derivatives. As an important by‐product of this theory we can deal easily with the case of some uncertain systems which do not satisfy the matching condition, and with systems presenting first order unmodelled dynamics.

Asymptotic linearity of minimax estimators

The celebrated local asymptotic minimax (LAM) theorem due to HÁjek (1972) also includes the statement that a LAM estimator Is necessarily asymptotically linear. A similar result. is true for semi‐parametric models, but Hájek's result doesn't apply to this case as the efficient influence function is often not contained in the (proper) tangent space. This note gives a simple, elementary proof of both the LAM theorem and the necessity of asymptotic linearity of a LAM estimator sequence.

Optimum Robust Estimation of Linear Aspects in Conditionally Contaminated Linear Models

P. J. Bickel's approach to and results on estimating the parameter vector $\beta$ of a conditionally contaminated linear regression model by asymptotically linear (AL) estimators $\hat\beta^\ast$ which have minimum trace of the asymptotic covariance matrix among all AL estimators with a given bound $b$ on their asymptotic bias (MT-AL estimators with bias bound $b$) is here extended to conditionally contaminated general linear models and in particular for estimating arbitrary linear aspects $\varphi(\beta) = C\beta$ of $\beta$ which are of actual interest in applications. Admitting that $\beta$ itself is not identifiable in the model (also a practically important situation), a complete characterization of MT-AL estimators with bias bound $b$ including the case where $b$ is smallest possible is presented here, which extends and sharpens H. Rieder's characterization of all AL estimators with minimum asymptotic bias. These characterizations (Theorem 1) represent generalizations (in different directions) of those which define Hampel-Krasker estimators for $\beta$ in linear regression models and admit (Theorem 2) explicit constructions of MT-AL estimators under generally applicable model assumption. Obviously, even in linear regression models, $\hat\varphi^\ast = C\hat\beta^\ast$ is not an MT-AL estimator for $\varphi$ if $\hat\beta^\ast$ is one for $\beta$ (there does not even exist an AL estimator nor an $M$ estimator for $\beta$, if $\beta$ is not identifiable in the model). Examples such as quadratic regression illustrate the not at all obvious relation between $\hat\beta^\ast$ and $\hat\varphi^\ast$, demonstrate the applicability of the general results and show explicitly the influence of the parametrization and the underlying design of the linear model.

Nonrelativistic versus ultra-relativistic description of Regge trajectories

Comparing the energy spectrum of the nonrelativistic description of hadrons as bound states of quarks to the one obtained in an ultra-relativistic treatment, we show that both approaches yield almost identical binding energies if the confining interaction is able to reproduce the observed asymptotic linearity of the Regge trajectories.

A weak convergence result useful in robust autoregression

Abstract This paper establishes the closeness, in a uniform sense, of a sequence of randomly weighted estimated residual empirical processes to a sequence of randomly weighted residual empirical processes via the weak convergence techniques. This result is used to obtain an asymptotic expansion of generalized M (G–M) estimators of autoregression parameters and the asymptotic uniform linearity (A.U.L.) of the sequence of ordinary residual empirical processes in a p -th order autoregression (AR( p )) model. The latter result is used to prove the A.U.L. of serial rank correlations of the residuals in an AR( p ) model and to yield a direct proof of the consistency of a class of estimators of the functional Γ( f ) ≔∫ f dϕ( F ), where f ( F ) is the error density (distribution function) in the AR( p ) model and ϕ is a nondecreasing bounded function on [0, 1]. These functionals appear in the asymptotic variances of various robust estimators of autoregression parameters. In summary, the paper offers a unified functional approach to some aspects of the robust estimation in AR( p ) models.