Hira L. Koul

Weighted empirical processes in dynamic nonlinear models

Introduction * Asymptotic Properties of W.E.P.s * Linear Rank and Signed Rank Statistics * M, R and Some Scale Estimators * Minimum Distance Estimators * Goodness-of-fit Tests in Regression * Autoregression * Nonlinear Autoregression

Large sample inference for long memory processes

Introduction Estimation Some Inference Problems Residual Empirical Processes Regression Models Nonparametric Regression with Heteroscedastic Errors Model Checking under Long Memory Long Memory under Infinite Variance.

Asymptotic normality of regression estimators with long memory errors

This paper discusses asymptotic normality of certain classes of M- and R-estimators of the slope parameter vector in linear regression models with long memory moving average errors, extending recent results of Koul (1992) and Koul and Mukherjee (1993). Like in the case of the long memory Gaussian errors, it is observed that all these estimators are asymptotically equivalent to the least squares estimator, a fact that is in sharp contrast with the i.i.d. errors case.

Martingale transforms goodness-of-fit tests in regression models

This paper discusses two goodness-of-fit testing problems. The first problem pertains to fitting an error distribution to an assumed nonlinear parametric regression model, while the second pertains to fitting a parametric regression model when the error distribution is unknown. For the first problem the paper contains tests based on a certain martingale type transform of residual empirical processes. The advantage of this transform is that the corresponding tests are asymptotically distribution free. For the second problem the proposed asymptotically distribution free tests are based on innovation martingale transforms. A Monte Carlo study shows that the simulated level of the proposed tests is close to the asymptotic level for moderate sample sizes.

M-estimators in linear models with long range dependent errors

This paper discusses the asymptotic behavior of a class of M-estimators in linear models when errors are Gaussian, or a function of Gaussian random variables, that are long range dependent. The asymptotics are discussed when the design variables are either i.i.d. or long range dependent, independent of the errors, or known constants. It is observed that in the latter two cases, the leading r.v.s in the approximation of the class M-estimators of the regression parameter vector corresponding to the skew symmetric scores and symmetric errors is proportional to the least squares estimator in the Gaussian errors case. Moreover, if the design variables are either i.i.d. or the known constants then the limiting distributions are normal. But if the design variables are long range dependent then the limiting distributions are nonnormal.

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

SummaryThis 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.

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.

Minimum distance regression model checking

Abstract This paper discusses a class of minimum distance tests for fitting a parametric regression model to a regression function when the underlying d-dimensional design variable is random, d⩾1, and the regression model is possibly heteroscedastic. These tests are based on certain minimized L2 distances between a nonparametric regression function estimator and the parametric model being fitted. The paper establishes the asymptotic normality of the proposed test statistics and that of the corresponding minimum distance estimators under the fitted model. These estimators turn out to be n1/2-consistent. Some simulations are also included.

A test for new better than used

In this paper a Kolmogorov-Smirnov type test is proposed for testing that a life distribution is exponential against that it is new better than used. The test is shown to be consistent and unbiased. A rejection region of size , n≥2, based on the statistic is exhibited which has power 1 against a subclass of alternatives. Some small sample null tail probabilities are derived, and selected critical values are tabulated, using Monte Carlo techniques. Finally, the asymptotic null distribution and the distribution under alternatives is discussed.

Testing for new is better than used in expectation

A class of tests which are usually believed to be the tests of exponentiality vs. increasing failure rate (IFR) alternatives are shown to be consistent against a class of NBUE alternatives - a much larger class than IFR. Moreover a comparison is made of the Bahadur efficiency of the total time on test statistic W with the Kolmogorov-Smirnov statistic D. It is found that the D-test has larger exact slope at least favorable IFR distributions which are close to the exponential and also at pure NBUE distributions close to the exponential than does the W-test statistic, whereas the reverse is the case at Weibull (8), 6 > 1.