Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

http://library.um.ac.id/free-contents/downloadpdf.php/buku/laws-of-small-numbers-extremes-and-rare-events-michael-falk-jurg-husler-rolf-dieter-reiss-42718.pdf

Laws of small numbers: extremes and rare events / Michael Falk, Jurg Husler, Rolf-Dieter Reiss

SUMMARY Robust analogues of the Wald and the Rao score statistics are presented for testing composite hypotheses in time series models involving dependent observations. Asympto- tic efficiency and empirical power comparisons using moderate size samples are given for first-order autoregressive processes. Our method of robustifying the estimating equation gives a simple and unified approach which is useful in a large class of problems including regression with dependent errors, and autoregressive and moving average processes. Large-sample properties of the robust procedures are given for these processes. Standard methods of estimation and testing based on the likelihood function, least squares and moments are used extensively in the analysis of time series data. The performance of these methods is known to be sensitive to changes in the assumptions regarding the error distribution. Moreover, these procedures are affected adversely by the presence of outliers in the data. Robustified versions of the standard statistical procedures are now well documented in the case of independent observations; see Huber (1981) and references therein. Robust estimation for time series has been considered by several authors. Gastwirth & Rubin (1975) have discussed robust estimation of location parameters in the presence of autoregressive dependence. Denby & Martin (1979) and Martin (1980, 1982) have studied efficiency robustness of M-estimators for autoregressive processes. The recent survey article of Martin & Yohai (1985) gives a good review of the literature on robust estimation for autoregressive and moving average processes, and contains several references to published and as yet unpublished papers. Work on robust tests for time series, on the other hand, appears to be scarce. Large- sample tests based on the score and the Wald statistic for time series using the likelihood framework are well known; see, for example, Basawa, Billard & Srinivasan (1984). Generalizations of the score and the Wald statistics using arbitrary estimating functions in place of the likelihood function were introduced by Basawa (1985). In the present paper we use a similar approach to formulate robustified versions of the score and the Wald statistics in specific time series models. Our main aim is to give a simple general method of constructing robust score functions and robust M-estimators for time series models, and to discuss the efficiency of tests derived from this method. In ?2 we introduce robustified score and Wald statistics, and discuss their limit distributions both under the null and a sequence of alternative hypotheses. We also

https://www.jstor.org/stable/pdfplus/2336728.pdf

Robust tests for time series with an application to first-order autoregressive processes

Publisher Summary This chapter discusses the concept of outlier detection in stationary and nonstationary time series models and the effects of outliers on unit root tests. It also discusses the different robust unit root tests suggested in the literature and the robust estimation of nonstationary data. Many of the time series in economics and finance are characterized by errors with heavy tailed distributions. There are two approaches to outlier detection. The first is the classical approach starting with the paper by Fox, and later extended by several others. The second is the Bayesian approach, first started by Abraham and Box. Outlier detection is important for methods based on discarding the outliers. The other alternative of leaving the outliers in and using robust methods does not always depend on outlier detection; the chapter describes the way the robust methods can be used for outlier detection.

10 Outliers, unit roots and robust estimation of nonstationary time series

We compare the behavior of several bootstrap procedures for monitoring changes in the error distribution of autoregressive time series. The proposed procedures are designed to control the overall significance level and include classical tests based on the empirical distribution function as well as Fourier-type methods that utilize the empirical characteristic function, both functions being computed on the basis of properly estimated residuals. The Monte Carlo study incorporates different estimators and a variety of sampling situations with and without outliers.

Bootstrap Procedures for Online Monitoring of Changes in Autoregressive Models

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/F85C9D04E0DF27DB97232FF2544E74AD/S0021900200024050a.pdf/div-class-title-the-robust-estimation-of-autoregressive-processes-by-functional-least-squares-div.pdf

C. R. Heathcote

Papers

The robust estimation of autoregressive processes by functional least squares