Mathematical Sciences: Optimal Inference in Non-Linear Regression Models with Long Range Dependent Errors and in Non-Linear Time Series

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A discrete time stationary stochastic process is said to be long range dependent if its correlations decrease to zero like a power of the lag, as the lag tends to infinity, but their sum diverges. Part I of the proposal proposes to investigate the large sample behavior of three classes of robust estimators and to develop asymptotically efficient and adaptive estimators in non-linear regression models when the errors are long range dependent with possibly unknown joint distributions. The Part II of the proposal is concerned with studying the asymptotic behavior of several classes of robust estimators and devloping asymptoticlaly optimal inference procedures in non-linear time series in the presence of regression component, in a semi-parametric setup. In particular, the P.I. plans to develop asymptotically efficient and adaptive estimators in random coefficient and threshold autoregression models when there may be a regression variable present in these models, in a llinear or non-linear fashion, and when the distributions of the random coefficient and the error variable are unknown. The optimal estimators would be developed a l a Hajek - Le Cam theory. A data set where an association between distant observations is slowly decaying but persistent, as distance between observations increases, is called long range dependent. Such data arise often in astronomy, economics, geophysics, hydrology, meteorology, and many other disciplines. An example worth mentioning is the data of 289 high-precision measurements on the 1-kg check standard weight made between 1963 to 1975 by the U.S. National Bureau of Standards. In spite of ideal conditions for preserving independence, the observations turned out to be long range dependent. The first part of the proposal is concerned with developing some optimal statistical procedures for analyzing the long range dependent data in the presence of a covariate. The second part of the proposal is concerned with developing efficient inferential procedures in some complicated time series models that often arise in econometrics. It is anticipated that these procedures will be broadly applicable and not very sensitive to model departures.