Box and Jenkins (1970) made the idea of obtaining a stationary time series by differencing the given, possibly nonstationary, time series popular. Numerous time series in economics are found to have this property. Subsequently, Granger and Joyeux (1980) and Hosking (1981) found examples of time series whose fractional difference becomes a short memory process, in particular, a white noise, while the initial series has unbounded spectral density at the origin, i.e. exhibits long memory.Further examples of data following long memory were found in hydrology and in network traffic data while in finance the phenomenon of strong dependence was established by dramatic empirical success of long memory processes in modeling the volatility of the asset prices and power transforms of stock market returns.At present there is a need for a text from where an interested reader can methodically learn about some basic asymptotic theory and techniques found useful in the analysis of statistical inference procedures for long memory processes. This text makes an attempt in this direction. The authors provide in a concise style a text at the graduate level summarizing theoretical developments both for short and long memory processes and their applications to statistics. The book also contains some real data applications and mentions some unsolved inference problems for interested researchers in the field.

This book presents a unified approach for obtaining the limiting distributions of minimum distance, M and R estimators corresponding to non-smooth underlying scores in a large class of dynamic non-linear models including ARCH models. It also discusses classes of goodness-of-t tests for fitting an error distribution in some of these models and/or fitting a regression-autoregressive function without assuming the knowledge of the error distribution. The main tool is the asymptotic equicontinuity of certain basic weighted residual empirical processes in the uniform and L2 metrics. The contents of this monograph should be useful to graduate students and research scholars in statistics, econometrics, and finance. This book is a an updated edition of the author's monograph Weighted Empirical Processes and Liner Models (IMS Lecture Notes-Monograph 21, 1992). The new edition differs from the previous one in many ways. To mention just a few: It includes asymptotically distribution free tests for fitting a regression and/or an autoregressive models; the asymptotic distributions of auto-regression quantiles and rank scores; and above all the weak convergence of the residual empirical processes useful in nonlinear ARCH models. Hira L. Koul is a professor of statistics at Michigan State University. He is a Fellow of the IMS and an Elected Member of the International Statistical Institute. He was awarded the prestigious Humboldt Research Award for Senior Researchers in 1995. He has been on the editorial boards of the Annals of Statistics, Sankhya, and J. Indian Statistical Association. Currently he is a Coordinating Editor of the Journal of Statistical Planning and Inference, and an Associate Editor of Statistics and Probability Letters.