Advances In Time Series Analysis With Hydrological Applications

The objective of this thesis is to develop and refine statistical methods which can be used for solving a variety of challenging problems which arise in the field of stochastic hydrology and elsewhere. Four areas in stochastic hydrology where more research is required are long term memory processes and the Hurst phenomenon, seasonal geophysical processes, bootstrapping time series models and nonlinear and/or nonGaussian features present in geophysical time series.;For addressing the first problem, we study the family of fractionally differencing autoregressive and moving-average processes (FARMA). Specifically, we characterize the efficiency of the sample mean and the efficiency of ordinary least squares estimates of regression parameters when the statistical error belongs to the FARMA family. Also, we prove that for FARMA models it is true that the logarithm of the determinant of the autocovariance matrix, divided by the number of observations converges to the logarithm of the variance of the white noise. Finally, the finite sample properties of the maximum likelihood method of estimation are compared with the average likelihood method.;For the second problem in stochastic hydrology stated above, we investigate the family of periodic autoregressive and moving-average models (PARMA). We characterize these processes from the point of view of the state-space formulation. We give an exact maximum likelihood algorithm for fitting a PARMA model. Also, we give algorithms for computing the autocovariance matrix, the inverse autocorrelations, and the information matrix associated with the PARMA model.;An additional problem that initially was formulated for its relevance to stochastic hydrology is an improved bootstrapping procedure to investigate parameter uncertainty. We generalize and study some of the asymptotic properties of this procedure.;With respect to the fourth problem in stochastic hydrology, we investigate nonparametric function fitting when applied to time series processes. We give strong approximation results for these estimates, subject to certain mixing properties of the stochastic process. We extend a data based method for choosing the smoothing constant, and propose another method for doing this. We investigate a convex combination of a parametric estimate plus a nonparametric function estimate. Also, we study within a time series context a semi-parametric approach for function estimation. Finally, we use the kernel method for estimating state-dependent models