D.Phil. / During the last few years the number of known variable stars which show periodic light level changes has grown by several tens of thousands. The aim of the research reported here was to extend the suite of statistical methods available for the analysis of periodic variable star time series. Solution techniques for five problems are discussed. The first is an automated method for detecting periodic variable stars from a database containing of the order of 100 000 time series of observations. Typically only 100-200 brightness measurements of each star were obtained, spread irregularly over an interval of about 3 years. The proposed method is based on a signal to noise ratio. Percentiles for the statistic are found by studying randomisations of a large number of the observed time series. It is shown that the percentiles depend strongly on the number of observations in a given dataset, and the dependence is calibrated empirically. The estimation of the frequency, amplitude and phase of a sinusoid from observations contaminated by correlated noise is the second problem considered. The study of the observational noise properties of nearly 200 real datasets of the relevant type is reported: noise can almost always be characterised as a random walk with superposed white noise. A scheme for obtaining weighted nonlinear least squares estimates of the parameters of interest, as well as standard errors of these estimates, is described. Simulation results are presented for both complete and incomplete data, and an application to real observations is also shown. In the third topic discussed it is assumed that contemporaneous measurements of the light in-tensity of a pulsating star is obtained in several colours. There is strong theoretical interest in a comparison of the amplitudes and phases of the variations in the different colours. A general scheme for calculating the covariance matrix of the estimated amplitude ratios and phase differences is described. The first step is to fit a time series model to the residuals after subtracting the best-fitting sinusoid from the observations. The residuals are then crosscorrelated to study the interdependence between the errors in the different colours. Once the multivariate time series structure can be modelled, the covariance matrix can be found by bootstrapping. An illustrative application is described in detail. The times between successive instances of maximum brightness, or the times between successive brightness minima, serve as estimates for the periods of the so-called "long period variables" (stars with pulsation periods of the order of months). The times between successive maxima (or minima) vary stochastically, and are also subject to measurement errors, which poses a problem for tests for systematic period changes — the topic of the fourth problem studied. A simple statistical model for the times between successive maxima, or minima, of such stars is used to calculate the auto-correlation properties of a new time series, which is non-stationary in its variance. The new series consists of an alternation of cycle lengths based on respectively the times between maxima, and those between minima of the light curve. Two different approaches to calculating the theoretical spectrum of the non-stationary time series, as required in the proposed statistical hypothesis test, are given. Illustrative applications complete the relevant chapter.

D.Phil. / We propose solutions to two statistical problems using the frequency domain approach to time series analysis. In both problems the data at hand can be described by the well known signal plus noise model. The first problem addressed is the estimation of the underlying variance of a process for the use in a Shewhart or CUSUM control chart when the mean of the process may be changing. We propose an estimator for the underlying variance based on the periodogram of the observed data. Such estimators have properties which make them superior to some estimators currently used in Statistical Quality Control. We also present a CUSUM chart for monitoring the variance which is based upon the periodogram-based estimator for the variance. The second problem, stimulated by a specific problem in Variable Star Astronomy, is to test whether or not the mean of a bivariate time series is constant over the span of observations. We consider two periodogram-based tests for constancy of the mean, derive their asymptotic distributions under the null hypothesis and under local alternatives and show how consistent estimators for the unknown parameters in the proposed model can be found