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The purpose of this thesis is to develop a valid statistical procedure for the estimation of the seasonal component of an economic time series when the seasonal component is suspected to be partly additive and partly multiplicative to the trend. The proposed procedure is based on a three-way classification analysis of variance model, where the first classification is used to represent the long term trend of the series, the second classification is used to represent any regular trend or cycle within the long term trend, and the third classification is used to represent the seasonal. The interaction term between the long term trend and the seasonal may be used to represent any long term change in the nature of the seasonal. However, as the standard analysis of variance significance tests assume independently distributed residuals, it is necessary to develop a test for independence of residuals against the very likely alternative of first order (positive) serial correlation. This is done by calculating the mean and variance of the Durbin-Watson d statistic for the three-way classification analysis of variance model. A numerical example is given to illustrate the procedure. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis considers an approach to evaluate the effectiveness of risk communications for prescription drugs by performing interrupted time series analysis of prescription drug volumes prior to and after the risk communication date. The paper presents methods for detecting change in the presence of autocorrelation and techniques to reduce bias in estimation. Statistical results and data plots are presented for 63 data series. Size and power of the statistical techniques are considered, and a correspondence analysis between these statistical techniques and a small group of physicians is performed. The methods considered in this thesis correspond weakly with physician sentiment, and exhibit inflated type I errors in the presence of significant autocorrelation.

This thesis examines the statistical properties of the Poisson AR(1) model of Al-Osh and Alzaid (1987) and McKenzie (1988). The analysis includes forecasting, estimation, testing for independence and specification and the addition of regressors to the model. The Poisson AR(1) model is an infinite server queue, and as such is well suited for modeling short-term disability claimants who are waiting to recover from an injury or illness. One of the goals of the thesis is to develop statistical methods for analyzing series of monthly counts of claimants collecting short-term disability benefits from the Workers' Compensation Board (WCB) of British Columbia. We consider four types of forecasts, which are the k-step ahead conditional mean, median, mode and distribution. For low count series the k-step ahead conditional distribution is practical and much more informative than the other forecasts. We consider three estimation methods: conditional least squares (CLS), generalized least squares (GLS) and maximum likelihood (ML). In the case of CLS estimation we find an analytic expression for the information and in the GLS case we find an approximation for the information. We find neat expressions for the score function and the observed Fisher information matrix. The score expressions leads to new definitions of residuals. Special care is taken to test for independence since the test is on the boundary of the parameter space. The score test is asymptotically equivalent to testing whether the CLS estimate of the correlation coefficient is zero. Further we define a Wald and likelihood ratio test. Then we use the general specification test of McCabe and Leybourne (1996) to test whether the model is sufficient to explain the variation found in the data. Next we add regressors to the model and update our earlier forecasting, estimation and testing results. We also show the model is identifiable. We conclude with a detailed application to monthly WCB claims counts. The preliminary analysis includes plots of the series, autocorrelation function and partial autocorrelation function. Model selection is based on the preliminary analysis, t-tests for the parameters, the general specification test and residuals. We also include forecasts for the first six months of 1995. / Business, Sauder School of / Graduate

Ph.D. / Harmonic time series are often used to describe the periodic nature of a time series, for example the periodic nature of a variable star’s observed light curve. Statistical methods for determining the number of harmonic components to include in harmonic time series are limited. In this thesis a stepwise bootstrap procedure based on a F-type statistic is suggested. The performance of the stepwise procedure is compared to that of Schwartz’s Bayesian Criterion (SBC) and a procedure based on a statistic described by Siegel (1980). Harmonic series with correlated noise terms and irregularly spaced observations are also considered. Tests to detect changes in harmonic parameters are also derived in this thesis. A cumulative sum statistic to test for constant amplitude is derived. It is shown that testing for constant amplitude is equivalent to testing for constant slope in simple linear regression. We also derive a likelihood ratio statistic to test for constant amplitude. It is shown that the latter likelihood ratio statistic is asymptotically equivalent to the cumulative sum statistic. These statistics are compared to a quadratic form statistic used by Koen (2009). Likelihood ratio tests are also derived for detecting changes in the frequency or phase of harmonic time series. Graphical devices to aid in diagnostic checking are suggested.

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In order to optimise the Resonance Enhanced Drilling (RED) performance in different rock formations, it is important to understand both the influence of the system parameters on the drilling dynamics, and other measures that are involved in the drilling operation. This work studies the dynamic behaviour of the drilling system. It also investigates the influence of various system parameters on the drilling module dynamics in order to identify in real-time the formation being drilled from the dynamical responses of the drilling assembly. It also aims to optimise the selection of the operating parameters for a drilled formation, resulting in an improvement in the Rate Of Penetration (ROP). A nonlinear time series analysis approach has been used to infer the changes in the system parameters from subtle changes in the system dynamics. Using the acceleration time-series as a measurement of simulated and experimental impact oscillators representing a model for drilling conditions with intermittent impacts, the systems attractors were reconstructed and characterised. It is shown that the stiffness correlates well with the topology of the reconstructed attractor. Nonimpacting trajectories formed an approximate plane within the three dimensional reconstructed phase-space. Contact with the constraint caused a systematic deviation from the linear subspace, the inclination of which, measured by statistics of the tangent vector, can be used to infer the stiffness. Based on the developed framework it is now possible to classify stiffness of the impacted material from a single variable in a simple way and in real-time. An experimental impact drilling rig was designed, built and used to study the influence of the system parameters on the high-frequency impact drilling. The newly designed rig is a smaller and simpler apparatus. It is designed to mimic the actual RED apparatus in terms of providing controlled axial vibration into a conventional rotary drilling whilst avoiding complications that might arise from including all RED rig elements. An instrumentation and sensing system was also developed to measure motions and forces resulting from the dynamic interactions between the drill-bit and the rock formations.

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published_or_final_version / Statistics / Doctoral / Doctor of Philosophy