In this thesis two general problems concerning goodness-of- fit statistics based on the empirical distribution are considered. The first concerns the problem of adapting Kolmogorov-Smirnov type statistics to test for discrete populations. The significance points of the statistics are given and various power comparisons made. The second problem concerns testing for goodness-of-fit with censored data using the Cramér-von Mises type statistics. The small and large sample distributions are given and the tests are modified so that they can be used to test for the normal and the exponential distributions. The asymptotic theory is developed. Percentage points for the statistics are given and various small sample and large sample power studies are made, for the various cases.

Many geological formations consist of crystalline rocks that have very low matrix permeability but allow flow through an interconnected network of fractures. Understanding the flow of groundwater through such rocks is important in considering disposal of radioactive waste in underground repositories. A specific area of interest is the conditioning of fracture transmissivities on measured values of pressure in these formations. This is the process where the values of fracture transmissivities in a model are adjusted to obtain a good fit of the calculated pressures to measured pressure values. While there are existing methods to condition transmissivity fields on transmissivity, pressure and flow measurements for a continuous porous medium there is little literature on conditioning fracture networks. Conditioning fracture transmissivities on pressure or flow values is a complex problem because the measured pressures are dependent on all the fracture transmissivities in the network. This thesis presents two new methods for conditioning fracture transmissivities in a discrete fracture network on measured pressure values. The first approach adopts a linear approximation when fracture transmissivities are mildly heterogeneous; this approach is then generalised to the minimisation of an objective function when fracture transmissivities are highly heterogeneous. This method is based on a generalisation of previous work on conditioning transmissivity values in a continuous porous medium. The second method developed is a Bayesian conditioning method. Bayes’ theorem is used to give an expression of proportionality for the posterior distribution of fracture log transmissivities in terms of the prior distribution and the data available through pressure measurements. The fracture transmissivities are assumed to be log normally distributed with a given mean and covariance, and the measured pressures are assumed to be normally distributed values each with a given error. From the expression of proportionality for the posterior distribution of fracture transmissivities the modes of the posterior distribution (the points of highest likelihood for the fracture transmissivities given the measured pressures) are numerically computed. Both algorithms are implemented in the existing finite element code NAPSAC developed and marketed by Serco Technical Services, which models groundwater flow in a fracture network.

The thesis consists of six chapters. The introductory first chapter considers some of the more general problems involved in the detection and rejection of outlying observations, and.describes the general form of the tests discussed in detail in the later chapters. In Chapter 2. likelihood-based criteria are derived for testing for single and multiple outliers at both the upper and the lower ends of samples from gamma distributions. The null distributions of these criteria are obtained by use of a recursive algorithm and the methods are extended to criteria appropriate for testing for multiple outliers occurring at both ends of the sample and to various 'Dixon' criteria. The results are applied to some practical examples. In Chapter 3 likelihood-based tests and criteria for single outliers in univariate normal samples are considered. The null distributions of the criteria are obtained by recursive algorithms. The cases of known and unknown mean and variance are considered separately and the methods are extended to cases where independent estimates of the variance are available. These methods and results are extended in Chapter 4 to tests and criteria for multiple outliers in univariate normal samples. The extensions of the results of both of these chapters to single and multiple outliers in multivariate normal samples are considered in Chapter 6. In Chapter 5 problems of single and multiple outliers in data following a linear model are discussed. A likelihood-based criterion is derived and the extreme tail of the null distribution of this criterion is obtained. Some practical examples on data from a series of chemical experiments are given.

Given a second order stationary time series it can be shown that there exists an optimum linear predictor of Xk, say X*k which is constructed from {Xt ,t=O,-l,-2 …} the mean square error of prediction being given by ek = E [|Xk- X*k|2]. In some cases however a series can be considered to have started at a point in the past and an attempt is made to see how well the optimum linear form of the predictor behaves in this case. Using the fundamental result due to Kolmogorov relating the prediction error e1 to the power spectrum f(w) e1 = exp. {1/2 pi Log from – pi to p log 2 pi f(w) dw} estimates of e1 are constructed using the estimated periodogram and power spectrum estimates. As is argued in some detail the quantity e1 is a natural one to look at when considering prediction and estimation problems and the estimates obtained are non-parametric. The characteristic functions of these estimates are obtained and it is shown that asymptotically they have distributions which are approximately normal. The rate of convergence to normality is also investigated. A previous author has used a similar estimate as the basis of a test of white noise and the published results are extended and in the light of the simulation results obtained some modifications are suggested. To increase the value of the estimates e1 their small sample distribution is approximated and extensive tables of percentage points are provided. Using these approximations one can construct a more powerful and versatile test for white noise and simulation results confirm that the theoretical results work well. The same approximation technique is used to derive the small sample distribution of some new estimates of the coefficients in the model generating {Xt}. These estimates are also based on the power spectrum. While it is shown small sample theory is limited in this situation the asymptotic results are very interesting and useful. Several suggestions are made as to further fields of investigation in both the univariate and multivariate cases.

M.Sc. (Mathematical Statistics) / While a typical regression model describes how the mean value of a response variable varies with a set of explanatory variables, quantile regression describes the variation in the quantiles of the response. When the response distribution di ers substantially from normality the quantiles provide a substantially richer description of the distribution than can be obtained by standard regression, and is obtainable without making any assumptions on the form of the underlying distribution. In this dissertation we study the theory of quantile regression models, with particular focus on the application of quantile regression methods to censored survival data. While the statistical literature on censored quantile regression methods is extensive, the computational di culties and complicated inferential and asymptotic arguments associated with many of these approaches present a considerable stumbling block in the routine application of the methodology. We discuss in detail a more recent approach which is based on counting processes and martingale properties associated with counting processes. The inferential and asymptotic properties of this method provides some notable advantages over comparable methods. The performance of the method is examined using Monte Carlo Simulation, as well as an application to a large loan portfolio of a nancial institution.

by Yuen Wah-Kong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1980. / Bibliography: leaves 37-38.

by May So-Ching Lam. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 87-88). / Abstracts in English and Chinese. / Chapter Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- The importance of Vector Time Series Analysis --- p.1 / Chapter 1.2 --- Objective --- p.3 / Chapter Chapter 2 --- Vector Autoregressive Models --- p.5 / Chapter 2.1 --- The VAR(p) models --- p.5 / Chapter 2.2 --- Least square estimation method --- p.7 / Chapter 2.3 --- VAR forecast --- p.9 / Chapter Chapter 3 --- Model Selection Criteria --- p.12 / Chapter 3.1 --- VAR order selection methods --- p.12 / Chapter 3.2 --- Hsiao's sequential method --- p.17 / Chapter 3.2.1 --- Two variables case --- p.19 / Chapter 3.2.2 --- Three variables case --- p.24 / Chapter Chapter 4 --- Illustrative Examples --- p.32 / Chapter Chapter 5 --- A Simulation Study --- p.37 / Chapter 5.1 --- Designs of experiments --- p.37 / Chapter 5.2 --- Simulation results --- p.47 / Chapter Chapter 6 --- Summary --- p.53 / Tables --- p.55 / References --- p.87

Typescript, etc. / Digitized by Kansas Correctional Industries

Photocopy of typescript. / Digitized by Kansas Correctional Industries

Digitized by Kansas Correctional Industries