In many applications, the quality of process outputs is described by more than one characteristic variable. These quality variables usually follow a multivariate normal (MN) distribution. This dissertation discusses the monitoring of the mean vector and the covariance matrix of MN processes. The first part of this dissertation develops a statistical process control (SPC) chart based on a generalized likelihood ratio (GLR) statistic to monitor the mean vector. The performance of the GLR chart is compared to the performance of the Hotelling Χ² chart, the multivariate exponentially weighted moving average (MEWMA) chart, and a multi-MEWMA combination. Results show that the Hotelling Χ² chart and the MEWMA chart are only effective for a small range of shift sizes in the mean vector, while the GLR chart and some carefully designed multi-MEWMA combinations can give similarly better overall performance in detecting a wide range of shift magnitudes. Unlike most of these other options, the GLR chart does not require specification of tuning parameter values by the user. The GLR chart also has the advantage in process diagnostics: at the time of a signal, estimates of change-point and out-of-control mean vector are immediately available to the user. All these advantages of the GLR chart make it a favorable option for practitioners. For the design of the GLR chart, a series of easy to use equations are provided to users for calculating the control limit to achieve the desired in-control performance. The use of this GLR chart with a variable sampling interval (VSI) scheme has also been evaluated and discussed. The rest of the dissertation considers the problem of monitoring the covariance matrix. Three GLR charts with different covariance matrix estimators have been discussed. Results show that the GLR chart with a multivariate exponentially weighted moving covariance (MEWMC) matrix estimator is slightly better than the existing method for detecting any general changes in the covariance matrix, and the GLR chart with a constrained maximum likelihood estimator (CMLE) gives much better overall performance for detecting a wide range of shift sizes than the best available options for detecting only variance increases. / Ph. D.

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<p> Acid rain continues to be a major environmental problem. Canada has been monitoring indicators of acid rain in various ecosystems since the 1970s. This project focuses on the analysis of a selected subset of data generated by the Turkey Lakes Watershed (TLW) monitoring program from 1980 to 1997. TLW consists of a series of connected lakes where 6 monitoring stations are strategically located to measure the input from an upper stream lake into a down stream lake. Segment regression models with AR(1) errors and unknown point of change are used to summarize the data. Relative likelihood based methods are applied to estimate the point of change. For pH, all the regression parameters except autocorrelation have been found to change significantly between the model segments. This was not the case for SO4 2- where a single model was found to be adequate. In addition pH has been found to have a moderate increasing trend and pronounced seasonality while SO4 2- showed a dramatic decreasing trend but little seasonality. Multivariate dimension reduction methods are used to provide an overall graphical summary of the changes in TLW water system. We also report the result of applying segment regression for the analysis of first two principal components in selected stations. The results show that the efforts of the Canadian and US governments to reduce the emission of SO2 have been successful in controlling the acid rain problem in Eastern Canada. The project ends with suggestions for various extensions of the present work.</p> / Thesis / Master of Science (MSc)

The focus of this work is the development of testing procedures for the existence of change-points in parametric hazard models of various types. Hazard functions and the related survival functions are common units of analysis for survival and reliability modeling. We develop a methodology to test for the alternative of a two-piece hazard against a simpler one-piece hazard. The location of the change is unknown and the tests are irregular due to the presence of the change-point only under the alternative hypothesis. Our approach is to consider the profile log-likelihood ratio test statistic as a process with respect to the unknown change-point. We then derive its limiting process and find the supremum distribution of the limiting process to obtain critical values for the test statistic. We first reexamine existing work based on Taylor Series expansions for abrupt changes in exponential data. We generalize these results to include Weibull data with known shape parameter. We then develop new tests for two-piece continuous hazard functions using local asymptotic normality (LAN). Finally we generalize our earlier results for abrupt changes to include covariate information using the LAN techniques. While we focus on the cases of no censoring, simple right censoring, and censoring generated by staggered-entry; our derivations reveal that our framework should apply to much broader censoring scenarios. / Ph. D.

Title: Detection of change in some special regression models Author: Bc. Petra Exnarová Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Marie Hušková, DrSc. Abstract: Presented thesis deals with testing of change in three special cases of change-point analysis. First of them is case of continuous change in linear regression (so-called broken-line model), the other two are related to change in parameters of discrete value distributions - simple case of Bernoulli distributed variables is studied first and then the approach is generalized for case of Multi- nomial distribution. Both situations of known and unknown change point are described for all three cases. Beside approximation by using limit theorems, the bootstrap method and permutation test are described for all studied cases as well. The comparison of critical values gained by different approaches for the particular tests and small power analysis is done using simulations. Keywords: change-point analysis, broken-line model, discrete distribution, boot- strap, permutation test 1

We provide a flexible means of estimating time-varying parameter models in a Bayesian framework. By specifying the state innovations to be characterized trough a threshold process that is driven by the absolute size of parameter changes, our model detects at each point in time whether a given regression coefficient is con stant or time-varying. Moreover, our framework accounts for model uncertainty in a data-based fashion through Bayesian shrinkage priors on the initial values of the states. In a simulation, we show that our model reliably identifies regime shifts in cases where the data generating processes display high, moderate, and low numbers of movements in the regression parameters. Finally, we illustrate the merits of our approach by means of two applications. In the first application we forecast the US equity premium and in the second application we investigate the macroeconomic effects of a US monetary policy shock. / Series: Research Report Series / Department of Statistics and Mathematics

Data from the 1995 National Health Interview Survey (NHIS) indicate that, due to chronic conditions, the onset of activity limitation typically occurs between age 40-70 years (i.e., the proportion of young adults with activity limitation is small and roughly constant with age and then it starts to change, roughly increasing). We use a Bayesian hierarchical model to detect the change point of a positive activity limitation status (ALS) across twelve domains based on race, gender, and education. We have two types of data: weighted and unweighted. We obtain weighted binomial counts using a regression analysis with the sample weights. Given the proportion of individuals in the population with positive ALS, we assume that the number of individuals with positive ALS at each age group has a binomial probability mass function. The proportions across age are different, and have the same beta distribution up to the change point (unknown), and the proportions after the change point have a different beta distribution. We consider two different analyses. The first considers each domain individually in its own model and the second considers the twelve domains simultaneously in a single model to“borrow strength" as in small area estimation. It is reasonable to assume that each domain has its own onset.In the first analysis, we use the Gibbs sampler to fit the model, and a computation of the marginal likelihoods, using an output analysis from the Gibbs sampler, provides the posterior distribution of the change point. We note that a reversible jump sampler fails in this analysis because it tends to get stuck either age 40 or age 70. In the second analysis, we use the Gibbs sampler to fit only the joint posterior distribution of the twelve change points. This is a difficult problem because the joint density requires the numerical computation of a triple integral at each iteration. The other parameters of the process are obtained using data augmentation by a Metropolis sampler and a Rao-Blackwellization. We found that overall the age of onset is about 50 to 60 years.

在存活分析中，含有轉折點的指數分配（又稱二段式指數分配）的模式,常被拿來研究某些疾病的復發率，以決定其治療方式是否有效。然而在文獻上，對這一個模式的探討大都局限在單一母體上，其問題不外乎有兩個：一是檢定此一轉折點是否存在；二是估計此一轉折點。 　　本文將此一問題擴充，從一個母體提昇至兩個母體，比較兩個母體是否具有相同的轉折點、起始危險率或轉換率。基本上，我們使用了貝氏方法和古典方法來分析。 　　我們利用貝氏方法，推導出兩個母體在不同的已知條件下，各母數比值或差值的事後分配。但是他們的形式幾乎都很複雜，使得欲做進一步的分析，困難重重。因此，我們引進了 Gibbs 抽樣法，利用各完全條件事後分配，萃取出各邊際事後分配，以供推論之用。 　　而在古典分析中，我們係採用概似比值檢定法。而最大的問題在於轉折點未知時，我們不知其對數概似比的分配為何。我們除了介紹兩個文獻中估計轉折點的方法，我們更利用了自助法 (bootstrap) 來估計其對數概似比的分配，以供檢定之用。 　　對於這樣兩母體的比較，在醫學上、工業上甚具意義。本文不僅推導出其供比較用的統計架構，更提供了具體而實用的抽樣方法， 對這問題的分析，頗具貢獻。