Doctor of Philosophy / Department of Statistics / Haiyan Wang / Advances in technology easily collect a large amount of data in scientific research such as agricultural screening and micro-array experiments. We are particularly interested in data from one-way and crossed two-way designs that have a large number of treatment combinations but small replications with heteroscedastic variances. In this framework, several test statistics have been proposed in the literature. Even though the form of these proposed test statistics may be different, they all use limiting normal or chi-square distribution to conduct their tests. Such approximation approaches the true distribution very slowly when the sample size ni is small while the number of levels of treatments a gets large. A strategy to obtain better accuracy in the classical large sample size setting is to use the bootstrap procedure with studentized statistic. Unfortunately, the available bootstrap method fails when the number of treatment level combinations is large while the number of replications is small. The Fisher and Hall (1990) asymptotic pivotal statistic under large sample size setting is no longer pivotal under small sample size setting with large number of treatment levels. In the first part of this dissertation, we start with describing suitable bootstrap statistics and procedures for hypothesis tests in one- and two-way ANOVA with a large number of levels and small sample sizes. We prove that the theoretical type I error-rate of Akritas and Papadatos (2004) and Wang and Akritas (2006) test statistics and their corresponding bootstrap versions have accuracy of order O(1/√a). We then modify their statistics to obtain asymptotically pivotal statistics in our current framework. We prove that the theoretical type I error-rate of the bootstrap version of the pivotal statistics is accurate up to order O(1/√a). In the second part of the dissertation, we propose a new test statistic in one-way ANOVA which is asymptotically pivotal in the current setting. We improve the accuracy of approximation of the distribution of the test statistic by deriving asymptotic expansion of the statistic under the current framework and define a new test rejection region through Cornish-Fisher expansion of quantiles. The type I error-rate of the new test has a faster convergence rate and is accurate up to order O(1/a). Simulation studies show that our tests performs better in terms of type I error-rate but comparable power with that of Akritas and Papadatos (2004) in the large a small ni setting. The connection between our asymptotic expansions and bootstrap distribution in the large a small ni setting is discussed. Our proposed test based on asymptotic expansion and Cornish-Fisher expansion of quantiles have both the advantage of higher accuracy and computational efficiency due to no resampling is needed.

D. Phil. / There have been two rather distinct approaches to the analysis of time series: the time domain approach and frequency domain approach. The former is exemplified by the work of Quenouille (1957), Durbin (1960), Box and Jenkins (1970) and Ljung and Box (1979). The principal names associated with the development of the latter approach are Slutsky (1929, 1934), Wiener (1930, 1949), Whittle (1953), Grenander (1951), Bartlett (1948, 1966) and Grenander and Rosenblatt (1957). The difference between these two methods is discussed in Wold (1963). In this thesis, we are concerned with a frequency domain approach. Consider a model of the "signal plus noise" form yt = g (2t — 1 2n ) + 77t t= 1,2,—. ,n (1.1) where g is a function on (0, 1) and Ti t is a white noise process. Our interest is primarily in testing the hypothesis that g is constant, that is, that it does not change over time. There is a vast literature related to this problem in the special case where g is a step function. In that case (1.1) specifies an abrupt change model. Such abrupt change models are treated extensively by Csorgo and Horvath (1997), where an exhaustive bibliography can also be found. The methods associated with the traditional abrupt change models are, almost without exception, time domain methods. The abrupt change model is in many respects too restrictive since it confines attention to signals g that are simple step functions. In practical applications the need has arisen for tests of constancy of the mean against a less precisely specified alternative. For instance, in the study of variables stars in astronomy (Lombard (1998a)) the appropriate alternative says something like: "g is non-constant but slowly varying and of unspecified functional form". To accommodate such alternatives within a time domain approach seems to very difficult, if at all possible. They can, however, be accommodated within a frequency domain approach quite easily, as shown by, for example, Lombard (1998a and 1998b). Tests of the constancy of g using the frequency domain characteristics of the observations have been investigated by a number of authors. Lombard (1988) proposed a test based on the maximum of squared Fourier cosine coefficients at the lowest frequency oscillations. Eubank and Hart (1992) proposed a test which is based on the maximum the averages of Fourier cosine coefficients. The essential idea underlying these tests is that regular variation in the time domain manifests itself entirely at low frequencies in the frequency domain. Consequently, when g is "high frequency" , that is consists entirely of oscillations at high frequencies, the tests of Lombard (1988) and of Eubank and Hart (1992) lose most of their power. The fundamental tool used in frequency domain analysis is the periodogram; see Chapter 2 below for the definition and basic properties of the latter. A new class of tests was suggested by Lombard (1998b) based on the weighted averages of periodogram ordinates. When 7i t in model (1.1) are i.i.d. random variables with zero mean and variance cr-2 , one form of the test statistic is T1r, = Etvk fiy (A0/0-2 - (1.2) k=1 where wk is a sequence of constants that decrease as k increases and m = [i]. The rationale for such tests is discussed in detail in Lombard (1998a and 1998b). The greater part of the present Thesis consists of an investigation of the asymptotic null distributions, and power, of such tests. It is also shown that such tests can be applied directly to other, seemingly unrelated problems. Three instances of the latter type of application that are investigated in detail are (i) frequency domain competitors of Bartlett's test for white noise, (ii) frequency domain-based tests of goodness-of-fit and (iii) frequency domain-based tests of heteroscedasticity in linear or non-linear regression. regression. The application of frequency domain methods to these problems are, to the best of our knowledge, new. Until now, most research has been restricted to the case where m in (1.1) are i.i.d. random variables. As far as the correlated data are concerned, the changepoint problem was investigated by, for instance, Picard (1985), Lombard and Hart (1994) and Bai (1994) using time domain methods. Kim and Hart (1998) proposed two test statistics derived from frequency domain considerations and that are modeled along the lines of the statistics considered by Eubank and Hart (1992) in the white noise case. An analogue of the type of test statistic given in (1.2) for use with correlated data was proposed, and used, by Lombard (1998a). The latter author does not, however, provide statements or proofs regarding the asymptotic properties of the proposed test.

D.Phil. / We consider two problems relating to location-scale families of distributions. Firstly, we consider methods of parameter estimation when two samples come from the same type of distribution, but possibly differ in terms of location and spread. Although there are methods of estimation that are asymptotically efficient, our interest is in fi…nding methods which also have good small-sample properties. Secondly, we consider tests for the hypothesis that two samples come from the same location-scale family. Both these problems are addressed using methods based on empirical distribution functions and empirical characteristic functions.

Biology is changing and becoming more quantitative. Research is creating new challenges that need to be addressed in education as well. New educational initiatives focus on combining laboratory procedures with mathematical skills, yet it seems that most curricula center on a single relationship between scientific knowledge and scientific method: that of the validity of knowledge claims, judged in terms of their consistency with data. Collecting data and obtaining results (however quantitative) are commonly part of science, but are not science itself. We envision that the operative use of the complete scientific method will play a critical role in providing the necessary underpinning for the integration of math and biology at various professional levels.

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The one-way analysis of variance (ANOVA) is mainly based on several assumptions and can be used to compare the means of two or more independent groups of a factor. To relax the normality assumption in one-way ANOVA, recent studies have considered exponential distortion or tilt of a reference distribution. The reason for the exponential distortion was not investigated before; thus the main objective of the study is to closely examine the reason behind it. In doing so, a new generalized semi-parametric approach for one-way ANOVA is introduced. The proposed method not only compares the means but also variances of any type of distributions. Simulation studies show that proposed method has favorable performance than classical ANOVA. The method is demonstrated on meteorological radar data and credit limit data. The asymptotic distribution of the proposed estimator was determined in order to test the hypothesis for equality of one sample multivariate distributions. The power comparison of one sample multivariate distributions reveals that there is a significant power improvement in the proposed chi-square test compared to the Hotelling's T-Square test for non normal distributions. A bootstrap paradigm is incorporated for testing equidistributions of multiple samples. As far as power comparison simulations for multiple large samples are considered, the proposed test outperforms other existing parametric, nonparametric and semi-parametric approaches for non normal distributions.

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Ph. D.