Doctor of Philosophy / Department of Statistics / Weixing Song / In this dissertation, goodness-of-fit tests are proposed for checking the adequacy of parametric distributional forms of the regression error density functions and the error-prone predictor density function in measurement error models, when replications of the surrogates of the latent variables are available. In the first project, we propose goodness-of-fit tests on the density function of the regression error in the errors-in-variables model. Instead of assuming that the distribution of the measurement error is known as is done in most relevant literature, we assume that replications of the surrogates of the latent variables are available. The test statistic is based upon a weighted integrated squared distance between a nonparametric estimate and a semi-parametric estimate of the density functions of certain residuals. Under the null hypothesis, the test statistic is shown to be asymptotically normal. Consistency and local power results of the proposed test under fixed alternatives and local alternatives are also established. Finite sample performance of the proposed test is evaluated via simulation studies. A real data example is also included to demonstrate the application of the proposed test. In the second project, we propose a class of goodness-of-fit tests for checking the parametric distributional forms of the error-prone random variables in the classic additive measurement error models. We also assume that replications of the surrogates of the error-prone variables are available. The test statistic is based upon a weighted integrated squared distance between a non-parametric estimator and a semi-parametric estimator of the density functions of the averaged surrogate data. Under the null hypothesis, the minimum distance estimator of the distribution parameters and the test statistics are shown to be asymptotically normal. Consistency and local power of the proposed tests under fixed alternatives and local alternatives are also established. Finite sample performance of the proposed tests is evaluated via simulation studies.

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.

Testing uniformity in the univariate case has been studied by many researchers. Many papers have been published on this issue, whereas the multi-dimensional uniformity test seems to have received less attention in the literature. A new test statistic for the multi-dimensional uniformity is proposed in this thesis. The proposed test statistic can be used to test whether an underlying multivariate probability distribution differs from a multi-dimensional uniform distribution. Some important properties of the proposed test statistic are discussed. As a special case, the bivariate test statistic is discussed in detail and the critical values of test statistic are obtained. By performing Monte Carlo simulation, the power of the new test is compared with the Distance to Boundary test, which was a recently proposed statistical test for multi-dimensional uniformity by Berrendero, Cuevas and Vazquez-Grande (2006). It has been shown that the test proposed in this thesis is more powerful than the Distance to Boundary test in some cases.

This thesis investigates Goodness-of-Fit tests for parametric regression models. With the help of sufficient dimension reduction techniques, we develop adaptive-to-model tests using projection in both the fixed dimension settings and the diverging dimension settings. The first part of the thesis develops a globally smoothing test in the fixed dimension settings for a parametric single index model. When the dimension p of covariates is larger than 1, existing empirical process-based tests either have non-tractable limiting null distributions or are not omnibus. To attack this problem, we propose a projected adaptive-to-model approach. If the null hypothesis is a parametric single index model, our method can fully utilize the dimension reduction structure under the null as if the regressors were one-dimensional. Then a martingale transformation proposed by Stute, Thies, and Zhu (1998) leads our test to be asymptotically distribution-free. Moreover, our test can automatically adapt to the underlying alternative models such that it can be omnibus and thus detect all alternative models departing from the null at the fastest possible convergence rate in hypothesis testing. A comparative simulation is conducted to check the performance of our test. We also apply our test to a self-noise mechanisms data set for illustration. The second part of the thesis proposes a globally smoothing test for parametric single-index models in the diverging dimension settings. In high dimensional data analysis, the dimension p of covariates is often large even though it may be still small compared with the sample size n. Thus we should regard p as a diverging number as n goes to infinity. With this in mind, we develop an adaptive-to-model empirical process as the basis of our test statistic, when the dimension p of covariates diverges to infinity as the sample size n tends to infinity. We also show that the martingale transformation proposed by Stute, Thies, and Zhu (1998) still work in the diverging dimension settings. The limiting distributions of the adaptive-to-model empirical process under both the null and the alternative are discussed in this new situation. Simulation examples are conducted to show the performance of this test when p grows with the sample size n. The last Chapter of the thesis considers the same problem as in the second part. Bierens's (1982) first constructed tests based on projection pursuit techniques and obtained an integrated conditional moment (ICM) test. We notice that Bierens's (1982) test performs very badly for large p, although it may be viewed as a globally smoothing test. With the help of sufficient dimension techniques, we propose an adaptive-to-model integrated conditional moment test for regression models in the diverging dimension setting. We also give the asymptotic properties of the new tests under both the null and alternative hypotheses in this new situation. When p grows with the sample size n, simulation studies show that our new tests perform much better than Bierens's (1982) original test.

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This dissertation has three parts. The first part proposes a two-stage estimation procedure for a copula-based model with semi-competing risks data, where the nonterminal event is subject to dependent censoring by the terminal event. Under a copula-based model, the marginal survival functions of individual event times are specified by semiparametric transformation models, and a parametric copula function specifies the between-event dependence. The parameters associated with the marginal of the terminal event are first estimated, and the marginal parameters for the non-terminal event time and the copula parameter are second estimated via maximizing a pseudo-likelihood function based on the joint distribution of the bivariate event times. We derived the asymptotic properties of the proposed estimator and provided an analytic variance estimator for inference. We showed that our approach leads to consistent estimates with less computational cost and more robustness compared to the one-stage procedure developed by Chen (2012). In addition, our approach demonstrates more desirable finite-sample performances over another existing two-stage estimation method proposed by Zhu et al. (2021). The second part develops a goodness-of-fit t est f or t he copula specification under semi-parametric copula models with semi-competing risks data. We constructed an information ratio (IR) statistic by comparing consistent estimates of the two information matrices, the sensitivity matrix and the variability matrix. The information matrices are derived from the log-likelihood function, which is a function of the marginal distribution of the terminal event time, the marginal distribution of the time to the first event, and the copula parameter. We established the asymptotic distribution of the IR statistic and examined the finite-sample performance of the IR test via a simulation study. The third part develops a class of models to characterize the effects of factors that vary with the age at baseline and the age at the event. This project is motivated by the Childhood Cancer Survivor Study. The age-specific effects of the covariates are estimated via an inverse probability weighted kernel smoothing method. We conducted simulation studies to evaluate the performance of the proposed estimator.

Master of Science / Department of Statistics / Suzanne Dubnicka / To help with identifying distributions to effectively model univariate continuous data, the R function diagnostic is proposed. The function will aid in determining reasonable candidate distributions that the data may have come from. It uses a combination of the Pearson goodness of fit statistic, Anderson-Darling statistic, Lin’s concordance correlation between the theoretical quantiles and observed quantiles, and the maximum difference between the theoretical quantiles and the observed quantiles. The function generates reasonable candidate distributions, QQ plots, and histograms with superimposed density curves. When a simulation study was done, the function worked adequately; however, it was also found that many of the distributions look very similar if the parameters are chosen carefully. The function was then used to attempt to decipher which distribution could be used to model weekly grocery expenditures of a family household.