▼ Flexible parametric distribution models that can represent both skewed and symmetric distributions, namely skew symmetric distributions, can be constructed by skewing symmetric kernel densities by using weighting distributions. In this dissertation, we study a multivariate skew family that have either centrally symmetric or spherically symmetric kernel. Specifically, we define multivariate skew symmetric forms of uniform, normal, Laplace, and Logistic distributions by using the cdf's of the same distributions as weighting distributions. Matrix variate extensions of these distributions are also introduced herein. To bypass the unbounded likelihood problem related to the inference about this model, we propose an estimation procedure based on the maximum product of spacings method. This idea also leads to bounded model selection criteria that can be considered as alternatives to Akaike's and other likelihood based criteria when the unbounded likelihood may be a problem. Applications of skew symmetric distributions to data are also considered. Advisors/Committee Members: Gupta, Arjun K.

▼ Ordinal data are common in the academic area such as a student grade, A, B, C, D, orF, also ordinal data are common in other area such as customer satisfaction survey. It is straightforward to fit a regression model to reflect the relationship between the response and the predictors. Since the response in an ordinal data set is a vector, it is not clear how the traditional statistics define residuals and detect outliers because of the dimension of response. Since the introduction of latent variable, we can model the data using the latent variable and we have a new type of residual called latent residual. With the help of introduction of latent variable into the model, it is easy to define residuals and detect outliers. In practice there are usually more than one predictor in the data set and we need to decide to choose variable that should be included in the model. We look at from a frequentist's perspective and a Bayesian perspective. Also when we fit a model to a data set, we care about how well this model fit the data set, and we look from both a frequentist's perspective and a Bayesian perspective. Usually methods from a frequentist's perspective rely on the asymptotic distribution to draw a conclusion and sometime this will become a problem especially when the sample size is small, on the contrary, methods from a Bayesian perspective use simulation and thus it removes the reliance on the asymptotic distribution. Chapter 3 talks about methods for outlier detection problems and Chapter 4 talks about goodness-of-fit and model selection problems, in Chapter 5 we apply the methods from Chapter 3 and Chapter 4 to the BGSU student data set. Chapter 6 summarized the whole dissertation and possible future research interest and applications. Advisors/Committee Members: Albert, James.

▼ Characterization of the normal distribution related to k samples based on regression is given. This characterization has been transformed to a characterization based on Student's law. With the help of these characterization results, the composite hypothesis of testing the distributional assumptions in one-way regression model can be replaced by an equivalent simple hypothesis. This simple hypothesis can then be tested using any of the EDF goodness-of-fit tests. The powers of these tests are studied using Monte Carlo methods. Multivariate normal distribution, with covariance matrix of the form σ2Σ0, is characterized based on UMVU estimator of the density function. Using this result with the transformation proposed by Rincon-Gallardo, Quesenberry and O'Reilly (1979), the composite hypothesis of testing k-variate normality with covariance matrix σ2Σ0 has been transformed to an equivalent simple hypothesis. It is shown that the transformation proposed here can also be used in changing the above composite hypothesis to an equivalent simple hypothesis. These transformations are compared using Monte Carlo methods. Approximate EDF goodness of fit tests for testing the distributional assumptions in one-way regression model are studied using the Monte Carlo simulations. Advisors/Committee Members: Gupta, Arjun K.

▼ We study the long-time asymptotic growth rate of mean occupation times of certain multidimensional continuous strong Markov processes. This problem for one-dimensional diffusions has been studied by many authors, also in the context of the limiting distributions of occupation times. The existing results for linear diffusions use analytical methods. They rely on Krein's correspondence from one-dimensional theory of strings. We extend the one-dimensional results of Kasahara, Kotani and Watanabe, (1975-1982) improved by Zirbel (1997), to multidimensional diffusions which are time changed Wiener processes. The difficulties in generalizing the existing methods to higher dimensions arise from the fact that the Wiener process in higher dimensions does not revisit single points, therefore we need a new approach. The method we propose is probabilistic and it works in multiple dimensions. Our approach uses reversibility and the scaling property of the Wiener process. We find a decomposition for the mean occupation time of a reversible multidimensional diffusion in terms of a function h which reflects recurrence properties of the process and a factor depending on the function used to measure the occupation time. For time changed Wiener process having radial speed density given by a power function, we find the recurrence function h exactly. This gives us the asymptotics of mean occupation times of radially time changed d-dimensional Wiener process with speed density m(x) = c|x|δ. We prove a comparison result which allows us to determine the asymptotic behavior of the function h in some non-radial cases. The bounds on hitting times combined with the comparison result allow us to determine the mean occupation times also in some non-radial cases. Advisors/Committee Members: Zirbel, Craig L.

▼ Multiple comparison inference is simultaneous inference ona comparison of the treatment means. The focus of this research was to develop simultaneous confidence interval methods for different types of multiple comparison inference when the equality of variances can not be assumed and when prior knowledge of the ratios of variance is available. Under the usual normality and equal variance assumptions, Dunnet's (1955) method provided the simultaneous inference on the difference between each new treatment mean and the control mean, which is useful for estimating the minimum effective dose (MED) and the maximum safe dose (MSD) in does-response studies. In practice, however, homogeneity of variances is seldom satisfied. In this research, an exact method for multiple comparisons with a control was developed when prior knowledge of the ratios of variance among treatments is available but without equal variance assumption. An example was considered and a simulation study on error rate was conducted. The results indicated that Dunnet's method has inflated error rate and may lead to erroneous inference when the equal variance assumption is not satisfied. In addition, robustness of the exact method was also examined through a simulation study. Tamhane and Logan (2003) proposed a simultaneous confidence interval method for identifying the MED and MSD, while assuming equal variance. By the same motivation, a method was proposed for identifying the doses which are both effective and safe when the variances are different among treatments and the ratios of variance are known. We suggested a simulation based approach to estimate the critical values. The power of the approach was estimated using a simulation study. In addition, all-pairwise comparisons are of interest in many circumstances. Under the unbalanced design, the Tukey-Kramer method provided a set of conservative simultaneous confidence intervals for all-pairwise differences assuming equal variances and normal distribution. When the variances are different among treatments and previous knowledge of the ratios of variance is available, we proposed an approximate approach which provides simultaneous inference on all-pairwise differences. A simulation study was conducted to evaluate the error rate of the Tukey-Kramer method and the proposed method. The results showed that the error rates of the Tukey-Kramer method are excessive and much larger than the nominal level when the equal variance assumption is invalid. The error rates of the proposed approach were all within the normal level. Similarly, Hsu's (1984) multiple comparisons with the best (MCB) is a good choice when it is of interest to compare each treatment with the best of the other treatments. We extended Hsu's constrained MCB to the unequal variance case and assumed known ratios of variance in this dissertation. When there is no information on the ratios of variances and the equality of variances can not be assumed, we proposed some approximate approaches for difference types of multiple comparison procedures in the last part of this dissertation. Advisors/Committee Members: Ning, Wei.

▼ The focus of this research was to develop simultaneous confidence bounds for all contrasts of several regression lines with a constrained explanatory variable. The pioneering work of Spurrier provided a set of simultaneous confidence bounds for all contrasts of several regression lines with an unrestricted explanatory variable. However, in many applications, the explanatory variable is constrained to smaller intervals. Spurrier clearly stated in the article (JASA, 1999) that the inference problem becomes much more complicated when the explanatory variable is bounded to a given interval. In fact, Wei Liu et al. (JASA, 2004) have investigated this issue, but were unable to solve the problem. Instead, they were obliged to rely on simulation based methods which produced approximate probability points for simultaneous comparisons. In this research, a set of simultaneous confidence bounds for all contrasts of several linear regression lines was constructed for when the explanatory variable is restricted to a fixed interval. These results greatly improve those of Spurrier since restricting the explanatory variable to a smaller interval results in narrower confidence bounds. Further, since the methods of this research are exact, they are superior to the earlier work of Wei Liu et al. A significant area of this research concerned a certain statistic that plays a crucial role in constructing confidence bounds with a constrained explanatory variable, and a pivotal quantity that aids in the discovery of critical values for determining the confidence bounds. In this research, the exact distribution of the statistic was found; in fact, amazingly, it has also been shown that the statistic is independent of the pivotal quantity. These research results shed surprising new light on long standing knotty problems in biostatistics. Applications of this method to drug stability studies were examined. In situations where multiple batches of a drug product are manufactured, it is desired to pool data from different batches to obtain a single shelf-life for all batches. This research provided a new pooling method that was demonstrated to be more versatile and efficient than the existing pooling procedures. Advisors/Committee Members: Chen, John T.

▼ Let X be a random variable with piecewise continuous and bounded probability density function. We consider the sequence of fractional parts of multiples of X. We prove that this sequence of random variables is a strongly mixing sequence and hence is asymptotically independent of any other random variable. This is the basis for the main investigation, which we describe now.Let T follow the Uniform distribution on (0,1). We consider the sequences of sine and cosine of multiples of T. We show that these random variables are identically distributed, uncorrelated but dependent, non-Markovian and non-exchangeable. To understand the dependence, we investigate the sum of the first n terms of each sequence and let n go to infinity. We would like to derive the asymptotic distribution of sums of these sequences. In classical convergence results, one either considers ergodic type results, involving dividing the sum by n or looking at rarer subsequences and derive a Normal limit. In our non-classical results, we show in case of the sine sequence, that the sum converges in distribution to the Cauchy distribution without normalization and in the case of the cosine sequence, the limiting distribution of the sum is heavy-tailed and non-normal but not Cauchy. The strong dependence in the sequence is the explanation for why no normalization is needed. We discuss pointwise convergence and the Cesaro convergence of the corresponding non-random series. We derive the asymptotic distribution of sums of arithmetic subsequences of the sine and cosine sequences. We derive convergence results for sums of weighted sine and cosine sequences and finally we extend our result to derive the asymptotic distribution of sums of a finite linear combination of sine and cosine terms. We discuss some open problems and future research directions. Advisors/Committee Members: Székely, Dr. Gábor.

▼ The logistic regression model is one of the popular mathematical models for the analysis of binary data with applications in physical, biomedical, and behavioral sciences, among others. The feature of this model is to quantify the effects of several explanatory variables on one dichotomous outcome variable. Normally, the asymptotic properties of the maximum likelihood estimates in the model parameters are used for statistical inference. However, logistic regression models have serious numerical problems if zero cells occur in the contingency table. For this scenario, this dissertation proposed a new approach to investigate the asymptotic properties of maximum likelihood estimators for the logistic regression models. In this dissertation, a generalization of the hybrid logistic regression model was introduced, which was originally proposed by Chen et al. (2003). These models deal with situations in which risk factors associated with the outcome are exceedingly rare in the control group. In principle, a two-stage hybrid procedure models the risks due to the rare factors in the first stage and models the residual risks due to the other factors in the second stage using the standard logistic regression model. Another highlight of this dissertation is on the multinomial logistic regression model, which handles the categorical dependent outcome variable with more than two levels. It extended the hybrid logistic regression model to the multinomial hybrid logistic regression model when the case group of the outcome variable has mutually exclusive and exhaustive subgroups. In the last part of the dissertation, we studied the bootstrap method to estimate the variances for the parameter estimates in the logistic regression model. Advisors/Committee Members: Chen, John.

▼ With the advent of faster computers and the availability of RNA crystal structures we can now use more information to align homologous RNA sequences. We can take a crystal structure and construct a probabilistic model, based on a SCFG, of an RNA molecule. We construct objects called nodes that modularize the model into small pieces that are more manageable. Using this model we can take sequences that are similar to the sequence in the 3D crystal structure and look for the most probable way that the model could have generated the sequence. Then we can get a detailed description of how each node of the model could have generated the sequence. Using this information we can align sequences. Given a seed alignment we give a procedure to construct a 3D structural alignment quickly. In addition we show how the parameters from the model can be estimated. We also have the ability to do motif swaps using objects called alternative nodes. We have developed an algorithm to quickly search through RNA 3D structures to find motifs. This is accomplished by taking a query motif with m bases and finding the center of the heavy atoms for each base and then rotating it onto candidate motifs that have the same number of bases. Then we measure how good a fit the candidate is to the query by using a discrepancy that we define which involves the distance between bases and their relative orientations. A simple inequality allows us to quickly identify candidates whose discrepancy with the query motif will exceed a cutoff discrepancy. We use this to screen out the vast majority quickly. Advisors/Committee Members: Zirbel, Craig.

▼ One of the primary issues in mathematical finance is the ability to construct portfolios that are optimal with respect to the risk. The stock price is subject to stochastic variability so the risk an investor encounters is due to the stock prices. A commonly used measure of risk is the expected maximum loss of a stock, in other words, how much one can lose. It can be defined informally as the largest drop from a stock peak to a stock nadir. Over a certain fixed length of time, a reasonably low expected maximum loss is as crucial to the success of any fund asa high maximum gain or maximum profit. The correlation coefficient of the maximum loss and the maximum gain indicates the relation between the gain and the risk using measures which are functions of the Sharpe ratio. The price of one share of the risky asset, the stock, is modeled by geometric Brownian motion. By taking the log of geometric Brownian motion, Brownian motion can be used as basis of the calculations related to the geometric Brownian motion. In this dissertation work, we present analytical results related to the joint distribution of the maximum loss and maximum gain of a Brownian motion and the correlation of them, and detailed explanation of this theoretical result which requires a review of standard but difficult literature. We have given an analytical expression for the correlation of the supremum and the infimum of standard Brownian motion up to an independent exponential time, we have shown convexity of the maximum gain and the maximum loss, and we have calculated some bounds for the expected values of maximum gain and maximum loss. We also search for a relation between the Sharpe ratio and the correlation coefficient for Brownian motion with drift and geometric Brownian motion with drift. Using the scaling property, we have shown that the correlation coefficient does not depend on the diffusion coefficient for Brownian motion. And finally, using real-life data, we have presented the correlation of maximum gain and maximum loss and the correlation of the supremum and the infimum of stock prices. Advisors/Committee Members: Szekely, Gabor.

▼ Maximal correlation has several desirable properties as a measure of dependence, including the fact that it vanishes if and only if the variables are independent. Except for a few special cases, it is hard to evaluate maximal correlation explicitly. In this dissertation, we focus on two-dimensional contingency tables and discuss a procedure for estimating maximal correlation, which we use for constructing a test of independence. For large samples, we present the asymptotic null distribution of the test statistic. For small samples or tables with sparseness, we use exact inferential methods, where we employ maximal correlation as the ordering criterion. We compare the maximal correlation test with other tests of independence by Monte Carlo simulations. When the underlying continuous variables are dependent but uncorrelated, we point out some cases for which the new test is more powerful. Advisors/Committee Members: Szekely, Gabor J.