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.

Doctor of Philosophy / Department of Statistics / Paul I. Nelson / The null hypothesis of equal distributions, H0 : F1[equals]F2[equals]...[equals]FK , is commonly used to compare two or more treatments based on data consisting of independent random samples. Using this approach, evidence of a difference among the treatments may be reported even though from a practical standpoint their effects are indistinguishable, a longstanding problem in hypothesis testing. The concept of effect size is widely used in the social sciences to deal with this issue by computing a unit-free estimate of the magnitude of the departure from H0 in terms of a change in location. I extend this approach by replacing H0 with hypotheses H0* that state that the distributions {Fi} are possibly different in location and or scale, but close, so that rejection provides evidence that at least one treatment has an important practical effect. Assessing statistical significance under H0* is difficult and typically requires inference in the presence of nuisance parameters. I will use frequentist, Bayesian and Fiducial modes of inference to obtain approximate tests and carry out simulation studies of their behavior in terms of size and power. In some cases a bootstrap will be employed. I will focus on tests based on independent random samples arising from K[greater than and equals]3 normal distributions not required to have the same variances to generalize the K[equals]2 sample parameter P(X1>X2) and non-centrality type parameters that arise in testing for the equality of means.