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Most distribution free nonparametric methods depend on the ranks or orderings of the individual observations. This dissertation develops methods for the situation when there is only partial information about the ranks available. A random-linear-extension exact test and an empirical version of the random-linear-extension test are proposed as a new way to compare groups of data with partial orders. The basic computation procedure is to generate all possible permutations constrained by the known partial order using a randomization method similar in nature to multiple imputation. This random-linear-extension test can be simply implemented using a Gibbs Sampler to generate a random sample of complete orderings. Given a complete ordering, standard nonparametric methods, such as the Wilcoxon rank-sum test, can be applied, and the corresponding test statistics and rejection regions can be calculated. As a direct result of our new method, a single p-value is replaced by a distribution of p-values. This is related to some recent work on Fuzzy P-values, which was introduced by Geyer and Meeden in Statistical Science in 2005. A special case is to compare two groups when only two objects can be compared at a time. Three matching schemes, random matching, ordered matching and reverse matching are introduced and compared between each other. The results described in this dissertation provide some surprising insights into the statistical information in partial orderings. / Statistics

Since the mid 1980's many statisticians have studied methods for combining parametric and nonparametric esimates to improve the quality of fits in a regression problem. Notably in 1987, Einsporn and Birch proposed the Model Robust Regression estimate (MRR1) in which estimates of the parametric function, ƒ, and the nonparametric function, 𝑔, were combined in a straightforward fashion via the use of a mixing parameter, λ. This technique was studied extensively at small samples and was shown to be quite effective at modeling various unusual functions. In 1995, Mays and Birch developed the MRR2 estimate as an alternative to MRR1. This model involved first forming the parametric fit to the data, and then adding in an estimate of 𝑔 according to the lack of fit demonstrated by the error terms. Using small samples, they illustrated the superiority of MRR2 to MRR1 in most situations. In this dissertation we have developed asymptotic convergence rates for both MRR1 and MRR2 in OLS and GLS (maximum likelihood) settings. In many of these settings, it is demonstrated that the user of MRR1 or MRR2 achieves the best convergence rates available regardless of whether or not the model is properly specified. This is the "Golden Result of Model Robust Regression". It turns out that the selection of the mixing parameter is paramount in determining whether or not this result is attained. / Ph. D.

Physiological process models, productivity studies, and wildlife abundance studies all require accurate representations of tree crowns. In the past, geometric shapes or flexible mathematical equations approximating geometric shapes were used to represent crown profiles. Crown profile of loblolly pine (<I>Pinus taeda</I> L.) was described using single-regressor, nonparametric regression analysis in an effort to improve crown representations. The resulting profiles were compared to more traditional representations. Nonparametric regression may be applicable when an underlying parametric model cannot be identified. The modeler does not specify a functional form. Rather, a data-driven technique is used to determine the shape a curve. The modeler determines the amount of local curvature to be depicted in the curve. A class of local-polynomial estimators which contains the popular kernel estimator as a special case was investigated. Kernel regression appears to fit closely to the interior data points but often possesses bias problems at the boundaries of the data, a feature less exhibited by local linear or local quadratic regression. When using nonparametric regression, decisions must be made regarding polynomial order and bandwidth. Such decisions depend on the presence of local curvature, desired degree of smoothing, and, for bandwidth in particular, the minimization of some global error criterion. In the present study, a penalized PRESS criterion (PRESS*) was selected as the global error criterion. When individual- tree, crown profile data are available, the technique of nonparametric regression appears capable of capturing more of the tree to tree variation in crown shape than multiple linear regression and other published functional forms. Thus, modelers should consider the use of nonparametric regression when describing crown profiles as well as in any regression situation where traditional techniques perform unsatisfactorily or fail. / Ph. D.

Most traditional parametric and nonparametric regression methods operate under the assumption that the true function is continuous over the design space. For methods such as ordinary least squares polynomial regression and local polynomial regression the functional estimates are constrained to be continuous. Fitting a function that is not continuous with a continuous estimate will have practical scientific implications as well as important model misspecification effects. Scientifically, breaks in the continuity of the underlying mean function may correspond to specific physical phenomena that will be hidden from the researcher by a continuous regression estimate. Statistically, misspecifying a mean function as continuous when it is not will result in an increased bias in the estimate. One recently developed nonparametric regression technique that does not constrain the fit to be continuous is the jump preserving M-smooth procedure of Chu, Glad, Godtliebsen & Marron (1998),`Edge-preserving smoothers for image processing', Journal of the American Statistical Association 93(442), 526-541. Chu et al.'s (1998) M-smoother is defined in such a way that the noise about the mean function is smoothed out while jumps in the mean function are preserved. Before the jump preserving M-smoother can be used in practice the choice of the bandwidth parameters must be addressed. The jump preserving M-smoother requires two bandwidth parameters h and g. These two parameters determine the amount of noise that is smoothed out as well as the size of the jumps which are preserved. If these parameters are chosen haphazardly the resulting fit could exhibit worse bias properties than traditional regression methods which assume a continuous mean function. Currently there are no automatic bandwidth selection procedures available for the jump preserving M-smoother of Chu et al. (1998). One of the main objectives of this dissertation is to develop an automatic data driven bandwidth selection procedure for Chu et al.'s (1998) M-smoother. We actually present two bandwidth selection procedures. The first is a crude rule of thumb method and the second is a more sophistocated direct plug in method. Our bandwidth selection procedures are modeled after the methods of Chu et al. (1998) with two significant modifications which make the methods robust to possible jump points. Another objective of this dissertation is to provide a nonparametric hypothesis test, based on Chu et al.'s (1998) M-smoother, to test for a break in the continuity of an underlying regression mean function. Our proposed hypothesis test is nonparametric in the sense that the mean function away from the jump point(s) is not required to follow a specific parametric model. In addition the test does not require the user to specify the number, position, or size of the jump points in the alternative hypothesis as do many current methods. Thus the null and alternative hypotheses for our test are: H0: The mean function is continuous (i.e. no jump points) vs. HA: The mean function is not continuous (i.e. there is at least one jump point). Our testing procedure takes the form of a critical bandwidth hypothesis test. The test statistic is essentially the largest bandwidth that allows Chu et al.'s (1998) M-smoother to satisfy the null hypothesis. The significance of the test is then calculated via a bootstrap method. This test is currently in the experimental stage of its development. In this dissertation we outline the steps required to calculate the test as well as assess the power based on a small simulation study. Future work such as a faster calculation algorithm is required before the testing procedure will be practical for the general user. / Ph. D.

I. Small-Sample Power of the One-Sample Sign Test for Approximately Normal Distributions. The power function of the one-sided, one-sample sign test is studied for populations which deviate from exact normality, either by skewness, kurtosis, or both. The terms of the Edgeworth asymptotic expansion of order more than N^-3/2 are used to represent the population density. Three sets of hypotheses and alternatives, concerning the location of (1) the median, (2) the median as approximated by the mean and coefficient of skewness, and (3) the mean, are considered in an attempt to make valid comparisons between the power of the sign test and Student's t test under the same conditions. Numerical results are given for samples of size 10, significance level .05, and for several combinations of the coefficients of skewness and kurtosis. II. Power of Two-Sample Rank Teats on the Equality of Two Distribution Functions. A comparative study is made of the power of two-sample rank tests of the hypothesis that both samples are drawn from the same population. The general alternative is that the variables from one population are stochastically larger than the variables from the other. One of the alternatives considered is that the variables in the first sample are distributed as the smallest of k variates with distribution F, and the variables in the second sample are distributed as the largest of these k – H₁ : H = 1 - (1 - F)^k, G = F^k. These two alternative distributions are mutually symmetric if F is symmetrical. Formulae are presented, which are independent of F, for the evaluation of the probability under H₁ of any joint arrangement of the variables from the two samples. A theorem is proved concerning the equality of the probabilities of certain pairs of orderings under assumptions of mutually symmetric populations. The other alternative is that both samples are normally distributed with the same variance but different means, the standardized difference between the two extreme distributions in the first alternative corresponding to the difference between the means. Numerical results of power are tabulated for small sample sizes, k = 2, 3 and 4, significance levels .01, .05 and .10. The rank tests considered are the most powerful rank test, the one and two-sided Wilcoxon tests, Terry's c₁ test, the one and two-aided median tests, the Wald-Wolfowitz runs test, and two new tests called the Psi test and the Gamma test. The two-sample rank test which is locally most powerful against any alternative·expressing an arbitrary functional relationship between the two population distribution functions and an unspecified parameter θ is derived and its asymptotic properties studied. The method is applied to two specific functional alternatives, H₁* : H = (1-θ)F^k + θ[1 - (1-F)^k], G = F^k, and H₁**: H = 1 - (1-F)^1+θ, G = F^1+θ, where θ ≥ 0, which are similar to the alternative of two extreme distributions. The resulting test statistics are the Gamma test and the Psi test, respectively. The latter test is shown to have desirable small-sample properties. The asymptotic power functions of the Wilcoxon and WaldWolfowitz tests are compared for the alternative of two extreme distributions with k = 2, equal sample sizes and significance level .05. / Ph. D.

Emulation of computer experiments via surrogate models can be difficult when the number of input parameters determining the simulation grows any greater than a few dozen. In this dissertation, we explore dimension reduction in the context of computer experiments. The active subspace method is a linear dimension reduction technique which uses the gradients of a function to determine important input directions. Unfortunately, we cannot expect to always have access to the gradients of our black-box functions. We thus begin by developing an estimator for the active subspace of a function using kernel methods to indirectly estimate the gradient. We then demonstrate how to deploy the learned input directions to improve the predictive performance of local regression models by ``undoing" the active subspace. Finally, we develop notions of sensitivities which are local to certain parts of the input space, which we then use to develop a Bayesian optimization algorithm which can exploit locally important directions. / Doctor of Philosophy / Increasingly, scientists and engineers developing new understanding or products rely on computers to simulate complex phenomena. Sometimes, these computer programs are so detailed that the amount of time they take to run becomes a serious issue. Surrogate modeling is the problem of trying to predict a computer experiment's result without having to actually run it, on the basis of having observed the behavior of similar simulations. Typically, computer experiments have different settings which induce different behavior. When there are many different settings to tweak, typical surrogate modeling approaches can struggle. In this dissertation, we develop a technique for deciding which input settings, or even which combinations of input settings, we should focus our attention on when trying to predict the output of the computer experiment. We then deploy this technique both to prediction of computer experiment outputs as well as to trying to find which of the input settings yields a particular desired result.

Data sets from research in clinical practice professions often do not meet assumptions necessary for appropriate use of parametric statistics (Lezak and Gray, 1984). When assumptions underlying the use of the parametric tests are violated or cannot be documented, the power of the parametric test may be invalidated and consequently, the significance levels inaccurate (Gibbons, 1976). Much research has investigated the relative merits of parametric versus nonparametric procedures using simulation studies, but little has been done using actual data sets from a particular discipline. This study compared the application of parametric and nonparametric statistics using a body of literature in clinical occupational therapy. The most common parametric procedures in occupational therapy research literature from 1980 - 1984 were identified using methodology adapted from Goodwin and Goodwin (1985). Five small sample size data sets from published occupational therapy research articles typifying the most commonly used univariate parametric procedures were obtained, and subjected to exploratory data analyses (Tukey, 1977) in order to evaluate whether or not assumptions underlying appropriate use of the respective parametric procedures had been met. Subsequently, the nonparametric analogue test was identified and computed. Results revealed that in three of the five cases (paired t-test, one factor ANOVA and Pearson Correlation Coefficient) assumptions underlying the use of the parametric test were not met. In one case (independent t-test) the assumptions were met with a minor qualification. In only one case (simple linear regression) were assumptions clearly met. It was also found that in each of the two cases where parametric assumptions were met, no significant differences in p values between the parametric and the nonparametric tests were found. And conversely, in each of the three cases where parametric assumptions were not met, significant differences between the parametric and nonparametric results were found. These findings indicate that if cases were considered as a whole, there was a one hundred percent agreement between whether or not parametric assumptions were violated and whether or not differences were discovered regarding parametric versus nonparametric results. Other findings regarding (a) non-normality, (b) outliers, (c) multiple violation of assumptions for a given procedure, and (d) research designs employed are discussed and implications identified. Suggestions for future research are put forth. / Ph. D.