In this dissertation we develop a theory which offers a unified approach to the problem of obtaining stochastic versions of deterministic rearrangement inequalities. / To develop the theory we first define two new classes of functions and establish preservation properties of these functions under various statistical and mathematical operations. / Next we introduce the notion of stochastically similarly arranged (SSA) pairs of random vectors. We prove that if the random vectors (X,Y) are SSA and the function f from R('n) x R('n) into R('n) is monotone with respect to a certain partial ordering on R('n) x R('n) then for every permutation (pi) the stochastic inequalities / (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) / hold. This result yields a unified way of obtaining stochastic versions of rearrangement inequalities. / We then show that many multivariate densities of interest in statistical practice govern pairs of random vectors which are SSA. / Next we show that under certain statistical operations on pairs of SSA random vectors the property of being SSA is preserved. For example, we show that the rank order of SSA random variables is SSA. We also show that the SSA property is preserved under certain contamination models. / Finally, we show how the results we obtain can be applied to problems in hypothesis testing. / Source: Dissertation Abstracts International, Volume: 42-10, Section: B, page: 4112. / Thesis (Ph.D.)--The Florida State University, 1981.

Source: Dissertation Abstracts International, Volume: 40-09, Section: B, page: 4372. / Thesis (Ph.D.)--The Florida State University, 1979.

Source: Dissertation Abstracts International, Volume: 41-07, Section: B, page: 2669. / Thesis (Ph.D.)--The Florida State University, 1980.

Three of the basic questions of Statistics may be stated as follows: (A) Which portion of the data X is actually informative about the parameter of interest (theta)? (B) How can all the relevant information about (theta) provided by the data X be extracted? (C) What kind of information about (theta) do the data X possess? / The perspective of this dissertation is that of a Bayesian. / Chapter I is essentially concerned with question A. The theory of conditional independence is explained and the relations between ancillarity, sufficiency, and statistical independence are discussed in depth. Some related concepts like specific sufficiency, bounded completeness, and splitting sets are also studied in some details. The language of conditional independence is used in the remaining Chapters. / Chapter II deals with question B for the particular problem of analysing categorical data with missing entries. It is demonstrated how a suitably chosen prior for the frequency parameters can streamline the analysis in the presence of missing entries due to non-response or other causes. The two cases where the data follow the Multinomial or the Multivariate Hypergeometric model are treated separately. In the first case it is adequate to restrict the prior (for the cell probabilities) to the class of Dirichlet distributions. In the Hypergeometric case it is convenient to select a prior (for the cell population frequencies) from the class of Dirichlet-Multinomial (DM) distributions. The DM distributions are studied in detail. / Chapter III is directly related to question C. Conditions on the likelihood function and on the prior distribution are presented in order to assess the effect of the sample on the posterior distribution. More specifically, it is shown that under certain conditions, the larger the observations obtained, the larger (stochastically in terms of the posterior distribution) is the appropriate parameter. / Finally, Chapter IV deals with the characterization of distributions in terms of Blackwell comparison of experiments. It is shown that a result (for the Hypergeometric model) obtained in Chapter II is actually a consequence of a property of complete families of distributions. / Source: Dissertation Abstracts International, Volume: 41-11, Section: B, page: 4175. / Thesis (Ph.D.)--The Florida State University, 1980.

Consider a Markov chain x(t), t = 0, 1, 2, ..., with a finite state space, N = {1, 2, ..., n}, transition probability matrix P = (p(,ij)) i, j (epsilon) N, and an initial probability vector V = (v(,i)) i (epsilon) N. For m (LESSTHEQ) n let A = {A(,1), A(,2), ..., A(,m)} be a partition on the set N. Define the process / (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) / The new process y(t), called a function of Markov chain, need not be Markov. If y(t) is again Markov, whatever the initial probability vector of x(t), x(t) is said to be lumped to y(t) with respect to the partition A. If y(t) is again Markov for only certain initial probability vectors of x(t), x(t) is said to be weakly lumped to y(t) with respect to the partition A. / Conditions under which x(t) can be lumped or weakly lumped to y(t) with respect to A, are introduced. Relationships between the two processes x(t) and y(t) and the properties of the new process y(t) are discussed. / Criteria are developed to determine whether a given Markov chain can be weakly lumped with respect to a given partition in terms of an analysis of systems of linear equations. Necessary and sufficient conditions on the transition probability matrix of a Markov chain, a partition, A, on N and a subset S of probability vectors for weak lumpability to occur are given in terms of the solution classes to these systems of linear equations. Finally, given that weak lumping occurs, the class S of all initial probability vectors which allow weak lumping is determined as is the transition probability matrix of the lumped process, y(t). / Lumpability and weak lumpability are also studied for Markov chains which are not irreducible. This involves a study of the interplay between two partitions of the state space N, the partition C, induced by the closed sets of states of the Markov chain and the partition A, with respect to which lumpability is to be considered. Under the assumptions that lumpability occurs the relationships which must exist between sets of the two partitions A and C are obtained in detail. It is found, for example that if neither partition is a refinement of the other and (A,C) form an irreducible pair of partitions over N then for each A (epsilon) A and C (epsilon) C, A (INTERSECT) C (NOT=) (phi). Further conditions which the transition probability matrix P must satisfy if lumpability is to hold are obtained as are relationships which must exist between P and P*. / Suppose a process y(t) is known to arise as a result of a weak lumping or lumping from some unknown Markov chain x(t). Let (chi)(t) be the class of all Markov chains x(t) with n states which yield this weak lumping or lumping. The problem of characterizing this class and a class S of initial probability vectors which allow this lumping is considered. A complete solution is given when n = 3 and m = 2. / The importance of lumpability in application is discussed. / Source: Dissertation Abstracts International, Volume: 41-11, Section: B, page: 4172. / Thesis (Ph.D.)--The Florida State University, 1980.

This dissertation is a contribution to the use of inequalities in reliability theory. Specifically, we study three partial orderings, develop some useful properties of these orderings, and apply them to obtain several applications in reliability. / The first partial ordering is the notion of convex-ordering among life distributions. This is in the spirit of Hardy, Littlewood, and Polya (1952) who introduced the concept of relative convexity. Many parametric families of distribution functions encountered in reliability theory are convex-ordered. Different coherent structures can also be compared with respect to this partial ordering. / The second partial ordering is the ordering of majorization among integrable functions. This ordering is a generalization of the majorization ordering of Hardy, Littlewood, and Polya (1952) for vectors in n-dimensional Euclidean spaces. The concept of majorization among vectors plays a fundamental role in establishing various inequalities. These inequalities can be recast as statements that certain functions are increasing with respect to the ordering of majorization. Such functions are called Schur-convex functions. An important result in the theory of majorization is the Schur-Ostrowski Theorem, which characterizes Schur-convex functions. A functional defined on the space of integrable functions is said to be Schur-convex if it is increasing with respect to the ordering of majorization. We obtain an analogue of the Schur-Ostrowski theorem which characterizes Schur-convex functionals in terms of their Gateaux differentials. / The third partial ordering is the ordering of unrestricted majorization among integrable functions. This partial ordering is similar to majorization but does not involve the use of decreasing rearrangements. We establish another analogue of the Schur-Ostrowski Theorem for functionals increasing with respect to the partial ordering of unrestricted majorization. / Source: Dissertation Abstracts International, Volume: 46-03, Section: B, page: 0891. / Thesis (Ph.D.)--The Florida State University, 1985.

The Kaplan-Meier estimator of a survival function is well-known to be asymptotically efficient when cause of failure (censored or non-censored) is always observed. We consider the problem of finding an estimator when the failure indicators are missing completely at random. Under this assumption, it is known that the method of nonparametric maximum likelihood fails to work in this problem. We introduce a new estimator that is a smooth functional of the Nelson-Aalen estimators of certain cumulative transition intensities. The asymptotic distribution of the estimator is derived using the functional delta method. Simulation studies reveal that this estimator competes well with the existing estimators. The idea is extended to the Cox model, and estimators are introduced for the regression parameter and the cumulative baseline hazard function. / Source: Dissertation Abstracts International, Volume: 57-01, Section: B, page: 0441. / Major Professor: Ian W. McKeague. / Thesis (Ph.D.)--The Florida State University, 1995.

A function $f(\underline{x})$ which increases each time we transpose an out of order pair of coordinates, $x\sb{j} > x\sb{k}$ for some $j x\sb{k}$ by transposing the two x coordinates. The theory of AI functions is tailor made for ranking and selection problems, in which case we assume that the density $f(\underline{\theta}$,$\underline{x})$ of observations with respective parameters $\theta\sb1, \..., \theta\sb{n}$ is AI, and the goal is to determine the largest or smallest parameters. / In this dissertation we present new applications of AI functions in such areas as biology and reliability, and we generalize the notion of AI functions. We consider multivector extensions, some with and one without respect to parameter vectors, and we connect these. Another generalization (TEGO) is motivated by the connection between total positivity (TP) and AI. TEGO results are shown to imply AI and TP results. We also define and develop a partial ordering on densities of rank vectors. The theory, which involves finding the extreme points of the convex set of AI rank densities, is then used to establish some power results of rank tests. / Source: Dissertation Abstracts International, Volume: 50-08, Section: B, page: 3563. / Major Professor: Fred Leysieffer. / Thesis (Ph.D.)--The Florida State University, 1989.

Under the imperfect repair model of Brown and Proschan (1983), a failed item is replaced by a new item (perfect repair) with probability p, and with probability 1 $-$ p, a minimal repair is performed; that is, the failed item is replaced by a working item of the same age. This procedure is repeated at each subsequent failure. Block, Borges, and Savits (1985) extend this model by allowing p to be a function of the age of the item. In both of these models, imperfect repairs are thus assumed to be minimal. Whitaker and Samaniego (1989) propose an estimator for the life distribution, F, of a new item, when either of these processes is observed until the time of the $m\sp{\rm th}$ perfect repair. / In this dissertation, we extend to the case of possibly discontinuous F, a result of Block, Borges, and Savits identifying the distribution of the waiting time until the first perfect repair. We then use a martingale approach to rederive and extend the weak convergence results of Whitaker and Samaniego. These results are used to derive asymptotic confidence bands for F, and an extension of the Wilcoxon two-sample test for data collected under these models. Finally, we propose a test of the minimal repair assumption, and the limiting distribution of the proposed test statistic is derived. / Source: Dissertation Abstracts International, Volume: 51-02, Section: B, page: 0831. / Major Professors: Myles Hollander; Jayaram Sethuraman. / Thesis (Ph.D.)--The Florida State University, 1989.

Several procedures are available for analyzing cross-classified data under the Poisson model. When data suggest the presence of "non-Poisson" variation an alternative model is desirable. Often a negative binomial model is useful as an alternative. In this dissertation methodology for analyzing data under a two-parameter negative binomial model is provided. A conditional likelihood approach is suggested to simplify estimation and inference procedures. Large sample properties of the conditional likelihood approach are derived. Based on simulations these properties are examined for small samples. The suggested methodology is applied to two sets of data from ecological research studies. / Source: Dissertation Abstracts International, Volume: 51-02, Section: B, page: 0832. / Major Professor: Duane Meeter. / Thesis (Ph.D.)--The Florida State University, 1989.