In this thesis we discuss some of the tools used in the study of the number of primes in short intervals. In particular, we discuss a large sieve density estimate due to Gallagher and two classical delay equations. We also show how these tools have been used by Maier and Stewart and provide computational data to their result.

by Wai-leung Li. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 74-77). / Chapter Chapter 1 --- Introduction --- p.1 / Chapter § 1.1 --- Why transformation of variables in regression analysis is needed? --- p.1 / Chapter § 1.2 --- Suggested functional transformation -- Box-Cox Transformation --- p.3 / Chapter § 1.3 --- Methodology --- p.5 / Chapter § 1.4 --- General theory of constructing asymptotic confidence intervals and confidence regions --- p.9 / Chapter § 1.4.1 --- Method based on the log-likelihood ratio statistic --- p.9 / Chapter § 1.4.2 --- Method based on the asymptotic normality of the maximum likelihood estimates --- p.13 / Chapter § 1.4.3 --- Method based on the score statistic --- p.15 / Chapter § 1.5 --- General theory of constructing exact confidence intervals and confidence regions --- p.17 / Chapter § 1.6 --- Summary --- p.23 / Chapter Chapter 2 --- Confidence Intervals for the non-linear parameter λ in the Box-Cox transformation models --- p.24 / Chapter § 2.1 --- Confidence intervals based on the log-likelihood ratio statistics --- p.26 / Chapter § 2.1.1 --- Asymptotically equivalent forms for constructing confidence intervals based on the log-likelihood ratio statistics --- p.30 / Chapter § 2.2 --- Confidence intervals based on the asymptotic normality of the maximum likelihood estimates --- p.31 / Chapter § 2.3 --- Confidence intervals based on the score statistics --- p.35 / Chapter § 2.4 --- Confidence intervals based on the exact test --- p.36 / Chapter § 2.5 --- Small simulation studies of constructing confidence intervals for A based on the four different methods --- p.37 / Chapter § 2.5.1 --- Design of the simulation studies --- p.40 / Chapter § 2.5.2 --- Simulation results --- p.41 / Chapter § 2.6 --- Summary --- p.44 / Chapter Chapter 3 --- Confidence Regions for the parameters in the Box-Cox transformation models --- p.45 / Chapter § 3.1 --- Confidence regions based on the log-likelihood ratio statistics --- p.45 / Chapter § 3.1.1 --- "Confidence region for (λ,ζ1）based on the log-likelihood ratio statistics" --- p.46 / Chapter § 3.1.2 --- Confidence region for (ζ1）based on the log-likelihood ratio statistics --- p.51 / Chapter § 3.2 --- Confidence regions based on the asymptotic normality of the maximum likelihood estimates --- p.53 / Chapter § 3.2.1 --- "Confidence region for (λ,ζ1）based on the asymptotic normality of the maximum likelihood estimates" --- p.53 / Chapter § 3.2.2 --- Confidence region for (ζ1）based on the asymptotic normality of the maximum likelihood estimates --- p.57 / Chapter § 3.3 --- Confidence regions based on the score statistics --- p.58 / Chapter § 3.3.1 --- "Confidence region for (λ,ζ1） based on the score statistic" --- p.59 / Chapter § 3.3.2 --- Confidence region for (ζ1 ) based on the score statistic --- p.60 / Chapter § 3.4 --- Confidence region based on the exact test --- p.61 / Chapter § 3.5 --- Small simulation studies of constructing confidence regions for the parameters of interest based on the four different methods --- p.62 / Chapter Chapter 4 --- Robustness and Discussion --- p.67 / Chapter §4.1 --- Contamination normal distribution --- p.67 / Chapter § 4.1.1 --- Confidence intervals for the non- linear parameter λ based on the contamination normal distribution of error terms --- p.68 / Chapter § 4.1.2 --- Confidence regions for the parameters of interest based on the contamination normal distribution of the error terms --- p.70 / Chapter § 4.2 --- Summary --- p.72 / References --- p.74 / Figures / Appendix A / Appendix B / Appendix C / Appendix D

Ip Wing Yiu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaf 44). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.2 --- Background --- p.1 / Chapter 1.3 --- Objective --- p.3 / Chapter 1.4 --- Scope of the thesis --- p.3 / Chapter 2 --- Basic Concepts --- p.5 / Chapter 2.1 --- Inverse Sampling --- p.5 / Chapter 2.2 --- Equivalence/ Non-inferiority Testing --- p.6 / Chapter 3 --- Inference for Risk Ratio --- p.8 / Chapter 3.1 --- Introduction --- p.8 / Chapter 3.2 --- Test Statistics for Risk Ratio --- p.8 / Chapter 3.3 --- Consistent Estimators of π --- p.12 / Chapter 4 --- Confidence Interval --- p.16 / Chapter 4.1 --- Introduction --- p.16 / Chapter 4.2 --- Tost-Based Confidence Interval --- p.17 / Chapter 4.3 --- Using sample-based estimates --- p.18 / Chapter 5 --- Simulation --- p.21 / Chapter 5.1 --- Introduction --- p.21 / Chapter 5.2 --- Simulation Procedures --- p.21 / Chapter 5.3 --- Simulation Results --- p.23 / Chapter 6 --- Conclusion --- p.27 / Appendix --- p.29 / Chapter A. --- Equation derviation --- p.29 / Chapter A1. --- Equation derviation 1 --- p.29 / Chapter A2. --- Equation derviation 2 --- p.31 / Chapter B. --- Table --- p.32 / References --- p.44

The role of spatial reasoning in the development of systems in the domain of Artificial Intelligence is increasing. One particular approach, qualitative spatial reasoning, investigates the usage of abstract representation to facilitate the representation of and the reasoning with spatial information.This thesis investigates the usage of intervals along global axes as the under-lying representational and reasoning mechanism for a spatial reasoning system. Aspects that are unique to representing spatial information (flow and multi-dimensionality) are used to provide a method for classifying relations between objects at multiple levels of granularity. The combination of these two mechanisms (intervals and classification) provide the basis for the development of a querying system that allows qualitative queries about object relations in multi-dimensional space to be performed upon the representation.The second issue examined by this thesis is the problem of representing intervals when all the interval relations may not be known precisely. A three part solution is proposed. The first shows how the simplest situation, where all relations are explicit and primitive, can be represented and integrated with the above mentioned querying system. The second situation demonstrates how, for interval relations that are primitive but are not all explicitly known, an effective point based representation may be constructed. Finally, when relations between intervals are disjunctions of possible primitive interval relations, a representation is presented which allows solutions to queries to be constructed from consistent data.Our contribution is two-fold:1. a method of classifying the spatial relations and the means of querying these relations;2. a process of efficiently representing incomplete interval information and the means of efficiently querying this information.The work presented ++ / in this thesis demonstrates the utility of a multi-dimensional qualitative spatial reasoning system based upon intervals. It also demonstrates how an interval representation may be constructed for datasets that have variable levels of information about relationships between intervals represented in the dataset.

Measuring the source and magnitude of components of variation has important applications in industrial, environmental and biological studies. This thesis considers the problem of constructing confidence intervals for variance components in Gaussian mixed linear models. A number of methods based on the usual ANOVA mean squares have been proposed for constructing confidence intervals for variance components in balanced mixed models. Some authors have suggested extending balanced model procedures to unbalanced models by replacing the ANOVA mean squares with mean squares from an unweighted means ANOVA. However, the unweighted means ANOVA is only defined for a few specific mixed models. In Chapter 2 we define a generalization of the unweighted means ANOVA for the three variance component mixed linear model and illustrate how the mean squares from this ANOVA may be used to construct confidence intervals for variance components. Computer simulations indicate that the proposed procedure gives intervals that are generally consistent with the stated confidence level, except in the case of extremely unbalanced designs. A set of statistics that can be used as an alternative to the generalized unweighted mean squares is developed in Chapter 3. The intervals constructed with these statistics have better coverage probability and are often narrower than the intervals constructed with the generalized unweighted mean squares. / Graduation date: 1998

In this thesis we discuss some of the tools used in the study of the number of primes in short intervals. In particular, we discuss a large sieve density estimate due to Gallagher and two classical delay equations. We also show how these tools have been used by Maier and Stewart and provide computational data to their result.

The bootstrap methods are widely used for constructing confidence intervals. However, the conventional bootstrap fails to be consistent under some nonstandard circumstances. The m out of n bootstrap is usually adopted to restore consistency, provided that a correct convergence rate can be specified for the plug-in estimators. In this thesis, we re-investigate the asymptotic properties of the bootstrap in a moving-parameter framework in which the underlying distribution is allowed to depend on n. We consider the problem of setting uniformly consistent confidence intervals for two non-regular cases: (1) the smooth function models with vanishing derivatives; and (2) the M-estimation with non-regular conditions. Under the moving-parameter setup, neither the conventional bootstrap nor the m out of n bootstrap is shown uniformly consistent over the whole parameter space. The results reflect to some extent finite-sample anomalies that cannot be explained by conventional, fixed-parameter, asymptotics. We propose a weighted bootstrap procedure for constructing uniformly consistent bootstrap confidence intervals, which does not require explicit specification of the convergence rate of the plug-in estimator. Under the smooth function models, we also propose a modified n out of n bootstrap procedure in special cases where the smooth function is applied to estimators that are uniformly bootstrappable. The estimating function bootstrap is also successfully employed for the latter model and enjoys computational advantages over the weighted bootstrap. We illustrate our findings by comparing the finite-sample coverage performances of the different bootstrap procedures. The stable performance of the proposed methods, contrasts sharply with the erratic coverages of the n out of n and m out of n bootstrap intervals, a result in agreement with our theoretical findings. / published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

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Thesis (Ph. D.)--Auburn University. / Abstract. Vita. Includes bibliographical references (p. 132-135).