Thesis (Ph. D.)--University of Washington, 2005. / Vita. Includes bibliographical references (p. 72-78).

Thesis (Ph. D.)--University of Washington, 1987. / Vita. Bibliography: leaves [192]-194.

Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (p. 140-145).

Although Wald tests form one of the major classes of hypothesis tests, they suffer from the well-known major drawback that they are not invariant under reparameterisation. This thesis uses the differential-geometric concept of a yoke to introduce one-parameter families of geometric Wald statistics, which are parameterisation-invariant statistics in the spirit of the traditional Wald statistics. Both the geometric Wald statistics based on the expected likelihood yokes and those based on the observed likelihood yokes are investigated. Bartlett-type adjustments of the geometric Wald statistics are obtained, in order to improve the accuracy of the chi-squared approximations to their distributions under the null hypothesis.

In this thesis we compare various methods for estimating the unknown parameters in mixtures of circular and spherical distributions. We study the von Mises distribution on the circle and the Fisher distribution on the sphere. We propose a new method of estimation based on the characteristic function and compare it with the classical methods based on maximum likelihood and moments. Thus far these methods have only been successfully applied to distributions on the line. Here we show that the extension to circular and spherical distributions is reasonably straightforward and convergence to the final estimates is fairly rapid. We apply these methods to various simulated and real data sets and show that the results obtained for the mixture of two von Mises distributions are satisfactory but generally depend on the sample size and method of estimation used. However, results obtained for the mixture of two Fisher distributions show that maximum likelihood performs best overall.

Statistical aspects of the population regulation of a migratory brown trout population are investigated. The life cycle of the trout, the study area and the sampling routine are described in Chapter 1. Models of numerical changes in fish populations are reviewed in Chapter 2. Measures that assess the nonlinear behaviour of nonlinear regression models are described in Chapter 3. The additive error Ricker model describes the relationship between the number of 0+ parr in May/June and the number of eggs. The nonlinear behaviour of the model is investigated in Chapter 4. The parameter effects nonlinearity of the model is reduced by a reparameterisation. Chapter 5 investigates the effect of errors in the egg variable on the distributions of the least squares estimators of the additive error and the multiplicative error Ricker models. The errors-in-variables considerably increase the variances of the least squares estimators. Models of the relationships between the numbers of 0+ parr in August/September, the number of 1+ parr, the egg production of a year class and the number of eggs are developed in Chapter 6. These models account for the effect of summer drought on survival. Survival is density dependent during the first summer of the life cycle and density independent thereafter. Standard measures of nonlinearity can seriously underestimate the nonlinear behaviour of piecewise linear change-point models. New measures of nonlinearity appropriate for piecewise linear change-point models are developed in Chapter 7. Chapter 8 develops a model of the growth of brown trout fed on maximum rations as a function of time, body weight and water temperature. Chapter 9 develops a model that relates the survival rate of 0+ parr between May/June and August/September to the length distribution of the trout in May/June. The results of the Thesis are discussed in Chapter 10.

Transformation is a powerful tool for model building. In regression the response variable is transformed in order to achieve the usual assumptions of normality, constant variance and additivity of effects. Here the normality assumption is replaced by the Laplace distributional assumption, appropriate when more large errors occur than would be expected if the errors were normally distributed. The parametric model is enlarged to include a transformation parameter and a likelihood procedure is adopted for estimating this parameter simultaneously with other parameters of interest. Diagnostic methods are described for assessing the influence of individual observations on the choice of transformation. Examples are presented. In distribution methodology the independent responses are transformed in order that a distributional assumption is satisfied for the transformed data. Here the interest is in the family of distributions which are not dependent on an unknown shape parameter. The gamma distribution (known order), with special case the exponential distribution, is a member of this family. An information number approach is proposed for transforming a known distribution to the gamma distribution (known order). The approach provides an insight into the large-sample behaviour of the likelihood procedure considered by Draper and Guttman (1968) for investigating transformations of data which allow the transformed observations to follow a gamma distribution. The information number approach is illustrated for three examples end the improvement towards the gamma distribution introduced by transformation is measured numerically and graphically. A graphical procedure is proposed for the general case of investigating transformations of data which allow the transformed observations to follow a distribution dependent on unknown threshold and scale parameters. The procedure is extended to include model testing and estimation for any distribution which with the aid of a power transformation can be put in the simple form of a distribution that is not dependent on an unknown shape parameter. The procedure is based on a ratio, R(y), which is constructed from the power transformation. Also described is a ratio-based technique for estimating the threshold parameter in important parametric models, including the three-parameter Weibull and lognormal distributions. Ratio estimation for the weibull distribution is assessed and compared with the modified maximum likelihood estimation of Cohen and Whitten (1982) in terms of bias and root mean squared error, by means of a simulation study. The methods are illustrated with several examples and extend naturally to singly Type 1 and Type 2 censored data.

This thesis is concerned principally with the problem of estimating the parameters of the Weibull and Beta distributions using several different techniques. These distributions are used in the area of reliability testing and it is important to achieve the best estimates possible of the parameters involved. After considering several accepted methods of estimating the relevant parameters, it is considered that the best method depends on the aim of the analysis, and on the value of the shape parameter β. For estimating the two-parameter Weibull distribution, it is recommended that Generalized Least Squares (GLS) is the best method to use for values of β between 0.5 and 30. However, Maximum Likelihood Estimator (MLE) is a good method for estimating quantiles. On this basis, the three-parameter Weibull distribution is investigated. The traditional parametrization is compared with a new parametrization developed in this work. By considering parameter effects and intrinsic curvature it is shown that the new parametrization results in a linear effect of the shape parameter. Also it has advantages in quantile estimation because of its ability to provide estimates for a wider range of data sets. A less frequently used distribution in the field of reliability is the Beta distribution. The lack of frequency of its use is partly due to the difficulty in estimating its parameters. A simple, applicable method is developed here of estimating these parameters. This 'group method' involves estimating the two ends of the distribution. It is shown that this procedure can be used, together with other methods of estimating the two- parameter Beta distribution successfully to estimate the four-parameter Beta distribution.

As a classic parameter originated from the binomial distribution, the binomial pro- portion has been well studied in the literature due to its wide range of applications. In contrast, the reciprocal of the binomial proportion, also known as the inverse proportion, is often overlooked, although it plays an important role in sampling designs and clinical studies. To estimate the inverse proportion, a simple method is to apply the maximum likelihood estimation (MLE). This estimator is, however, not a valid estimator because it suffers from the zero-event problem, which occurs when there is no successful event in the trials. At first, we review a number of methods proposed to overcome the zero-event problem and discuss whether they are feasible to estimate the inverse proportion. Inspired by the Wilson (1927) and Agresti and Coull (1998), in this thesis, we focus on a family of shrinkage estimators of the inverse proportion and propose to derive the optimal estimator within this family. The shrinkage estimator overcomes the zero-event problem by including a positive shrinkage parameter, which is intrinsically related to the expected value of the resulting estimator. To find the best shrinkage parameter, the relationship between the shrinkage parameter and the estimation bias of the shrinkage estimator is investigated systematically. Note that the explicit expression of the expected value function of the estimator and the best shrinkage parameter are quite complicated to compute when the number of trials is large. Hence, we review three methods in the literature which were proposed to approximate the expected value function. And after being inspired, we propose a new approximate formula for the expected value function and derive an approximate solution of the optimal shrinkage parameter by the Taylor expansion. Because there still exist an unknown binomial proportion in the optimal shrinkage parameter, we suggest a plug-in estimator for the unknown proportion with an adaptive threshold. Finally, simulation studies are conducted to evaluate the performance of our new estimator. As baselines for comparison, we also include the Fattorini estimator, the Haldane estimator and a piecewise estimator in the simulations. According to the simulation results, the new estimator is able to achieve a better or equally good performance compared with the Fattorini estimators in most settings. Hence, our new estimator can be a reliable estimator for the inverse proportion in most practical cases

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