Let X and Y be two random variables such that Y depends on X=x. This is a very common situation in many real life applications. The problem is to estimate the location and scale parameters in the marginal distributions of X and Y and the conditional distribution of Y given X=x. We are also interested in estimating the regression coefficient and the correlation coefficient. We have a cost constraint for observing X=x, the larger x is the more expensive it becomes. The allowable sample size n is governed by a pre-determined total cost. This can lead to a situation where some of the largest X=x observations cannot be observed (Type II censoring). Two general methods of estimation are available, the method of least squares and the method of maximum likelihood. For most non-normal distributions, however, the latter is analytically and computationally problematic. Instead, we use the method of modified maximum likelihood estimation which is known to be essentially as efficient as the maximum likelihood estimation. The method has a distinct advantage: It yields estimators which are explicit functions of sample observations and are, therefore, analytically and computationally straightforward. In this thesis specifically, the problem is to evaluate the effect of the largest order statistics x(i) (i&gt / n-r) in a random sample of size n (i) on the mean E(X) and variance V(X) of X, (ii) on the cost of observing the x-observations, (iii) on the conditional mean E(Y|X=x) and variance V(Y|X=x) and (iv) on the regression coefficient. It is shown that unduly large x-observations have a detrimental effect on the allowable sample size and the estimators, both least squares and modified maximum likelihood. The advantage of not observing a few largest observations are evaluated. The distributions considered are Weibull, Generalized Logistic and the scaled Student&rsquo / s t.

<p> In this work, optimal censoring schemes are investigated for the non-parametric confidence intervals of population quantiles under progressive Type-II right censoring. The proposed inference can be universally applied to any probability distributions for continuous random variables. By using the interval mass as an optimality criterion, the optimization process is also independent of the actual observed values from a sample as long as the initial sample size n and the number of observations m are predetermined. This study is based on the fact that each (uncensored) order statistic observed from progressive Type-II censoring can be represented as a mixture of underlying ordinary order statistics with exactly known weights [11, 12]. Using several sample sizes combined with various degrees of censoring, the results of the optimization are tabulated here for a wide range of quantiles with selected levels of significance (i.e., α = 0.01, 0.05, 0.10). With the optimality criterion under consideration, the efficiencies of the worst progressive Type-II censoring scheme and ordinary Type-II censoring scheme are also examined in comparison with the best censoring scheme obtained for a given quantile with fixed n and m.</p> / Thesis / Master of Science (MSc)

<p>This thesis investigates nonparametric inference under multiple independent samples with various modes of censoring, and also presents results concerning Pitman Closeness under Progressive Type-II right censoring. For the nonparametric inference with multiple independent samples, the case of Type-II right censoring is first considered. Two extensions to this are then discussed: doubly Type-II censoring, and Progressive Type-II right censoring. We consider confidence intervals for quantiles, prediction intervals for order statistics from a future sample, and tolerance intervals for a population proportion. Benefits of using multiple samples over one sample are discussed. For each of these scenarios, we consider simulation as an alternative to exact calculations. In each case we illustrate the results with data from the literature. Furthermore, we consider two problems concerning Pitman Closeness and Progressive Type-II right censoring. We derive simple explicit formulae for the Pitman Closeness probabilities of the order statistics to population quantiles. Various tables are given to illustrate these results. We then use the Pitman Closeness measure as a criterion for determining the optimal censoring scheme for samples drawn from the exponential distribution. A general result is conjectured, and demonstrated in special cases</p> / Doctor of Philosophy (PhD)

<p>The joint censoring scheme is of practical significance while conducting comparative life-tests of products from different units within the same facility. In this thesis, we derive the exact distributions of the maximum likelihood estimators (MLEs) of the unknown parameters when joint censoring of some form is present among the multiple samples, and then discuss the construction of exact confidence intervals for the parameters.</p> <p>We develop inferential methods based on four different joint censoring schemes. The first one is when a jointly Type-II censored sample arising from $k$ independent exponential populations is available. The second one is when a jointly progressively Type-II censored sample is available, while the last two cases correspond to jointly Type-I hybrid censored and jointly Type-II hybrid censored samples. For each one of these cases, we derive the conditional MLEs of the $k$ exponential mean parameters, and derive their conditional moment generating functions and exact densities, using which we then develop exact confidence intervals for the $k$ population parameters. Furthermore, approximate confidence intervals based on the asymptotic normality of the MLEs, parametric bootstrap intervals, and credible confidence regions from a Bayesian viewpoint are all discussed. An empirical evaluation of all these methods of confidence intervals is also made in terms of coverage probabilities and average widths. Finally, we present examples in order to illustrate all the methods of inference developed here for different joint censoring scenarios.</p> / Doctor of Science (PhD)

<p>A survival distribution is developed for exponential two-component systems that can survive as long as at least one of the two components in the system function. It is assumed that the two components are initially independent and non-identical. If one of the two components fail (repair is impossible), the surviving component is subject to a different failure rate due to the stress caused by the failure of the other.</p> <p>In this paper, we consider such an exponential two-component system failure model when the observed failure time data are (1) complete, (2) Type-I censored, (3) Type-I censored with partial information on component failures, (4) Type-II censored and (5) Type-II censored with partial information on component failures. In these situations, we discuss the maximum likelihood estimates (MLEs) of the parameters by assuming the lifetimes to be exponentially distributed. The exact distributions (whenever possible) of the MLEs of the parameters are then derived by using the conditional moment generating function approach. Construction of confidence intervals for the model parameters are discussed by using the exact conditional distributions (when available), asymptotic distributions, and two parametric bootstrap methods. The performance of these four confidence intervals, in terms of coverage probabilities are then assessed through Monte Carlo simulation studies. Finally, some examples are presented to illustrate all the methods of inference developed here.</p> <p>In the case of Type-I and Type-II censored data, since there are no closed-form expressions for the MLEs, we present an iterative maximum likelihood estimation procedure for the determination of the MLEs of all the model parameters. We also carry out a Monte Carlo simulation study to examine the bias and variance of the MLEs.</p> <p>In the case of Type-II censored data, since the exact distributions of the MLEs depend on the data, we discuss the exact conditional confidence intervals and asymptotic confidence intervals for the unknown parameters by conditioning on the data observed.</p> / Thesis / Doctor of Philosophy (PhD)

In this thesis, we investigate several inferential issues regarding the lifetime data from exponential distribution under different censoring schemes. For reasons of time constraint and cost reduction, censored sampling is commonly employed in practice, especially in reliability engineering. Among various censoring schemes, progressive Type-I censoring provides not only the practical advantage of known termination time but also greater flexibility to the experimenter in the design stage by allowing for the removal of test units at non-terminal time points. Hence, we first consider the inference for a progressively Type-I censored life-testing experiment with k uniformly spaced intervals. For small to moderate sample sizes, a practical modification is proposed to the censoring scheme in order to guarantee a feasible life-test under progressive Type-I censoring. Under this setup, we obtain the maximum likelihood estimator (MLE) of the unknown mean parameter and derive the exact sampling distribution of the MLE through the use of conditional moment generating function under the condition that the existence of the MLE is ensured. Using the exact distribution of the MLE as well as its asymptotic distribution and the parametric bootstrap method, we discuss the construction of confidence intervals for the mean parameter and their performance is then assessed through Monte Carlo simulations. Next, we consider a special class of accelerated life tests, known as step-stress tests in reliability testing. In a step-stress test, the stress levels increase discretely at pre-fixed time points and this allows the experimenter to obtain information on the parameters of the lifetime distributions more quickly than under normal operating conditions. Here, we consider a k-step-stress accelerated life testing experiment with an equal step duration τ. In particular, the case of progressively Type-I censored data with a single stress variable is investigated. For small to moderate sample sizes, we introduce another practical modification to the model for a feasible k-step-stress test under progressive censoring, and the optimal τ is searched using the modified model. Next, we seek the optimal τ under the condition that the step-stress test proceeds to the k-th stress level, and the efficiency of this conditional inference is compared to the preceding models. In all cases, censoring is allowed at each change stress point iτ, i = 1, 2, ... , k, and the problem of selecting the optimal Tis discussed using C-optimality, D-optimality, and A-optimality criteria. Moreover, when a test unit fails, there are often more than one fatal cause for the failure, such as mechanical or electrical. Thus, we also consider the simple stepstress models under Type-I and Type-II censoring situations when the lifetime distributions corresponding to the different risk factors are independently exponentially distributed. Under this setup, we derive the MLEs of the unknown mean parameters of the different causes under the assumption of a cumulative exposure model. The exact distributions of the MLEs of the parameters are then derived through the use of conditional moment generating functions. Using these exact distributions as well as the asymptotic distributions and the parametric bootstrap method, we discuss the construction of confidence intervals for the parameters and then assess their performance through Monte Carlo simulations. / Thesis / Doctor of Philosophy (PhD)