| The skew-normal family of probability distributions is a fairly recent family of distributions that attracted wide attention in the literature due to its strict inclusion of the normal distribution, its mathematical tractability and because it reproduces some properties of the normal distribution. Since the normal distribution is still the most commonly used distribution both in statistical theory and applications, a family of distributions that possesses the above properties has a great potential impact in theoretical and applied probability and statistics. However, despite this potential impact, there are still relatively few statisticians who use this family in their theoretical and applied works. The main reason is because research on characterization and statistical inference for this family is still in its early stage. The problem of statistical estimation of its parameters, for instance, is still unresolved. No accepted procedure for the hypothesis tests for the parameters of this distribution has been developed except in one special case when the hypothesis being tested is that of normality against skew-normality. Even the basic problem of testing for the goodness-of-fit of a set of data for this family is relatively untouched.;This dissertation addressed the above issues. In this research, some contributions in the areas of characterization and statistical inference were made. In particular, two characterization results based on quadratic statistics were obtained. These results reduced to known characterizations of the normal distribution and generalized to a larger family of probability distributions. Related to these characterizations, a more general skew-normal family was formulated and briefly investigated. In the area of statistical estimation, the problem with the method of moments and the maximum likelihood estimators for the one-parameter skew-normal distribution was analyzed. Alternative estimators based on the sample proportion were proposed both for the one-parameter and three-parameter models. Some asymptotic properties of these alternative estimators were derived. These estimators, although not optimal, have the advantage of being computationally simple and can be utilized when the usual maximum likelihood and method of moments estimates are not satisfactory. The alternative estimator of the skewness parameter of the skew-normal distribution was also adopted and a new estimator of the correlation coefficient in the Roberts' correlation model was obtained and compared with the truncated version of Roberts' estimator. The bias of these estimators were examined and it was shown through simulation studies that none of this two estimators is uniformly better in terms of mean square error.;In the area of hypothesis testing, small sample percentage points for the exact efficient score test and for the approximate efficient score test for the skewness parameter of the skew-normal distribution in the absence and in the presence, respectively, of nuisance location and scale parameters were simulated. Finally, a simple goodness-of-fit procedure was proposed and a power study was conducted. The study showed that this goodness-of-fit procedure can achieve power comparable with those achieved by some common empirical distribution function tests. |
