It is known that record values and order statistics are closely related. When record values and order statistics are viewed as point processes, the two processes both share the order statistics property. The results of Beg and Balasubramanian(1990), Wu and Ouyang(1996), and Huang and Su(1999) about record values and order statistics motivated us to investigate more general results of characterization for order statistics point processes by using conditional expectations based on order statistics. On the other hand, in the class of point processes, there are a lot of characterizations of homogeneous Poisson processes based on the memoryless property of exponential distribution. The result of Asadi(1999) about characterization of the Gumble bivariate exponential or the bivariate geometric distribution inspired us be interested in investigating some similar results about non-independent bivarite homogeneous Poisson processes.

The study of properties of probability distributions has always been a persistent theme of statistics and of applied probability. This thesis deals with an investigation of distribution functions under the following two topics: (i) characterization of distributions based on record values and order statistics, (ii) properties of the skew-t distribution. Within the extensive characterization literature there are several results involving properties of record values and order statistics. Although there have been many well known results already developed, it is still of great interest to find new characterization of distributions based on record values and order statistics. In the first part, we provide the conditional distribution of any record value given the maximum order statistics and study characterizations of distributions based on record values and the maximum order statistics. We also give some characterizations of the mean value function within the class of order statistics point processes, by using certain relations between the conditional moments of the jump times or current lives. These results can be applied to characterize the uniform distribution using the sequence of order statistics, and the exponential distribution using the sequence of record values, respectively. Azzalini (1985, 1986) introduced the skew-normal distribution which includes the normal distribution and has some properties like the normal and yet is skew. This class of distributions is useful in studying robustness and for modeling skewness. Since then, skew-symmetric distributions have been proposed by many authors. In the second part, the so-called generalized skew-t distribution is defined and studied. Examples of distributions in this class, generated by the ratio of two independent skew-symmetric distributions, are given. We also investigate properties of the skew-symmetric distribution.