Two generalizations of the skew-normal distribution and two variants of McCarthy's theorem

The thesis is structured into two main parts. The first and major part is concerned with the skew-normal distribution, introduced by Azzalini (1985) [6], while the second one is connected with the scoring rules. In part one the problem of finding confidence intervals for the skewness parameter of the skew-normal distribution is addressed. Two new five-parameter continuous distributions which generalize the skew-normal distribution as well as some other well-known distributions are proposed and studied. Some mathematical properties of both distributions are derived. Part two is focused on the extension of the theorem of characterization of scoring rules, due to McCarthy (1956) ([16] of part 2), in two directions: for countable infinite sample spaces, but with bounded score and for finite sample spaces, but with unbounded score.