Semiparametric Bounds On Means And Variances For Truncated Random Variables

Between two extremes of complete knowledge and complete ignorance of the distribution of a random variable lies the situation in which incomplete information of that has been given,then we can address something which is non-trivial about the upper and lower bounds of the probability, expectation and variance of functions of this random variable. The known information about a distribution may be numerical, e.g. moments , or geometrical, e.g. unimode , monotonicity on distribution,etc.The problem of moments is of fairly old origin as a branch of probability theory. Since the classical Chebyshev inequality , Markov inequality , Gauss inequality,it has enjoyed much popularity and there is a considerable amount of literature on this area.The interest in the problem of moments remains strong up to the present day. The problem of moments occupy an important place in the theory of probability. It is very much connected with many branches of mathematics and other science areas,such as functional analysis,real analysis,operator spectrum,economic,finance , operation , decision science , continued fraction theory , orthogonal polynomial,positive semidefinite optimization,Bennett-Hoeffding inequality of nonindependent sum,large value and small value probability,medical imaging,receiver operating characteristic curve , option pricing , etc. Hence , further researches on moment problems are meaningful from both theoretical and practical points of view. In this dissertation,we derive first sharp bounds on tail distribution having its moments and mode known. Bounds on means and variances of functions of truncated random variables are then obtained . The main work is listed as follows:1. The problem of bounding distribution is formalized as the positive semidefinite optimization expression. Specially, sharp bounds on distribution are derived under the first two or three moments given,respectively,and these bounds are achievable and distribution free. To indicate different approaches,we extend the known comparison method and give an alternative proof on the second moments case. For the third moment case,semidefinite optimization method and discrete method are illustrated for comparison purpose. 2. The problems of bounding means of functions of truncated random variables are formalized as the positive semidefinite optimization expressions. Sharp bounds on means of three classes of functions of truncated random variables are derived under the first two moments given. These bounds are achievable and distribution free,and the optimal distributions are obtained.3. A totally new constructive method to bound variance of function of truncated random variable is presented basing on symmetric and dual theory. Upper bounds of three classes of functions of truncated random variable are obtained by employing this method,and these upper bounds are best known.4. The problem of bounding means of functions of truncated random variables having moments and unimode or bimode known is formalized as the positive semidefinite optimization expressions. Upper bounds on means of European call option and distribution are derived with unimode or bimode given. The feasible conditions are addressed in two cases.