Asymptotic Expansions for Moments of Functions of Stochastic Processes and Their Applications

Abstract

Let{X(t)> t eT be a stochastic process and g be a real function. Suppose that E|X(t) -0| q+2 -» 0 , q being a natural number and θ being a parameter. Under some smoothing conditions on g asymptotic expansions for Ε g(X(t)) and var g(X(t)) are derived with the remainder terms Ο(E|X(t) -6| g+1 ) and 0(E|x(t) -9| g+2 ) , respectively. The proofs use the idea of the method of statistical differentials. Function g may depend on t explicitely. A generalization to the multi-dimensional case is also given. The asymptotic formulas are applied to the problem of compa-AMS 1980 subject classifications: 62 G 05 , 62 G 20 Key words and phrases: asymptotic expansions, statistical differentials, reliability estimators, deficiency Brought to you by | Nanyang Technological University Authenticated Download Date | 6/10/15 5:10 AM 252 J. Hurt rison of some reliability estimators by deficiency and to investigation the asymptotic behaviour of complicated functions of statistics.