Empirical examination of Edgeworth series

The material presented is based on a numerical investigation that was made for five types of probability approximations which involve the first seven terms of the Edgeworth series expansion for the distribution of a continuous random variableT. For each approximation, the probability expressions considered in the investigation were Pr(T≦t), Pr(?t≦T≦t) and Pr(?t+1≦T≦t), whereT has zero mean, unit variance, and specified central momentsμ ₃,μ ₄,μ ₅. Computations were made for thoset values in the set ?4.00(0.25) 4.00 that are pertinent for the probability expression being considered and for all combinations of the following values forμ ₃,μ ₄,μ ₅∶μ ₃=?2.0, ?1.0, ?0.5,0.0, 0.5, 1.0, 2.0;μ ₄=1, 2, 3, 5, 10;μ ₅=0.0, 3μ ₃?6.0, 3μ ₃, 3μ ₃+6.0. The principal results of this paper consist of a specification (for each approximation, probability expression, andμ ₃,μ ₄,μ ₅ combination) of limits ont such that within these limits the computed values of the probability expression are meaningful; that is, satisfy required monotonicity properties as a function oft and are neither negative nor greater than unity. Also the values of Pr(T≦0) and of Pr(?1.75≦T≦1.75) are listed for the cases considered. These results indicate that the types of approximations investigated are of doubtful usefulness for the situations examined; that is, for cases where the third and higher order moments of the random variable considered differ substantially from those for the normal variable having the same mean and variance.