Method for constructing Stieltjes classes for M-indeterminate probability distributions

Abstract

Let F be a distribution function with density f, finite moments of any positive integer order and such that the problem of moments for F has a nonunique solution ( F is M-indeterminate). We are looking for explicit Stieltjes classes S = { f ε = f[1 + εh], ε ∈ [−1, 1]} of distributions (here in terms of their densities) all sharing the same moments as those of F. We suggest a general method for constructing such classes. Then we apply our method to describe the M-indeterminacy of power transformations of popular distributions such as lognormal, inverse Gaussian and logistic. The Stieltjes classes presented here are new. We also find numerical values of the index of dissimilarity for some particular cases.