Uniform error bounds for a continuous approximation of non-negative random variables

Abstract

In this work, we deal with approximations for distribution functions of non-negative random variables. More specifically, we construct continuous approximants using an acceleration technique over a well-know inversion formula for Laplace transforms. We give uniform error bounds using a representation of these approximations in terms of gamma-type operators. We apply our results to certain mixtures of Erlang distributions which contain the class of continuous phase-type distributions.

On the approximation of functions satisfying defective renewal equations

“…Our aim is to consider the renewal function given in (9) in order to obtain conditions on F and v such that m ∈ D 1 , with D 1 as defined (13). In this case, by Theorem 2.1, M [2] t m, as defined in (11)-(12) has order of convergence 1/t 2 .…”

“…A modification of the operator defined in (4) was used in [9] to approximate the distribution function F X of a nonnegative random variable X by means of its Laplace-Stieltjes transform, using that (4), when g = F X can be rewritten in terms of a well-known inversion formula for this transform (in the beginning of Section 3.2. we explain this connection). By considering the inversion formula (2) using the Laplace transform instead of the Laplace-Stieltjes transform, we can therefore widen the class of functions under consideration, using, at the same time the general convergence results given in [9]. The connection of both inversion formulas through the same operator was considered also in [2].…”

“…Our aim in this paper is to construct, as in [9] an accelerated approximation to (2) and apply it to functions satisfying a defective renewal equation of the form:…”

On the Minimal Value in Maier's Property Concerning Phase-Type Distributions

“…thus having (recall (9)) a jk (c + p j ) n +ā jk (c +p j ) n = j (c) n (2b jk cos(n j (c)) − 2c jk sin(n j (c))), so that, we have by (15) …”

“…The problem of studyingc is motivated, in our work, as a consequence of finding a representation of a given phase-type distribution in terms of a mixture of gamma densities (see [15] , Proposition 5.2 for more details). The selection of Equation (3) will provide the optimal scale parameter in the previous representation, in the sense of giving optimal error bounds in an approximation developed in the aforementioned paper.…”