A note on recovering the distributions from exponential moments

Abstract

The problem of recovering a cumulative distribution function of a positive random variable via the scaled Laplace transform inversion is studied. The uniform upper bound of proposed approximation is derived. The approximation of a compound Poisson distribution as well as the estimation of a distribution function of the summands given the sample from a compound Poisson distribution are investigated. Applying the simulation study, the question of selecting the optimal scaling parameter of the proposed Laplace transform inversion is considered. The behavior of the approximants are demonstrated via plots and table.

Entropy and density approximation from Laplace transforms

“…Analogous error estimate is found in Mnatsakanov-Sarkisian [15], Theorem 1. Using equispaced real valued L(α j = j) only, the authors construct an approximate cumulative distribution function F M converging uniformly to F X .…”

Approximation of the ruin probability using the scaled Laplace transform inversion

“…Our simulation study shows that the approximation rate derived in Mnatsakanov et al [2] is not optimal. This motivated our interest to improve the rate using the scaled Laplace transform inversion suggested by Mnatsakanov and Sarkisian [5]. …”

“…Note also that the Laplace transform inversions proposed in [5–12] do not require the claim size distribution F to be a completely monotone function. See, for example, Gzyl et al [6], where the ME method and MR-approach proposed in Mnatsakanov [12] are compared.…”

“…In Section 2 we introduce two MR-approximations of a function and its derivative, see (3) and (4), respectively, and provide the upper bound for MR-approximation (4); the upper bound for (3) has been already established in [5] (see also Theorem 1A in the Appendix). In Section 3, we propose two approximations (see (17) and (20)) of the ruin probability ψ , which are based on the finite number of values of Laplace (ℒ ψ ) and Laplace–Stieltjes ( L G ) transforms of ψ and G = 1 − ψ , respectively: $${\mathcal{L}}_{\psi }false(sfalse)={\int }_0^{\infty }{e}^{-s\tau }\psi false(\tau false)d\tau$$ $$L_{G}false(sfalse)={\int }_0^{\infty }{e}^{-s\tau }dGfalse(\tau false),s\in false[0,\infty false):={ℝ}_{+}.$$…”

Polynomial Approximations for Bivariate Aggregate Claims Amount Probability Distributions

“…One can note that the polynomial expansion with the first parametrization performs better than the moment-recovered based method. However, the moment-recovered based method has been enhanced in recent papers with the introduction of the scaled Laplace transform, see [21,22]. These improvements have been made in the univariate case but might be extended soon to the bivariate case and comparison would be interesting in the future.…”