The Rate of Convergence of some Asymptotic Expansions for Distribution Approximations via an Esseen Type Estimate

Abstract

Some asymptotic expansions not necessarily related to the central limit theorem are studied. We first observe that the smoothing inequality of Esseen implies the proximity, in the Kolmogorov distance sense, of the distributions of the random variables of two random sequences satisfying a sort of general asymptotic relation. We then present several instances of this observation. A first example, partially motivated by the the statistical theory of high precision measurements, is given by a uniform asymptotic approximation to (g(X + μ _n )) _n∈ℕ, where g is some smooth function, X is a random variable and (μ _n ) _n∈ℕ is a sequence going to infinity; a multivariate version is also stated and proved. We finally present a second class of examples given by a randomization of the interesting parameter in some classical asymptotic formulas; namely, a generic Laplace's type integral, randomized by the sequence (μ _n X) _n∈ℕ, X being a Gamma distributed random variable.

Keywords:

• High precision measurements
• Kolmogorov distance
• Rate of convergence
• Uniform asymptotic weak approximations

Mathematics Subject Classification:

• 41A60
• 60B10
• 60F05

Acknowledgment

This work is dedicated to Dinis D. F. Pestana, as a token of recognition for a lifelong enthusiasm for Statistics and for his never failing support of young mathematicians.

This work was partially supported by CMA/FCT/UNL under Financiamento Base 2009 ISFL-1-297 from FCT/MCTES/PT.

Notes

Any continuous solution of this functional equation is of the form g(x) = |x|^α or g(x) = sign(x)|x|^α, for some α (see (Itô, 1993, p. 1443)). For having g ∈ 𝒞²(ℝ) we may infer that g(x) = x ^α, for some α ∈ ℕ, is an acceptable solution