- Citation:
- Esquível, Manuel L., João Lita da Silva, João Tiago Mexia, and Luís Ramos. "Rate of Convergence of some Asymptotic Expansions for Distribution Approximations via an Esseen Type Estimate." Communications in Statistics - Theory and Methods. 43.2 (2014): 266-290.

### 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 gX + nn∈, where g is some smooth function, X is a random variable and nn∈ 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 nXn∈, X being a Gamma distributed random variable.

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