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On the rate of convergence of uniform approximations for sequences of distribution functions
João Lita da Silva,Luís Pedro Ramos 한국통계학회 2014 Journal of the Korean Statistical Society Vol.43 No.1
In this paper, we develop uniform bounds for the sequence of distribution functions ofg(Vn + μn), where g is some smooth function, {Vn, n ≥ 1} is a sequence of identicallydistributed random variables with common distribution having a bounded derivative and{μn} are constants such that μn → ∞. These bounds allow us to identify a suitablesequence of random variables which is asymptotically of the same type of g(Vn + μn)showing that the rate of convergence for these uniform approximations depends on theratio of the second derivative to the first derivative of g. The corresponding generalizationto the multivariate case is also analyzed. An application of our results to the STATIS-ACTmethod is provided in the final section.