# On the rate of convergence of uniform approximations for sequences of distribution functions

Research output: Contribution to journalArticle

### Abstract

In this paper, we develop uniform bounds for the sequence of distribution functions of g(Vn+μn), wheregis some smooth function,is a sequence of identically distributed random variables with common distribution having a bounded derivative and {μn} are constants such that μn→∞. These bounds allow us to identify a suitable sequence 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 the ratio of the second derivative to the first derivative ofg. The corresponding generalization to the multivariate case is also analyzed. An application of our results to the STATIS-ACT method is provided in the final section.
Original language Unknown 47-65 Journal Of The Korean Statistical Society 43 1 https://doi.org/10.1016/j.jkss.2013.06.001 Published - 1 Jan 2014

### Cite this

@article{4a99c28624824fe1aa6c0e9a4564a3cc,
title = "On the rate of convergence of uniform approximations for sequences of distribution functions",
abstract = "In this paper, we develop uniform bounds for the sequence of distribution functions of g(Vn+μn), wheregis some smooth function,is a sequence of identically distributed random variables with common distribution having a bounded derivative and {μn} are constants such that μn→∞. These bounds allow us to identify a suitable sequence 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 the ratio of the second derivative to the first derivative ofg. The corresponding generalization to the multivariate case is also analyzed. An application of our results to the STATIS-ACT method is provided in the final section.",
author = "Ramos, {Lu{\'i}s Pedro Carneiro} and {Lita da Silva}, {Jo{\~a}o Filipe}",
year = "2014",
month = "1",
day = "1",
doi = "10.1016/j.jkss.2013.06.001",
language = "Unknown",
volume = "43",
pages = "47--65",
journal = "Journal Of The Korean Statistical Society",
issn = "1226-3192",
publisher = "Elsevier",
number = "1",

}

In: Journal Of The Korean Statistical Society, Vol. 43, No. 1, 01.01.2014, p. 47-65.

Research output: Contribution to journalArticle

TY - JOUR

T1 - On the rate of convergence of uniform approximations for sequences of distribution functions

AU - Ramos, Luís Pedro Carneiro

AU - Lita da Silva, João Filipe

PY - 2014/1/1

Y1 - 2014/1/1

N2 - In this paper, we develop uniform bounds for the sequence of distribution functions of g(Vn+μn), wheregis some smooth function,is a sequence of identically distributed random variables with common distribution having a bounded derivative and {μn} are constants such that μn→∞. These bounds allow us to identify a suitable sequence 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 the ratio of the second derivative to the first derivative ofg. The corresponding generalization to the multivariate case is also analyzed. An application of our results to the STATIS-ACT method is provided in the final section.

AB - In this paper, we develop uniform bounds for the sequence of distribution functions of g(Vn+μn), wheregis some smooth function,is a sequence of identically distributed random variables with common distribution having a bounded derivative and {μn} are constants such that μn→∞. These bounds allow us to identify a suitable sequence 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 the ratio of the second derivative to the first derivative ofg. The corresponding generalization to the multivariate case is also analyzed. An application of our results to the STATIS-ACT method is provided in the final section.

U2 - 10.1016/j.jkss.2013.06.001

DO - 10.1016/j.jkss.2013.06.001

M3 - Article

VL - 43

SP - 47

EP - 65

JO - Journal Of The Korean Statistical Society

JF - Journal Of The Korean Statistical Society

SN - 1226-3192

IS - 1

ER -