A Theorem on Uniform Convergence of Stochastic Functions with Applications

✍ Scribed by Ke-Hai Yuan

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 239 KB
• Volume: 62
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1997.1674

✦ Synopsis

In a variety of statistical problems one needs to manipulate a sequence of stochastic functions involving some unknown parameters. The asymptotic behavior of the estimated parameters often depends on the asymptotic properties of such functions. Especially, the consistency of the estimated parameters follows from the uniform convergence of the sequence of stochastic functions. A theorem on uniform convergence of a sequence of vector valued random functions is presented. The forms of these functions are very general and the assumptions are rather natural. If the sequence of random functions is generated by a sequence of random vectors, these random vectors are only required to be independently distributed and can be of different dimensions. As applications, we consider the consistency of the estimated regression parameters in logistic regression and in M-estimation in a linear model.

1997 Academic Press

In many statistical problems, we need to get an estimate % n of % based on the observed variables y i through function G n ( y, %). In order to obtain the asymptotic behavior of % n , it is often sufficient to demonstrate uniform convergence of G n ( y, %). Especially, the strong consistency of % n follows from article no. MV971674 100 0047-259XÂ97 25.00