A Law of the Single Logarithm for Weighted Sums of Arrays Applied to Bootstrap Model Selection in Regression

• Activity Number: 180
• Type: Contributed
• Date/Time: Monday, July 30, 2012 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Christian Léger*+ and Pierre Lafaye de Micheaux
• Companies: Université de Montréal and Université de Montréal
• Keywords: BIC ; Linear Regression ; Variable Selection ; Law of the Iterated Logarithm ; Resampling ; Triangular arrays

Abstract

According to van der Vaart (1998) "The law of the iterated logarithm is an intriguing result but appears to be of less interest to statisticians." Whether or not this is true, statisticians sometimes need to know bounds on the almost sure size of means or linear combinations of random variables to establish certain statistical results, and laws of the iterated (or single) logarithm are then needed. This was the case when we studied the problem of bootstrapping all-subsets model selection in regression. We generalized a law of the single logarithm obtained by Qi (1994) and Li et al. (1995) to the case of weighted sums of triangular arrays of random variables. This result implies that the popular Bayesian Information Criterion of Schwarz (1978) is no longer asymptotically consistent at the bootstrap level whereas it is with the original variables because the linear combination satisfies a law of the iterated logarithm at that level.