2018/07/12 by Miles E. Lopes, Lopes, Miles E., Zhenhua Lin +3
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Methodology (stat.ME) #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1807.04429
openalex publication_date 2018/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In recent years, bootstrap methods have drawn attention for their ability to\napproximate the laws of "max statistics" in high-dimensional problems. A\nleading example of such a statistic is the coordinate-wise maximum of a sample\naverage of n random vectors in \ℝp. Existing results for this\nstatistic show that the bootstrap can work when n\≪ p, and rates of\napproximation (in Kolmogorov distance) have been obtained with only logarithmic\ndependence in p. Nevertheless, one of the challenging aspects of this setting\nis that established rates tend to scale like n-1/6 as a function of n.\n The main purpose of this paper is to demonstrate that improvement in rate is\npossible when extra model structure is available. Specifically, we show that if\nthe coordinate-wise variances of the observations exhibit decay, then a nearly\nn-1/2 rate can be achieved, independent of p. Furthermore, a surprising\naspect of this dimension-free rate is that it holds even when the decay is very\nweak. Lastly, we provide examples showing how these ideas can be applied to\ninference problems dealing with functional and multinomial data.\n