• False coverage rate - adjusted smoothed bootstrap simultaneous confidence intervals for selected parameters

      Sun, Jing; Department of Biostatistics and Epidemiology (Augusta University, 2020-05)
      Many modern applications refer to a large number of populations with high dimensional parameters. Since there are so many parameters, researchers often draw inferences regarding the most significant parameters, which are called selected parameters. Benjamini and Yekutieli (2005) proposed the false coverage-statement rate (FCR) method for multiplicity correction when constructing confidence intervals for only selected parameters. FCR for the confidence interval method is parallel to the concept of the false discovery rate for multiple hypothesis testing. In practice, we typically construct FCR-adjusted approximate confidence intervals for selected parameters either using the bootstrap method or the normal approximation method. However, these approximated confidence intervals show higher FCR for small and moderate sample sizes. Therefore, we suggest a novel procedure to construct simultaneous confidence intervals for the selected parameters by using a smoothed bootstrap procedure. We consider a smoothed bootstrap procedure using a kernel density estimator. A pertinent problem associated with the smoothed bootstrap approach is how to choose the unknown bandwidth in some optimal sense. We derive an optimal choice for the bandwidth and the resulting smoothed bootstrap confidence intervals asymptotically to give better control of the FCR than its competitors. We further show that the suggested smoothed bootstrap simultaneous confidence intervals are FCR-consistent if the dimension of data grows no faster than N^3/2. Finite sample performances of our method are illustrated based on empirical studies. Through these empirical studies, it is shown that the proposed method can be successfully applied in practice.