In large-scale problems, it is common practice to select important parameters by a procedure such as the BH procedure (Benjamini and Hochberg, 1995) and construct confidence intervals (CIs) for further investigation while the false coverage-statement rate (FCR) for the CIs is controlled at a desired level. Although the well-known BY CIs (Benjamini and Yekutieli, 2005) control the FCR, they are uniformly inflated. Weinstein et al. (2013) recently developed shorter CIs than BY CIs for those selected parameters with large estimated values. However, that method suffers from an uncontrollable number of intervals that are not compatible with the selection results, and some of the CIs are longer than BY CIs for those selected parameters with small estimated values.

In this talk, we propose two new procedures to construct CIs for selected parameters that are uniformly shorter than BY CIs with the FCR controlled asymptotically. Furthermore, the first method produces CIs that are compatible with the selection results, and the second method allows only a prefixed proportion of CIs that are incompatible with the selection results. The shorter CIs are achieved either by selecting a reduced number of “significant” parameters by the BH procedure or by forcing some of the significant parameters by the BH procedure to be “insignificant.”  The proposed procedures are strongly recommended if one’s primary interest is to assess the biological relevance of the selected “significant” parameters through more-precise CIs while being willing to potentially miss a small portion of “significant” discoveries or compatible CIs.

Prof. Xinping Cui