Power in High‐Dimensional Testing Problems

Abstract

Fan, Liao, and Yao (2015) recently introduced a remarkable method for increasing the asymptotic power of tests in high‐dimensional testing problems. If applicable to a given test, their power enhancement principle leads to an improved test that has the same asymptotic size, has uniformly non‐inferior asymptotic power, and is consistent against a strictly broader range of alternatives than the initially given test. We study under which conditions this method can be applied and show the following: In asymptotic regimes where the dimensionality of the parameter space is fixed as sample size increases, there often exist tests that cannot be further improved with the power enhancement principle. However, when the dimensionality of the parameter space increases sufficiently slowly with sample size and a marginal local asymptotic normality (LAN) condition is satisfied, every test with asymptotic size smaller than 1 can be improved with the power enhancement principle. While the marginal LAN condition alone does not allow one to extend the latter statement to all rates at which the dimensionality increases with sample size, we give sufficient conditions under which this is the case.

On Sobolev tests of uniformity on the circle with an extension to the sphere

“…We propose here a new test that uses together the data-driven selection of M explained above as well as the test φ (n) MRV . The idea underpinning the new test finds its roots in the power enhancement principle studied in [11] and [28]. The new test φ (n) is based on a combination of the two tests φ…”

A Dynamic Network of Arbitrage Characteristics

“…Therefore, a power enhanced test should be developed to strengthen the power of Wald tests and to detect the most relevant characteristics among a characteristic zoo included in h(X i ). Kock and Preinerstorfer (2019) [21] illustrated that if the number of coefficients diverges as the number of observations approaches infinity, the standard Wald test is power enhanceable. Meanwhile, Fan, Liao and Yao (2015) [10] proposed a power enhanced test after showing that if true coefficients have a sparse structure, the traditional Wald test has very low power.…”

Methods for High-Dimensional Data in Econometrics

“…However, in high-dimensional settings such tests are known to have low power against specific alternatives (Fan et al, 2015). This has provoked recent interest in constructing tests that direct power towards specific alternatives of interest (Fan et al, 2015;Kock and Preinerstorfer, 2019). In the chapter, I focus in particular on testing against specific alternatives that can be described as a (not necessarily convex) cone.…”

“…By rejecting if at least one of these tests rejects, the resulting test has asymptotic size equal to the initial test, and may be consistent against a strictly larger set of alternatives. The existence of power enhancement tests has recently been discussed by Kock and Preinerstorfer (2019).…”

“…This has provoked a recent interest in constructing tests that direct power towards parameter subspaces of interest (see e.g. Fan et al, 2015;Kock and Preinerstorfer, 2019).…”