More good news on the HKM test for multivariate reflected symmetry about an unknown centre

Abstract

We revisit the problem of testing for multivariate reflected symmetry about an unspecified point. Although this testing problem is invariant with respect to full-rank affine transformations, among the hitherto few proposed tests only the test studied in [12] respects this property. We identify a measure of deviation ∆ (say) from symmetry associated with the test statistic T n (say), and we obtain the limit normal distribution of T n as n → ∞ under a fixed alternative to symmetry. Since a consistent estimator of the variance of this limit normal distribution is available, we obtain an asymptotic confidence interval for ∆. The test, when applied to a classical data set, strongly rejects the hypothesis of reflected symmetry, although other tests even do not object against the much stronger hypothesis of elliptical symmetry.

Testing multivariate normality by zeros of the harmonic oscillator in characteristic function spaces

“…Since |CS ± (t,X j )| ≤ 2 and | j (t) + j (t)| ≤ 2||t|| 2 ||Δ j || 2 , the Cauchy-Schwarz inequality yields Proof of Theorem 11. The proof is similar to the proof of theorem 5 of Henze and Mayer (2020) and will therefore only be sketched. From ( 26), (23), and (30), we have 2 a = ∑ 4 i,j=1 i,j a , where i,j a = 4 ∫ ∫ L i,j (s, t)z(s)z(t)w a (s)w a (t) dsdt, and L i,j (s,t) is given in (32).…”

Testing multivariate normality by zeros of the harmonic oscillator in characteristic function spaces

“…The proof is similar to the proof of Theorem 5 of [33] and will therefore only be sketched. From ( 26), ( 23) and ( 30), we have σ 2 a = 4 i,j=1 σ i,j a , where σ i,j a = 4 L i,j (s, t)z(s)z(t)w a (s)w a (t) dsdt, and L i,j (s, t) is given in (32).…”

Testing semiparametric model-equivalence hypotheses based on the characteristic function

“…The general theory and specific conditions under which (2.4)-(2.6) hold true have been presented by using a Hilbert space approach, in Baringhaus et al (2017) and Henze and Mayer (2020), for distributional homogeneity and symmetry-equivalence testing, respectively.…”

“…On the other hand the problem of testing "model equivalence" has only recently been considered in a more general context, beyond that of testing about a single parameter. We refer to Dette and Munk (1998), Janssen (2000), Freitag et al (2007), Baringhaus et al (2017), Dette et al (2018), Möllenhoff et al (2019), and Henze and Mayer (2020), for testing model equivalence (or "neighborhood-of-model validation") in various settings.…”