Characterizations of Multinormality and Corresponding Tests of Fit, Including for Garch Models

Abstract

We provide novel characterizations of multivariate normality that incorporate both the characteristic function and the moment generating function, and we employ these results to construct a class of affine invariant, consistent and easy-to-use goodness-of-fit tests for normality. The test statistics are suitably weighted L 2 -statistics, and we provide their asymptotic behavior both for i.i.d. observations as well as in the context of testing that the innovation distribution of a multivariate GARCH model is Gaussian. We also study the finite-sample behavior of the new tests and compare the new criteria with alternative existing tests.

Testing for normality in any dimension based on a partial differential equation involving the moment generating function

“…Since β and γ are related by β = γ −1/2 , this corresponds to letting γ tend to infinity. The same linear combination 2b 1,d + 3 b 1,d also showed up as a limit statistic in [18]. Notice that, in the univariate case, the limit statistic figuring in Theorem 4.1 is nothing but three times squared sample skewness.…”

“…By Proposition 10.2 of [18], the integral is of order O P (Γ d+3 n ) exp(Γ 2 n /γ) (notice that 1 + γ in that paper corresponds to (our) γ). From display (10.6) and display (10.7) of [18] we have Γ n = O P ( √ log n) and exp Γ 2 n /γ = n 2/γ · (log n) (d−2)/γ O P (1). Since γ > 2, it follows that e t ⊤ X j · 1 √ n n j=1 X j .…”

“…By generalizing a characterization of normality involving both the characteristic and the moment generating function (see [35]) to the multivariate case, [18] proposed to base a test of H 0 on the weighted L 2 -statistic…”

A test for Gaussianity in Hilbert spaces via the empirical characteristic functional

“…It is thus not surprising that a myriad of tests for multivariate normality have been proposed. For some more recent approaches, see for example, Arcones (2007), Doornik and Hansen (2008), Ebner (2012), , Henze, Jiménez-Gamero, and Meintanis (2019), Henze and Visagie (2019), Kankainen, Taskinen, and Oja (2007), Pudelko (2005), Székely and Rizzo (2005), Thulin (2014), Villaseñor-Alva and Estrada (2009), and Voinov, Pya, Makarov, and Voinov (2016).…”

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

“…There is a continuing interest in this testing problem, as evidenced by a multitude of papers. The proposed tests may be roughly classified as follows: Arcones (2007), Baringhaus and Henze (1988), Henze and Wagner (1997), Henze and Zirkler (1990), Pudelko (2005), and Tenreiro (2009) consider tests based on the empirical characteristic function, while , Henze, Jiménez-Gamero, and Meintanis (2019), and Henze and Visagie (2019) employ the empirical moment generating function. A classical (and still popular) approach is to consider measures of multivariate skewness and kurtosis (see Doornik and Hansen, 2008;Kankainen, Taskinen, and Oja, 2007;Malkovich and Afifi, 1973;Mardia, 1970;Mardia, 1974;Móri, Rohatgi, and Székely, 1993), as supposedly diagnostic tools with regard to the kind of deviation from normality when this hypothesis has been rejected, but the deficiencies of those measures in this regard have been clearly demonstrated (see Baringhaus and Henze, 1991;Baringhaus and Henze, 1992;Henze, 1994aHenze, , 1994bHenze, , 1997b.…”