On combining the zero bias transform and the empirical
characteristic function to test normality

Ebner, Bruno

doi:10.30757/alea.v18-38

Cited by 7 publications

(4 citation statements)

References 39 publications

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“…As a starting point, we conjecture that for a sequence

{false(n_{d} false)}_{d \in ℕ}

, where n d ≥ d + 1 and

n_{d} = o true(true(\frac{2 a}{2 a + 1} true)^{prefix- \frac{d}{2}} true)

, we have under H 0 as d → ∞

\begin{matrix} alignleft \\ align-1 (\frac{a}{π})^{\frac{d}{2}} \frac{T_{n_{d}, a}}{d} \overset{italica.s.}{\to} 1 . & align-2 \end{matrix}

Finally, it would be of interest to consider a related family of test statistics, which is given by

S_{n, a} = n \int_{ℝ^{d}} {‖ \nabla ψ_{n} (t) + t ψ_{n} (t) ‖}_{ℂ}^{2} w_{a} (t) d t .

Thus, the theoretical CF in T n , a has been replaced by the empirical counterpart. Note that in the univariate case, this family is extensively studied in Ebner (2021), but the generalization to higher dimensions is still open. We conjecture that similar results as derived in Sections 2–4 hold for S n , a .…”

Section: Discussionmentioning

confidence: 99%

“…Analogously to Henze (1990) and Ebner (2021), we can now approximate the distribution of T ∞,a by that of a member of the system of Pearson distributions which has the same first four moments as T ∞,a . To this end, we used the statistical software R, see R Core Team (2019), and the package PearsonDS, see Becker & Klößner (2017).…”

Section: The Limit Null Distributionmentioning

confidence: 99%

“…In the spirit of the Stein–Tikhomirov method, see Formanov & Formanova (2013) and Arras et al (2016), and hence using the CFs

true{\exp false(i t^{⊺} x false), t \in ℝ^{d} true}

as test functions, a simple calculation shows the equivalence of the Stein equation to the initial value problem in (). In the case d = 1, the same initial value problem was motivated by a fixed point of the zero‐bias transform in Ebner (2021). For more information on the zero‐bias transform, see Goldstein & Reinert (1997) and Shevtsova (2013).…”

Section: Introductionmentioning

confidence: 99%

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Testing normality in any dimension by Fourier methods in a multivariate Stein equation

2021

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We study a novel class of affine-invariant and consistent tests for multivariate normality. The tests are based on a characterization of the standard d-variate normal distribution by way of the unique solution of an initial value problem connected to a partial differential equation, which is motivated by a multivariate Stein equation. The test criterion is a suitably weighted L 2 -statistic. We derive the limit distribution of the test statistic under the null hypothesis as well as under contiguous and fixed alternatives to normality. A consistent estimator of the limiting variance under fixed alternatives, as well as an asymptotic confidence interval of the distance of an underlying alternative with respect to the multivariate normal law, is derived. In simulation studies, we show that the tests are strong in comparison with prominent competitors and that the empirical coverage rate of the asymptotic confidence interval converges to the nominal level. We present a real data example and also outline topics for further research.

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“…As a starting point, we conjecture that for a sequence

{false(n_{d} false)}_{d \in ℕ}

, where n d ≥ d + 1 and

n_{d} = o true(true(\frac{2 a}{2 a + 1} true)^{prefix- \frac{d}{2}} true)

, we have under H 0 as d → ∞

\begin{matrix} alignleft \\ align-1 (\frac{a}{π})^{\frac{d}{2}} \frac{T_{n_{d}, a}}{d} \overset{italica.s.}{\to} 1 . & align-2 \end{matrix}

Finally, it would be of interest to consider a related family of test statistics, which is given by

S_{n, a} = n \int_{ℝ^{d}} {‖ \nabla ψ_{n} (t) + t ψ_{n} (t) ‖}_{ℂ}^{2} w_{a} (t) d t .

Section: Discussionmentioning

confidence: 99%

Section: The Limit Null Distributionmentioning

confidence: 99%

“…In the spirit of the Stein–Tikhomirov method, see Formanov & Formanova (2013) and Arras et al (2016), and hence using the CFs

true{\exp false(i t^{⊺} x false), t \in ℝ^{d} true}

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Testing normality in any dimension by Fourier methods in a multivariate Stein equation

2021

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show abstract

“…An alternative test of univariate normality based on T from Example 1 is proposed in Ebner (2021), but in this case test functions of the form {g t (x) = exp(itx) : t ∈ R} (i.e. related to characteristic functions) are used.…”

Section: Composite Goodness-of-fit Tests From Stein Operatorsmentioning

confidence: 99%

Stein's Method Meets Statistics: A Review of Some Recent Developments

Anastasiou¹,

Barp²,

Briol³

et al. 2021

Preprint

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Stein's method is a collection of tools for analysing distributional comparisons through the study of a class of linear operators called Stein operators. Originally studied in probability, Stein's method has also enabled some important developments in statistics. This early success has led to a high research activity in this area in recent years. The goal of this survey is to bring together some of these developments in theoretical statistics as well as in computational statistics and, in doing so, to stimulate further research into the successful field of Stein's method and statistics. The topics we discuss include: explicit error bounds for asymptotic approximations of estimators and test statistics, a measure of prior sensitivity in Bayesian statistics, tools to benchmark and compare sampling methods such as approximate Markov chain Monte Carlo, deterministic alternatives to sampling methods, control variate techniques, and goodness-of-fit testing.

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On the eigenvalues associated with the limit null distribution of the Epps-Pulley test of normality

Ebner

Henze

2022

Stat Papers

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The Shapiro–Wilk test (SW) and the Anderson–Darling test (AD) turned out to be strong procedures for testing for normality. They are joined by a class of tests for normality proposed by Epps and Pulley that, in contrast to SW and AD, have been extended by Baringhaus and Henze to yield easy-to-use affine invariant and universally consistent tests for normality in any dimension. The limit null distribution of the Epps–Pulley test involves a sequences of eigenvalues of a certain integral operator induced by the covariance kernel of a Gaussian process. We solve the associated integral equation and present the corresponding eigenvalues.

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On combining the zero bias transform and the empirical characteristic function to test normality

Cited by 7 publications

References 39 publications

Testing normality in any dimension by Fourier methods in a multivariate Stein equation

Testing normality in any dimension by Fourier methods in a multivariate Stein equation

Stein's Method Meets Statistics: A Review of Some Recent Developments

On the eigenvalues associated with the limit null distribution of the Epps-Pulley test of normality

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