Abstract
We propose two families of tests for the classical goodness-of-fit problem to univariate normality. The new procedures are based on \(L^2\)-distances of the empirical zero-bias transformation to the empirical distribution or the normal distribution function. Weak convergence results are derived under the null hypothesis, under contiguous as well as under fixed alternatives. A comparative finite-sample power study shows the competitiveness to classical procedures.
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Acknowledgements
The authors thank Norbert Henze for useful comments and also express their gratitude to three anonymous referees for careful reading and suggestions that helped improve the article.
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Appendices
A Preliminary results concerning the weight functions
We first prove that the density function of a centred normal distribution is an admissible weight function. Then, we give a general result for the asymptotic behaviour of integral terms involving weight functions of the type we consider. In the whole section, we adopt the setting and notation from Sect. 2.
Lemma 2
The functions \(\omega _a(s) = (2 \pi a)^{-1/2} \exp (- s^2 / (2 a))\), \(s \in \mathbb {R}\), \(a > 0\), satisfy the weight function conditions stated in Sect. 2.
Proof
The only non-trivial statement is that \(\omega _a\) satisfies (8). Let \(0< \varepsilon < 1/8\) be arbitrary. In the case \(|S_n^{-1} - 1| \le \varepsilon \) and \(|{\overline{X}}_n| / S_n \le \varepsilon \), a Taylor expansion gives
where \(\big |\xi _n(s) - s\big | \le \big | (s - {\overline{X}}_n) / S_n - s \big | \le (|s| + 1) / 8\). Consequently,
from which we conclude
Combining this with (24),
As \(\varepsilon \) was arbitrary, the claim follows from the boundedness in probability of \(\sqrt{n} \big ( S_n^{-1} - 1 \big )\) and \(\sqrt{n} \big ( {\overline{X}}_n / S_n \big )\). \(\square \)
Lemma 3
Let \({\mathcal {U}}_n\) be a random element of \({\mathcal {H}}\), \(n \in \mathbb {N}\), such that \(\left||\sqrt{n} \, {\mathcal {U}}_n \right||_{{\mathcal {H}}} = O_\mathbb {P}(1)\). Then,
If in addition \(\sup _{s \, \in \, \mathbb {R}} \big | {\mathcal {U}}_n(s) \big | \le C\)\(\mathbb {P}\)-a.s. for each \(n \in \mathbb {N}\) and some \(C > 0\),
Proof
By Hölder’s inequality (\(p = q = 2\)) and Slutsky’s lemma
where we used the assumption on \({\mathcal {U}}_n\) and the fact that (8) implies
The second claim also follows from Hölder’s inequality (\(p = 3/2, \, q = 3\)) and (8) since
B Asymptotic expansions
We adopt the setting from Sect. 2, that is, we let \(X, X_1, X_2, \ldots \) be iid. random variables with distribution function F and \(\mathbb {E}[X^2] < \infty \) as well as \(\mathbb {E}X = 0\), \(\mathbb {V}(X) = 1\). The following lemma collects basic facts about a quantity closely related to the empirical zero-bias distribution function.
Lemma 4
The function
is a continuous distribution function for each \(n \in \mathbb {N}\) (and on a set of measure one). Furthermore,
\(\mathbb {P}\)-a.s., as \(n \rightarrow \infty \), and
Proof
We fix \(n \in \mathbb {N}\) and notice that
Using the first representation when integrating over \(({\overline{X}}_n, \infty )\) and the second for \((- \infty , {\overline{X}}_n]\), we obtain
Now, we conclude from
that \({\widehat{F}}_n^X\) is a continuous distribution function. By the strong law of large numbers and the almost sure convergence \(({\overline{X}}_n, S_n^2) \rightarrow (0,1)\), we have
\(\mathbb {P}\)-a.s., as \(n \rightarrow \infty \), for any fixed \(s \in \mathbb {R}\). The proof of the classical Glivenko–Cantelli theorem applies to \({\widehat{F}}_n^X\) which yields (25). For the last claim, we set
Straightforward calculations using Tonelli’s theorem and the integrability condition (7) give
so \(\left||A_n \right||_{{\mathcal {H}}}^2 = o_{\mathbb {P}}(1)\). Together with \(\sqrt{n} \, {\overline{X}}_n = O_\mathbb {P}(1)\) and Slutsky’s lemma, this implies (26). \(\square \)
We proceed by proving further asymptotic expansions of the same type as (26).
Lemma 5
Assume, in addition to the above prerequisites, that X has a continuously differentiable density function p with
We have
and, with \(F^X\) as in Lemma 1,
Moreover,
which reads as \(\sqrt{n} \, S_n^2 \, \varPhi (s) \approx n^{- 1/2} \sum _{j=1}^{n} X_j^2 \, \varPhi (s)\) when \(\mathbb {P}^X = {\mathcal {N}}(0, 1)\) (cf. Theorem 1).
Proof
By Taylor’s theorem,
where
and \(\big | \xi _n(s) - s \big | \le \big | (s - {\overline{X}}_n) / S_n - s \big |\). Condition (7) assures that \(R_n \in {\mathcal {H}}\)\(\mathbb {P}\)-a.s. and with \(\sqrt{n} \big (S_n^{-1} - 1\big ) = O_\mathbb {P}(1)\), \(\sqrt{n} \, {\overline{X}}_n = O_\mathbb {P}(1)\) we conclude
Now, let \(0< \varepsilon < 1\) be arbitrary. In the case \(\big | S_n^{-1} - 1 \big | \le \varepsilon \) and \(\big | {\overline{X}}_n \big | / S_n \le \varepsilon \), we have
where
and \(\big |{\widetilde{\xi }}_n(s) - s \big | \le \big | (s - {\overline{X}}_n) / S_n - s \big | \le |s| + 1\). Using \(\big ( {\widetilde{\xi }}_n(s) \big )^2 \le (2 |s| + 1)^2\), we get
Since \(\sqrt{n} \big (S_n^{-1} - 1\big )\) and \(\sqrt{n} \big ( {\overline{X}}_n / S_n \big )\) are bounded in probability and \(\varepsilon \) was arbitrary, \(||{\widetilde{R}}_n||_{{\mathcal {H}}}^2 = o_\mathbb {P}(1)\). The last claim of the lemma follows from
\(\square \)
C Proof of the limit relations in (11) and (12)
We will give the proof of (11), using the notation from Sect. 2. The limit in (12) is obtained by the same argument. Set
as well as
Splitting the integral in the definition of \(G_{n, a}^{(1)}\) (see (5)) into integrals over \((- \infty , 0]\) and \((0, \infty )\), simple changes of variable yield
Since the integrals on the right-hand side of the above equation are Laplace transforms, and since we have
and
an Abelian theorem for the Laplace transform, as stated on p. 182 in the book by Widder (1959) (see also Baringhaus et al. 2000), implies the claim. Here, \(\varGamma (1/2) = \sqrt{\pi }\) denotes the Gamma function evaluated at 1 / 2.
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Betsch, S., Ebner, B. Testing normality via a distributional fixed point property in the Stein characterization. TEST 29, 105–138 (2020). https://doi.org/10.1007/s11749-019-00630-0
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DOI: https://doi.org/10.1007/s11749-019-00630-0