Testing for normality with neural networks

Abstract

In this paper, we treat the problem of testing for normality as a binary classification problem and construct a feedforward neural network that can act as a powerful normality test. We show that by changing its decision threshold, we can control the frequency of false non-normal predictions and thus make the network more similar to standard statistical tests. We also find the optimal decision thresholds that minimize the total error probability for each sample size. The experiments conducted on the samples with no more than 100 elements suggest that our method is more accurate and more powerful than the selected standard tests of normality for almost all the types of alternative distributions and sample sizes. In particular, the neural network was the most powerful method for testing normality of the samples with fewer than 30 elements regardless of the alternative distribution type. Its total accuracy increased with the sample size. Additionally, when the optimal decision-thresholds were used, the network was very accurate for larger samples with 250–1000 elements. With AUROC equal to almost 1, the network was the most accurate method overall. Since the normality of data is an assumption of numerous statistical techniques, the network constructed in this study has a very high potential for use in everyday practice of statistics, data analysis and machine learning.

Appendices

Appendices

A Standard statistical tests of normality

Throughout this Appendix, ${\mathbf {x}}=[x_1, x_2, \ldots , x_n]$ will denote a sample drawn from the distribution whose normality we want to test. The same holds for other Appendices.

2.1 A.1 The Shapiro–Wilk Test (SW)

Let ${\mathbf {b}} = [b_1, b_2, \ldots , b_n]^T$ denote the vector of the expected values of order statistics of independent and identically distributed random variables sampled from the standard normal distribution. Let ${\mathbf {V}}$ denote the corresponding covariance matrix.

The intuition behind the SW test is as follows. If a random variable X follows normal distribution $N(\mu ,\sigma ^2)$, and $Z\sim N(0, 1)$, then $X=\mu +\sigma Z$ [49]. For the ordered random samples ${\mathbf {x}}=[x_1,x_2,\ldots ,x_n]\sim N(\mu , \sigma ^2)$ and ${\mathbf {z}}=[z_1,z_2,\ldots ,z_n]\sim N(0,1)$, the best linear unbiased estimate of $\sigma$ is [49]:

$$\begin{aligned} {\hat{\sigma }}=\frac{{\mathbf {b}}^T{\mathbf {V}}^{-1}{\mathbf {z}}}{{\mathbf {b}}^T{\mathbf {V}}^{-1}{\mathbf {b}}} \end{aligned}$$

(12)

In that case, ${\hat{\sigma }}^2$ should be equal to the usual estimate of variance $\text {S}({\mathbf {x}})^2$:

$$\begin{aligned} \text {S}({\mathbf {x}})^2=\sum _{i=1}^{n}(x_i-\overline{x})^2 \end{aligned}$$

(13)

The value of the test statistic, W, is a scaled ratio of ${\hat{\sigma }}^2$ and $\text {S}({\mathbf {x}})^2$:

$$\begin{aligned} W = \frac{R^4}{C^2}\frac{{\hat{\sigma }}^2}{\text {S}({\mathbf {x}})^2} =\frac{\left( \sum _{i=1}^{n}a_ix_i\right) ^2}{\sum _{i=1}^{n}\left( x_i - \overline{x}\right) ^2} \end{aligned}$$

(14)

where:

$$\begin{aligned} R^2 = {\mathbf {b}}^T{\mathbf {V}}^{-1}{\mathbf {b}},\quad C^2 = {\mathbf {b}}^T{\mathbf {V}}^{-1}{\mathbf {V}}^{-1}{\mathbf {b}} \end{aligned}$$

(15)

and

$$\begin{aligned} \mathbf{a } = [a_1, a_2, \ldots , a_n] = \frac{\mathbf{b }^T\mathbf{V }^{-1}}{\left( {\mathbf{b }^T\mathbf{V }^{-1}\mathbf{V }^{-1}\mathbf{b }}\right) ^{1/2}} \end{aligned}$$

(16)

The range of W is [0, 1], with higher values indicating stronger evidence in support of normality. The original formulation required use of the tables of the critical values of W [52] at the most common levels of $\alpha$ and was limited to smaller samples with $n \in [3, 20]$ elements because the values of ${\mathbf {b}}$ and ${\mathbf {V}}^{-1}$ were known only for small samples at the time [6]. Royston [98] extended the upper limit of n to 2000 and presented a normalizing transformation algorithm suitable for computer implementation. The upper limit was further improved by Royston [99] who formulated algorithm AS 194 which allowed the test to be used for the samples with $n \in [3, 5000]$.

2.2 A.2 The Shapiro–Francia Test (SF)

The Shapiro–Francia test is a simplification of the SW test. It was initially proposed for the larger samples, for which the matrix ${\mathbf {V}}$ in the SW test was unknown [61]. By assuming that the order statistics are independent, we can substitute the identity matrix ${\mathbf {I}}$ for ${\mathbf {V}}$ in Equation (16) to obtain the Shapiro-Francia test statistic:

$$\begin{aligned} SF = \frac{\left( \sum _{i=1}^{n}a_ix_i\right) ^2}{\sum _{i=1}^{n}\left( x_i - \overline{x}\right) ^2} \end{aligned}$$

(17)

where:

$$\begin{aligned} \mathbf{a } = [a_1, a_2, \ldots , a_n] = \frac{\mathbf{b }^T}{\left( {\mathbf{b }^T\mathbf{b }}\right) ^{1/2}} \end{aligned}$$

(18)

Then, SF is actually the squared Pearson correlation between $\mathbf{a }$ and ${\mathbf {x}}$, i.e., the $R^2$ of the regression of ${\mathbf {x}}$ on $\mathbf{a }$ [2]. Only the expected values of the order statistics are needed to conduct this test. They can be calculated using algorithm AS 177 proposed by Royston [100].

As in the SW test, ${\mathbf {x}}$ is assumed to be ordered.

2.3 A.3 The Lilliefors Test (LF)

This test was introduced independently by Lilliefors [56] and van Soest [101]. The LF test checks if the given sample ${\mathbf {x}}$ comes from a normal distribution whose parameters $\mu$ and $\sigma$ are taken to be the sample mean ($\overline{x}$) and standard deviation ($\text {sd}({\mathbf {x}})$). It is equivalent to the Kolmogorov–Smirnov test of goodness-of-fit [102] for those particular choices of $\mu$ and $\sigma$ [13]. The LF test is conducted as follows. If the sample at hand comes from $N(\overline{x},\text {sd}({\mathbf {x}})^2)$, then its transformation ${\mathbf {z}}=[z_1,z_2,\ldots ,z_n]$, where:

$$\begin{aligned} z_i = \frac{x_i-\overline{x}}{\text {sd}({\mathbf {x}})}\quad (i=1,2,\ldots ,n) \end{aligned}$$

(19)

should follow N(0, 1). The difference between the EDF of ${\mathbf {z}}$, $edf_{{\mathbf {z}}}$, and the CDF of N(0, 1), $\Phi$, quantifies how well the sample ${\mathbf {x}}$ fits the normal distribution. In the LF test, that difference is calculated as follows

$$\begin{aligned} D = \max \limits _{i=1}^{n}\left| edf_{{\mathbf {z}}}(z_i)-\Phi (z_i)\right| \end{aligned}$$

(20)

Higher values of D indicate greater deviation from normality.

2.4 A.4 The Anderson–Darling Test (AD)

Whereas the Lilliefors test focuses on the largest difference between the sample’s EDF and the CDF of the hypothesized model distribution, the AD test calculates the expected weighted difference between those two [58], with the weighting function designed to make use of the specific properties of the model. The AD statistic is:

$$\begin{aligned} A= n\int _{-\infty }^{+\infty }\left[ F^*(x)-edf(x)\right] ^2\psi (F^*(x))dF^*(x) \end{aligned}$$

(21)

where $\psi$ is the weighting function, edf is the sample’s EDF, and $F^*$ is the CDF of the distribution we want to test. When testing for normality, the weighting function is chosen to be sensitive to the tails [13, 58]:

$$\begin{aligned} \psi (t)=\left[ t(1-t)\right] ^{-1} \end{aligned}$$

(22)

Then, the statistic (21) can be calculated in the following way [2, 103]:

$$\begin{aligned} A = -n -\frac{1}{n}\sum _{i=1}^{n}\left[ 2i-1\right] \left[ \ln \Phi (z_{(i)}) + \ln \left( 1-\Phi (z_{(n-i+1)})\right) \right] \end{aligned}$$

(23)

where $[z_{(1)},z_{(2)},\ldots ,z_{(n)}]$ ($z_{(i)} \le z_{(i+1)}, i=1,2,\ldots ,n-1$) is the ordered permutation of ${\mathbf {z}}=[z_1,z_2,\ldots ,z_n]$ that is obtained from ${\mathbf {x}}$ as in the LF test. The p-values are computed from the modified statistic [103, 104]:

$$\begin{aligned} A(1+0.75/n+2.25/n^2) \end{aligned}$$

(24)

Larger values of the AD statistic indicate stronger arguments in favor of non-normality.

2.5 A.5 The Cramér–von Mises Test (CVM)

As noted by Wijekularathna et al. [13], the AD test is a generalization of the Cramér–von Mises test [62,63,64,65]. When $\psi (\cdot )=1$, the AD test’s statistic reduces to that of the CVM test:

$$\begin{aligned} C= n\int _{-\infty }^{+\infty }\left[ F^*(x)-edf(x)\right] ^2dF^*(x) \end{aligned}$$

(25)

Because $\psi (\cdot )$ takes into account the specific properties of the model distribution, the AD test may be more sensitive than the CVM test [13]. As is the case with the AD test, larger values of the CVM statistic (25) are more compatible with departures from normality.

2.6 A.6 The Jarque–Bera Test (JB)

The Jarque–Bera test [59, 60], checks how much the sample’s skewness ($\sqrt{\beta _1}$) and kurtosis ($\beta _2$) match those of normal distributions. Namely, for each normal distribution it holds that $\sqrt{\beta _1}=0$ and $\beta _2=3$. The statistic of the test, as originally defined by Jarque and Bera [59], is computed as follows:

$$\begin{aligned} J = \frac{\left( \sqrt{\beta _1}\right) ^2}{6/n} + \frac{(\beta _2-3)^2}{24/n} \end{aligned}$$

(26)

We see that higher values of J indicate greater deviation from the skewness and kurtosis of normal distributions. The same idea was examined by Bowman and Shenton [105]. The asymptotic expected values of the estimators of skewness and kurtosis are 0 and 3, while the asymptotic variances are 6/n and 24/n for the sample of size n [105]. The J statistic is then a sum of two asymptotically independent standardized normals. However, the estimator of kurtosis slowly converges to normality, which is why the original statistic is not useful for small and medium-sized samples [106]. Urzua [106] adjusted the statistic by using the exact expressions for the means and variances of the estimators of skewness and kurtosis:

$$\begin{aligned} J = n\left( \frac{\left( \sqrt{\beta _1}\right) ^2}{c_1} + \frac{(\beta _2-c_2)^2}{c_3}\right) \end{aligned}$$

(27)

where:

$$\begin{aligned} c_1= & {} \frac{6(n-2)}{(n+1)(n+3)}\\ c_2= & {} \frac{3(n-1)}{n+1}\\ c_3= & {} \frac{24n(n-2)(n-3)}{(n+1)^2(n+3)(n+5)} \end{aligned}$$

which allowed the test to be applied to smaller samples.

2.7 A.7 The D’Agostino–Pearson Test (DP)

The DP test is a combination of the individual skewness and kurtosis tests of normality [66, 67]. Its statistic is:

$$\begin{aligned} K^2 = Z_1^2 + Z_2^2 \end{aligned}$$

(28)

where $Z_1$ and $Z_2$ are normal approximations to the skewness and kurtosis. The exact formulas for $Z_1$ and $Z_2$ can be found in D’Agostino et al. [67]. Under the null hypothesis of normality, $K^2$ has an approximate $\chi ^2$ distribution with two degrees of freedom [67].

The rationale behind the DP test is that the statistic combining both the sample skewness and kurtosis can detect departures from normality in terms of both moments, unlike the tests based on only one standardized moment.

B Robustified tests of normality

1.1 B.1 The Robustified Jarque–Bera Tests (RJB)

The statistic of the classical JB test, introduced in Section A.6, is based on the classical estimators of the first four moments. Those estimators are sensitive to outliers, which is why the JB test is not robust. The robustified Jarque–Bera tests are obtained when the non-robust moment estimators are replaced with their robust alternatives. Since there are multiple ways to define a robust estimator of a moment, there is not one, but a plethora of the RJB tests, which were defined by Stehlík et al. [70] and named the RT tests.

Let ${\mathbf {x}}=[x_1, x_2, \ldots , x_n]$ be the ordered sample under consideration .Let $T_0$ be the sample mean, $T_1$ its median, $T_2 \equiv T_{(2, s)}=\frac{1}{n-2s}\sum _{i=s+1}^{n-s}x_i$ the trimmed sample mean, and $T_3=\text {median}_{i\le j}\{(x_i+x_j)/2\}$ the Lehman–Hodges pseudo-median of ${\mathbf {x}}$. Except $T_0$, all the other location estimators are not sensitive to outliers. Then, the robust central moment estimators can be defined as follows:

$$\begin{aligned} M_j(r, T_k) = \frac{1}{n-2r}\sum _{i=r+1}^{n-r}\varphi _j(x_i-T_k) \end{aligned}$$

(29)

with

$$\begin{aligned} \varphi _j(z)={\left\{ \begin{array}{ll} \sqrt{\frac{\pi }{2}}|z|,&{} j=0 \\ z^j,&{} j \ge 1 \end{array}\right. } \end{aligned}$$

(30)

With the notation set up, the general statistic of the robustified Jarque–Bera tests can be defined as follows [16, 70]:

$$\begin{aligned} \begin{aligned} RJB =&\frac{n}{C_1}\left( \frac{M_{j_1}^{a_1}(r_1, T_{i_1})}{M_{j_2}^{a_2}(r_2, T_{i_2})}-K_1\right) ^2 \\ +&\frac{n}{C_2}\left( \frac{M_{j_3}^{a_3}(r_3, T_{i_3})}{M_{j_4}^{a_4}(r_4, T_{i_4})}-K_2\right) ^2 \end{aligned} \end{aligned}$$

(31)

With the right choice of the parameters ($C_1, j_1, a_1 \ldots$), the RJB statistic can be reduced to that of the classical Jarque–Bera test.

For the purpose of this study, we used the same four RJB tests that Stehlík et al. [16] evaluated in their study: MMRT$_1$, MMRT$_2$, TTRT$_1$, TTRT$_2$. We refer readers to Stehlík et al. [16] for more details on those particular tests’ parameters.

1.2 B.2 The Robustified Lin–Mudholkar Test (RLM)

The Lin–Mudholkar test (LM) is based on the fact that the estimators of mean and variance are independent if and only if the sample at hand comes from a normal distribution [50]. The correlation coefficient between $\overline{x}$ and $\text {sd}({\mathbf {x}})$ serves as the statistic of the LM test. Stehlík et al. [16] use the following bootstrap estimator of the coefficient:

$$\begin{aligned} LM = \frac{\sqrt{\beta _1}}{\sqrt{\beta _2-\frac{n-3}{n-1}}} \end{aligned}$$

(32)

The robustified LM tests rely on robust estimation of moments to obtain robust estimators of the skewness and kurtosis. As is the case with the RJB tests, RLM is a class of tests each defined by the particular choice of the estimators’ parameters. The RLM test considered in this study is the same as the one used by Stehlík et al. [16], in which

$$\begin{aligned} \frac{M_3(0.05n, T_0)}{M_2^{3/2}(0.05n, T_3)} \end{aligned}$$

(33)

estimates skewness and

$$\begin{aligned} \max \left( 1, \frac{M_4(0.05n, T_3)}{M_0^4(0.05n, T_0)}\right) \end{aligned}$$

(34)

estimates kurtosis.

1.3 B.3 The Robustified Shapiro–Wilk Test (RSW)

Let $J({\mathbf {x}})$ be the scaled average absolute deviation from the sample median $M=\text {median}({\mathbf {x}})$:

$$\begin{aligned} J({\mathbf {x}}) = \frac{1}{n}\sqrt{\frac{\pi }{2}}\sum _{i=1}^{n}|x_i-M| \end{aligned}$$

(35)

The RSW test statistic is the ratio of the usual standard deviation estimate $\text {sd}({\mathbf {x}})$ and $J(\mathbf{x})$ [53]:

$$\begin{aligned} RSW = \frac{\text {sd}({\mathbf {x}})}{J({\mathbf {x}})} \end{aligned}$$

(36)

Under the null hypothesis of normality, J is asymptotically normally distributed and a consistent estimate of the true deviation [53]. Therefore, the values of RSW close to 1 are expected when the sample at hand does come from a normal distribution.

C Machine-learning methods for normality testing

1.1 C.1 The FSSD kernel test of normality

As mentioned in Section 2, there are several kernel tests of goodness-of-fit. Just as our approach, they also represent a blend of machine learning and statistics. We evaluate the test of Jitkrittum et al. [42] against our network because that test is computationally less complex than the original kernel tests of goodness-of-fit proposed by Chwialkowski et al. [35], but comparable with them in terms of statistical power. Of the other candidates, the approach of Kellner and Celisse [45] is of quadratic complexity and requires bootstrap, and a drawback of the approach of Lloyd and Ghahramani [44], as Jitkrittum et al. [42] point out, is that it requires a model to be fit and new data to be simulated from it. Also, this approach fails to exploit our prior knowledge on the characteristics of the distribution for which goodness-of-fit is being determined. So, the test formulated by Jitkrittum et al. [42] was our choice as the representative of the kernel tests of goodness-of-fit. It is a distribution-free test, so we first describe its general version before we show how we used it to test for normality. The test is defined for multidimensional distributions, but we present it for the case of one-dimensional distributions because we are interested in one-dimensional Gaussians.

Let ${\mathcal {F}}$ be the RKHS of real-valued functions over ${\mathcal {X}}\in \mathrm{I\!R}$ with the reproducing kernel k. Let q be the density of model $\Psi$. As in Chwialkowski et al. [35], a Stein operator $T_{q}$ [36] can be defined over ${\mathcal {F}}$:

$$\begin{aligned} (T_{q}f)(x) = \frac{\partial \log q(x)}{\partial x}f(x)+\frac{\partial f(x)}{\partial x} \end{aligned}$$

(37)

Let us note that for

$$\begin{aligned} \xi _q(x, \cdot )=\frac{\partial \log q(x)}{\partial x}k(x, \cdot ) + \frac{\partial k(x, \cdot )}{\partial x} \end{aligned}$$

(38)

it holds that:

$$\begin{aligned} (T_q f)(x) = \langle {f}{\xi _q(x, \cdot )}\rangle _{{\mathcal {F}}} \end{aligned}$$

(39)

If $Z \sim \Psi$, then ${\mathrm{I\!E}}(T_{q}f)(Z)= 0$ [35]. Let X be the random variable which follows the distribution from which the sample ${\mathbf {x}}$ was drawn. The Stein discrepancy $S_q$ between X and Z is defined as follows [35]:

$$\begin{aligned} \begin{aligned} S_{q}(X)&= \sup _{\left\Vert f\right\Vert< 1} \left[ {\mathrm{I\!E}}(T_{q}f)(X)-{\mathrm{I\!E}}(T_{q}f)(Z) \right] \\&= \sup _{\left\Vert f\right\Vert<1}{\mathrm{I\!E}}(T_{q}f)(X)\\&= \sup _{\left\Vert f\right\Vert < 1}\langle {f}{{\mathrm{I\!E}}\xi _q(X,\cdot )}\rangle _{\mathcal {F^d}} = \left\Vert g(\cdot )\right\Vert _{{\mathcal {F}}} \end{aligned} \end{aligned}$$

(40)

where $g(\cdot )={\mathrm{I\!E}}\xi _q(X,\cdot )$ is called the Stein witness function and belongs to ${\mathcal {F}}$. Chwialkowski et al. [35] show that if k is a cc-universal kernel [107], then $S_{q}(X)=0$ if and only if $X \sim \Psi$, provided a couple of mathematical conditions are satisfied.

Jitkrittum et al. [42] follow the same approach as Chwialkowski et al. [43] for kernel two-sample tests and present the statistic that is comparable to the original one of Chwialkowski et al. [35] in terms of power, but faster to compute. The idea is to use a real analytic kernel k that makes the witness function g real analytic. In that case, the values of $g(v_1),g(v_2),\ldots ,g(v_m)$ for a sample of points $\{v_j\}_{j=1}^{m}$, drawn from X, are almost surely zero w.r.t. the density of X if $X\sim \Psi$. Jitkrittum et al. [42] define the following statistic which they call the finite set Stein discrepancy:

$$\begin{aligned} FSSD^2 = \frac{1}{m}\sum _{j=1}^{m}g(v_i)^2 \end{aligned}$$

(41)

If $X\sim \Psi$, $FSSD^2=0$ almost surely. Jitkrittum et al. [42] use the following estimate of $FSSD^2$:

$$\begin{aligned} FSSD^2 = \frac{2}{n(n-1)}\sum _{i=2}^{n}\sum _{j < i}\Delta (x_i,x_j) \end{aligned}$$

(42)

where

$$\begin{aligned} \Delta (x_i, x_j) = \frac{1}{m}\sum _{\ell =1}^{m}\xi _q(x_i,v_{\ell })\xi _q(x_j,v_{\ell }) \end{aligned}$$

(43)

In our case, we want to test if $\Psi$ is equal to any normal distribution. Similarly to the LF test, we can use the sample estimates of the mean and variance as the parameters of the normal model. Then, we can randomly draw m numbers from $N(\overline{x}, \text {sd}({\mathbf {x}})^2)$ and use them as points $\{v_j\}_{j=1}^{m}$, calculating the estimate (42) for ${\mathbf {x}}$ and $N(\overline{x}, \text {sd}({\mathbf {x}}))$. For kernel, we chose the Gaussian kernel as it fulfills the conditions laid out by Jitkrittum et al. [42]. To set its bandwidth, we used the median heuristic [108], which sets it to the median of the absolute differences $|x_i-x_j|$ ($x_i, x_j \in {\mathbf {x}}, 1\le i < j \le n$). The exact number of locations, m, was set to 10.

Since g is always zero if the sample comes from the normal distribution, the larger the value of $FSSD^2$, the more likely it is that the sample came from a non-normal distribution. We refer to this test as the FSSD test.

1.2 C.2 The statistic-based neural network

Since the neural networks are designed with a fixed-size input in mind, but samples can have any number of elements, Sigut et al. [3] represent the samples with the statistics of several normality tests which were chosen in advance. The rationale behind this method is that, taken together, the statistics of different normality tests examine samples from complementary perspectives, so a neural network that combines the statistics could be more accurate than individual tests. Sigut et al. [3] use the estimates of the following statistics:

• 1. skewness:

  $$\begin{aligned} \sqrt{\beta _1} = \frac{m_3}{m_2^{3/2}},\quad \text {where } m_u=\frac{1}{n}\sum _{i=1}^{n}(x_i-\overline{x})^u \end{aligned}$$

  (44)

• 2. kurtosis:

  $$\begin{aligned} \sqrt{\beta _2} = \frac{m_4}{m_2^2} \end{aligned}$$

  (45)

• 3. the W statistic of the Shapiro-Wilk test (see Equation (14)),

• 4. the statistic of the test proposed by Lin and Mudholkar [50]:

  $$\begin{aligned} \begin{aligned} Z_p&= \frac{1}{2}\ln \frac{1+r}{1-r} \\ r&= \frac{\sum _{i=1}^{n}(x_i-\overline{x})(h_i-\overline{h})}{\sqrt{\left( \sum _{i=1}^{n}(x_i-\overline{x})^2\right) \left( \sum _{i=1}^{n}(h_i-\overline{h})^2\right) }}\\ h_i&= \left( \frac{\sum _{j\ne i}^{n}x_j^2-\frac{1}{n-1}\left( \sum _{j\ne i}^{n}x_j\right) ^2}{n}\right) ^{\frac{1}{3}}\\ \overline{h}&= \frac{1}{n}h_i \end{aligned} \end{aligned}$$

  (46)

• 5. and the statistic of the Vasicek test [51]:

  $$\begin{aligned} K_{m,n} = \frac{n}{2m\times \text {sd}({\mathbf {x}})}\left( \prod _{i=1}^{n}(x_{(i+m)}-x_{(i-m)})\right) ^{\frac{1}{n}} \end{aligned}$$

  (47)

  where m is a positive integer smaller than n/2, $[x_{(1)},x_{(2)},\ldots ,x_{(n)}]$ is the non-decreasingly sorted sample ${\mathbf {x}}$, $x_{(i)}=x_{(1)}$ for $i<1$, and $x_{(i)}=x_{(n)}$ for $i>n$.

It is not clear which activation function Sigut et al. [3] use. They train three networks with a single hidden layer containing 3, 5, and 10 neurons, respectively. One of the networks is designed to take the sample size into account as well so that it can be more flexible. Just as the other two, the network showed that it was capable of modeling posterior Bayesian probabilities of the input samples being normal. Sigut et al. [3] focus on the samples with no more than 200 elements.

In addition to our network, presented in Section 4, we trained one that follows the approach of Sigut et al. [3]. We refer to that network as Statistic-Based Neural Network (SBNN) because it expects an array of statistics as its input. More precisely, prior to being fed to the network, each sample ${\mathbf {x}}$ is transformed to the following array:

$$\begin{aligned}{}[\sqrt{\beta _1}, \beta _2, W, Z_p, K_{3,n}, K_{5, n}, n] \end{aligned}$$

(48)

just as in Sigut et al. [3] (n is the sample size). ReLU was used as the activation function. To make comparison fair, we trained SBNN in the same way as our network which we design in Section 4.

D Results for set ${\mathcal {F}}$

Detailed results for each n in each subset of ${\mathcal {F}}$ are presented in Tables 15, 16, 17 and 18.

Table 15 Power against left contamination for different sample sizes, estimated using set ${\mathcal {F}}_{left}$

Table 16 Power against right contamination for different sample sizes, estimated using set ${\mathcal {F}}_{right}$

Table 17 Power against central contamination for different sample sizes, estimated using set ${\mathcal {F}}_{central}$

Table 18 Power against symmetric contamination for different sample sizes, estimated using set ${\mathcal {F}}_{symmetric}$