In this work, whether China Stock Market obeys Benford’s law is checked. To achieve this, some selected tests are carefully calculated and compared with other tests. We want to know which statistical test is the best one for assessing conformity to the law in “Big Data”. We use the existing R-Benford Tests libraries to make these calculations. Besides, some specific codes for the application are also written in the R language.

How to test a dataset conforming to Benford’s law is still an open question. So firstly, we give the definition of the law. But the law has various forms of definition. However, which definition should be chosen to test the benfordness of a dataset is flexible. In this study, we choose the two basic definitions, i.e., the distribution of first significant digit and the distribution of first two significant digit. We state these definition as follows:

$\text{Prob}\left(D_1=d_1,D_2=d_2,\cdots ,D_k=d_k\right)=\log \left(1+{\left(\underset{j=1}{\overset{k}{{\displaystyle \sum }}}{10}^{k-j}\cdot d_j\right)}^{-1}\right)$ (2.1)

where $D_1,D_2,\cdots ,D_k$ represent the first digits, the second digits…, the kth digits, respectively.

This is a general definition of Benford’s law.

Equation (2.1), when applied to the first digit, becomes the basic formula of Benford’s law as follows:

$\text{Prob}\left(D_1=d_1\right)=\log \left(1+\frac{1}{d_1}\right),\text{with}{d}_1\in \left\{1,2,\cdots ,9\right\}$ (2.2)

Equation (2.1), When applied to the first two digits is as follows:

$\begin{array}{l}\text{Prob}\left(D_1=d_1,D_2=d_2\right)\\ =\log \left(1+\frac{1}{10d_1+d_2}\right),\text{with}{d}_1\in \left\{1,2,\cdots ,9\right\}\text{and}\text{with}{d}_2\in \left\{0,1,\cdots ,9\right\}\end{array}$ (2.3)

where $D_1,D_2$ denotes the value of first two digits and $\text{Prob}\left(D_1=d_1,D_2=d_2\right)$ represent the probability that the first two digits are equal to $d_1{d}_2$.

Equation (2.1), When applied to the Second digits is as follows:

$\text{Prob}\left(D_2=d_2\right)=\underset{d_1=1}{\overset{9}{{\displaystyle \sum }}}\log \left(1+\frac{1}{10d_1+d_2}\right),\text{with}{d}_2\in \left\{0,1,\cdots ,9\right\}$ (2.4)

Table 1 shows the expected frequency of first digits, second digits and first two digits of Benford’s law. They are calculated based on Equations (2.2), (2.3), (2.4), respectively.

Source: Author’s calculation with Excel.

We assess conformity to Benford’s law based on Equations (2.2), (2.3); i.e., the expected frequency of first digits and first two digits.

In order to assess conformity to Benford’s law, we have collected Stock Exchanging Data published by Chinese companies from the year 2017 to 2021 listed on Shanghai A-Share Market, which were obtained from CSMAR (China Stock Market and Accounting Research Database). From these Trading Data, we choose two indices: Close Price and Daily Return as variables to be used to assess Benfordness of China Stock Market.

Table 2 shows the general information of those Data, including number of Trading Days, number of Companies, number of A-shares and Data size. From the last line, we can see the Data size is very big. Our calculation and analysis are based on these trading Data.

Source: Shanghai Stock Exchange Statistics Annual (2019) (2022), http://www.sse.com.cn/aboutus/publication/yearly/ .

Before our check, the hypotheses are given first. We assumed China Stock Market Obeys Benford’s law, that is a data should follow benford distribution. We choose two indices to analyze: Close Price and Daily Return to assess the benfordness. We calculate and assess 2017 to 2021 five years trading Data, respectively. Besides, we also combine these data as a whole to assess benfordness. Specifically, these hypotheses are as follows:

Hypothesis 1: The yearly distribution of first digits of Close Price follows Benford’s law;

Hypothesis 2: The yearly distribution of first digits of Daily Return follows Benford’s law;

Hypothesis 3: The overall distribution of first digits of Daily Return follows Benford’s law.

There are many statistical and non-statistical tests to be used to assess a dataset conforming to Benford’s law. We choose the tests based on Joenssen used in his paper [18] and make a comparison between them. According to these tests and comparison, some of tests which may be suitable to assess conformity to Benford’s law for large data size are selected.

The common goodness-of-fit test for discrete distributions is the Pearson Chi-Square test which is based on the comparison between observed frequencies and expected frequencies. The following is the calculating formula:

${\chi }^2=n\underset{i=1}{\overset{9}{{\displaystyle \sum }}}\frac{{\left(p_i-{\pi }_i\right)}^2}{{\pi }_i}$ (2.5)

From the above formula, ${\chi }^2$ is obviously sensitive to the data size n and it may be very large when the data size is big. ${\chi }^2$ statistic asymptotically follows ${\chi }^2$ distribution with 8 degrees of freedom. If the value of ${\chi }^2$ is large, the hypothesis, the data obeying Benford’s law should be rejected.

The Kolmogorov-Smirnov test is also commonly used goodness-of-fit test, which is based on the maximum vertical distance between the empirical and expected cumulative distribution functions. By using the supremum function, the test can be computed as follows:

$D=\sqrt{n}{\sup }_{j=1,\cdots ,9}\left|\underset{i=1}{\overset{j}{{\displaystyle \sum }}}\left(p_i-{\pi }_i\right)\right|$ (2.6)

Similar to the ${\chi }^2$, if the value of D is too large, the hypothesis is considered to be rejected.

The following test is used by Leemis et al. [5] to determine conformance to Benford’s law for some parametric survival distributions. The test is based on the distribution of Chebyshev distances between empirical distribution and theoretical distribution under the null-hypothesis. In order to derive asymptotic test statistics, Morrow modified the test statistics used by Leemis et al. and by Cho and Gaines as follows

$m=\sqrt{n}{\max }_{i=1,\cdots ,9}\left|p_i-{\pi }_i\right|$ (2.7)

Large values of m would reject the hypothesis like the statistical test D. The next test, based on the Euclidean distance between observed and expected frequencies, is also first modified and used by Morrow.

$d=\sqrt{n\sqrt{\underset{i=1}{\overset{9}{{\displaystyle \sum }}}{\left(p_i-{\pi }_i\right)}^2}}$ (2.8)

Large distances mean the data seriously deviating from Benford’s law. Another test is Freedman’s modification of Watson’s $U^2$ statistic for discrete distribution. If Deviation from Benford’s law is very large leading to larger $U^2$, which will reject the null hypothesis.

$U^2=\frac{n}{9\cdot {10}^{k-1}}\cdot \left[\underset{i={10}^{k-1}}{\overset{{10}^k-2}{{\displaystyle \sum }}}{\left(\underset{j=1}{\overset{i}{{\displaystyle \sum }}}\left(p_i-{\pi }_i\right)\right)}^2-\frac{1}{9\cdot {10}^{k-1}}\left(\underset{i={10}^{k-1}}{\overset{{10}^k-2}{{\displaystyle \sum }}}{\left(\underset{j=1}{\overset{i}{{\displaystyle \sum }}}\left(p_i-{\pi }_i\right)\right)}^2\right)\right]$ (2.9)

$J_P^2$ statistic is based on the Shapiro-Francia test for normality. The value of $J_P^2$ is within zero and one. The value closer to 1 means the data closer to Benford’s law.

$J_P^2=\mathrm{sgn}\left(\underset{i=1}{\overset{9}{{\displaystyle \sum }}}\left({\pi }_i-\frac{1}{9}\right)\cdot p_i\right)\cdot \frac{{\left({{\displaystyle \sum }}_{i=1}^9\left({\pi }_i-\frac{1}{9}\right)\cdot p_i\right)}^2}{{{\displaystyle \sum }}_{i=1}^9{\left({\pi }_i-\frac{1}{9}\right)}^2\cdot \left({{\displaystyle \sum }}_{i=1}^9{\left(p_i\right)}^2-\frac{1}{9}\right)}$ (3.0)

Besides the above test, the J-divergence is added to measure the goodness-of-fit between the empirical distribution of first digit and Benford’s law. J-divergence is defined as half the sum of two possible Kullback-Leibler distances as follows:

$KL\left(P,B\right)=\underset{k=1}{\overset{9}{{\displaystyle \sum }}}\text{Prob}\left(\text{first}\text{digit}\text{is}k\right)\times {\log }_2\left[\frac{\text{Prob}\left(\text{first}\text{digit}\text{is}k\right)}{B\left(k\right)}\right],$

$KL\left(B,P\right)=\underset{k=1}{\overset{9}{{\displaystyle \sum }}}B\left(k\right)\times {\log }_2\left[\frac{B\left(k\right)}{\text{Prob}\left(\text{first}\text{digit}\text{is}k\right)}\right]$

$J=\frac{1}{2}\times \left[KL\left(P,B\right)+KL\left(B,P\right)\right]$ (3.1)

B(k) denotes the Benford’s law expected frequencies. Kullback-Leibler distance, also known as relative entropy, has several good measure-theoretic properties, including non-negative and equaling to zero if and only if P and B are same distributions. The smaller value of J-div means closer to Benford’s law.