Correction: Cerqueti, R.; Lupi, C. Some New Tests of Conformity with Benford’s Law. Stats 2021, 4, 745–761

Text Correction

There were several errors in the original publication [1].

1—Convergence in Equation (10) is approximate.

A correction has been made to Proposition 2, in the first text line after Equations (9) and (10):

where ${\mathbf{e}}_n^{*}:={(e_{n1},\dots ,e_{n,k-1})}^{\prime }$ and ${\Sigma }^{*}$ is made of the first $k-1$ rows and columns of $\Sigma$. Furthermore, using the symbol $\stackrel{d}{⇝}$ to denote approximate convergence in distribution as $n\to \infty$,

$$MA{D}^{★}:={\displaystyle \frac{\sqrt{n}}{k}}{\displaystyle \sum _{j=1}^k}{\displaystyle \frac{\left|f_{nj}-p_j\right|}{\sqrt{p_j(1-p_j)}}}\stackrel{d}{⇝}\mathrm{N}\left(\sqrt{{\displaystyle \frac{2}{\pi }}},{\displaystyle \frac{1}{k^2}}{\displaystyle \sum _{i=1}^k}{\displaystyle \sum _{j=1}^k}{r}_{ij}\right)$$

(10)

2—Explicit reference to quantities $\mathbf{e}$ and ${\mathbf{e}}_n^{†}$ was missing.

A correction has been made to the text line before Equation (14):

To prove (10), first note that, since $\sqrt{n}\mathbf{e}\stackrel{d}{⟶}\mathrm{N}(\mathbf{0},\Sigma )$, ${\mathbf{e}}_n^{†}:={\left(e_{n1}^{†},\dots ,e_{nk}^{†}\right)}^{\prime }$ is such that the $e_{nj}^{†}$’s are individually $N(0,1)$ but are correlated. We also exploit the fact that if $Y\sim \mathrm{N}(0,1)$, then see [36].

3—The distribution in Equation (16) was wrong.

A correction has been made to Equation (16) and the following three text lines:

$$\sqrt{n}{e}_{nj}^{★}:={\displaystyle \frac{\sqrt{n}\left|f_{nj}-p_j\right|}{\sqrt{p_j(1-p_j)}}}\stackrel{d}{⟶}\mathrm{HN}\left(\sqrt{{\displaystyle \frac{2}{\pi }}},1-{\displaystyle \frac{2}{\pi }}\right)$$

(16)

where $\mathrm{HN}(·,·)$ denotes the half-normal random variable.

Furthermore, $\sqrt{n}E\left({\mathbf{e}}_n^{★}\right):=\sqrt{n}E{\left(e_{n1}^{★},\dots ,e_{nk}^{★}\right)}^{\prime }=ı\sqrt{{\displaystyle \frac{2}{\pi }}}$, with ı a k-vector of ones, and $\mathrm{Cov}\left(\sqrt{n}{\mathbf{e}}_n^{★}\right)=\mathbf{R}$, with $\mathbf{R}$ to be defined below.

4—A comma was missing in the LHS of Equation (20).

A comma was added in the LHS of Equation (20):

$$\begin{array}{cc}\hfill cov\left(|e_{ni}^{†}|,|e_{nj}^{†}|\right)& =E\left(|e_{ni}^{†}\left|\right|e_{nj}^{†}|\right)-E\left(|e_{ni}^{†}|\right)E\left(|e_{nj}^{†}|\right)\hfill \\ \hfill & ={\displaystyle \frac{2}{\pi }}\left({\rho }_{ij}arcsin\left({\rho }_{ij}\right)+\sqrt{1-{\rho }_{ij}^2}\right)-{\displaystyle \frac{2}{\pi }}.\hfill \end{array}$$

(20)

5—The reference to k large was missing.

A correction has been made to the text line before Equation (23), and two text lines were added following Equation (23):

Finally, as $n\to \infty$ and for k large enough

$${\displaystyle \frac{\sqrt{n}}{k}}{\displaystyle \sum _{j=1}^k}{e}_{nj}^{★}={\displaystyle \frac{1}{k}}{\displaystyle \sum _{j=1}^k}{\displaystyle \frac{\sqrt{n}\left|f_{nj}-p_j\right|}{\sqrt{p_j(1-p_j)}}}\stackrel{d}{⟶}\mathrm{N}\left(\sqrt{{\displaystyle \frac{2}{\pi }}},{\displaystyle \frac{1}{k^2}}{ı}^{\prime }\mathbf{R}ı\right).$$

(23)

Of course, k is such that it cannot tend to ∞. The usual values of k are $k\in \{9,10,90\}$. This implies that convergence to the normal can only be approximate, leading to (10).

6—Remark 6 was missing.

Remark 6 was added:

Remark 6.

Simulations (available from the authors upon request) show that, as $n\to \infty$, the convergence of $MA{D}^{★}$ to the normal distribution is very fast as k increases. Even for $k=9$ (first-digit case) or $k=10$ (second-digit case), the approximation is highly accurate. Since the distribution of $\sqrt{n}{e}_{nj}^{★}$ is asymmetric, a very slight skewness is still observed for small values of k; however, this vanishes as k increases. In practice, agreement with the asymptotic distribution is essentially perfect in the first two-digit case.

The authors state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.