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

Roy Cerqueti; Claudio Lupi

doi:10.3390/stats9040077

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

e_{n}^{*} : = {(e_{n 1}, \dots, e_{n, k - 1})}^{'}

and

Σ^{*}

is made of the first

k - 1

rows and columns of

Σ

. Furthermore, using the symbol

\overset{d}{⇝}

to denote approximate convergence in distribution as

n \to \infty

,

M A D^{★} : = \frac{\sqrt{n}}{k} \sum_{j = 1}^{k} \frac{|f_{n j} - p_{j}|}{\sqrt{p_{j} (1 - p_{j})}} \overset{d}{⇝} N (\sqrt{\frac{2}{π}}, \frac{1}{k^{2}} \sum_{i = 1}^{k} \sum_{j = 1}^{k} r_{i j})

(10)

2—Explicit reference to quantities

e

and

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} e \overset{d}{⟶} N (0, Σ)

,

e_{n}^{†} : = {(e_{n 1}^{†}, \dots, e_{n k}^{†})}^{'}

is such that the

e_{n j}^{†}

’s are individually

N (0, 1)

but are correlated. We also exploit the fact that if

Y \sim 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_{n j}^{★} : = \frac{\sqrt{n} |f_{n j} - p_{j}|}{\sqrt{p_{j} (1 - p_{j})}} \overset{d}{⟶} HN (\sqrt{\frac{2}{π}}, 1 - \frac{2}{π})

(16)

where

HN (\cdot, \cdot)

denotes the half-normal random variable.

Furthermore,

\sqrt{n} E (e_{n}^{★}) : = \sqrt{n} E {(e_{n 1}^{★}, \dots, e_{n k}^{★})}^{'} = ı \sqrt{\frac{2}{π}}

, with ı a k-vector of ones, and

Cov (\sqrt{n} e_{n}^{★}) = R

, with

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{matrix} cov (| e_{n i}^{†} |, | e_{n j}^{†} |) & = E (| e_{n i}^{†} | | e_{n j}^{†} |) - E (| e_{n i}^{†} |) E (| e_{n j}^{†} |) \\ = \frac{2}{π} (ρ_{i j} arcsin (ρ_{i j}) + \sqrt{1 - ρ_{i j}^{2}}) - \frac{2}{π} . \end{matrix}

(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

\frac{\sqrt{n}}{k} \sum_{j = 1}^{k} e_{n j}^{★} = \frac{1}{k} \sum_{j = 1}^{k} \frac{\sqrt{n} |f_{n j} - p_{j}|}{\sqrt{p_{j} (1 - p_{j})}} \overset{d}{⟶} N (\sqrt{\frac{2}{π}}, \frac{1}{k^{2}} ı^{'} R ı) .

(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

M A 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_{n j}^{★}

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.

Reference

Cerqueti, R.; Lupi, C. Some New Tests of Conformity with Benford’s Law. Stats 2021, 4, 745–761. [Google Scholar] [CrossRef] [Scilit]

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