# On the Mistaken Use of the Chi-Square Test in Benford’s Law

## Abstract

**:**

**Table of Contents**

- The First Digit on the Left Side of Numbers
- Benford’s Law and the Predominance of Low Digits
- Second Digits, Third Digits, and Higher Order Digits
- A Robust Measure of Order of Magnitude (ROM)
- Two Essential Requirements for Benford Behavior
- Three Generic Causes of the Benford Phenomenon
- The chi-square Test in the Context of Benford’s Law
- The First Paradox
- The Second Paradox
- The Proper Use and Context of the chi-square Application
- The Statistical Theory Forming the Basis for the Chi-Square Test
- Comparison of the chi-square Statistic with SSD
- Testing the Lognormal via the Chi-Square Statistic
- Testing the Fibonacci Series via the Chi-Square Statistic
- Testing Partial US Population Data Sets via the Chi-Square Statistic
- The Nature of Hypothetical Real-Life-Like Revenue Data
- The Nature of Hypothetical Real-Life-Like Population Data
- Conclusions

## 1. The First Digit on the Left Side of Numbers

Digit Index: | {1, 2, 3, 4, 5, 6, 7, 8, 9} |

Digits Count totaling 40 values: | {15, 8, 6, 4, 4, 0, 2, 1, 0} |

Proportions of Digits with “%” sign omitted: | {38, 20, 15, 10, 10, 0, 5, 3, 0} |

**Not all digits are created equal**”, or rather “Not all first digits are created equal”, even though this seems to be contrary to intuition and against all common sense.

## 2. Benford’s Law and the Predominance of Low Digits

Probability[First Leading Digit is d] = LOG_{10}(1 + 1/d) | ||

LOG_{10}(1 + 1/1) = LOG(2.00) | = | 0.301 |

LOG_{10}(1 + 1/2) = LOG(1.50) | = | 0.176 |

LOG_{10}(1 + 1/3) = LOG(1.33) | = | 0.125 |

LOG_{10}(1 + 1/4) = LOG(1.25) | = | 0.097 |

LOG_{10}(1 + 1/5) = LOG(1.20) | = | 0.079 |

LOG_{10}(1 + 1/6) = LOG(1.17) | = | 0.067 |

LOG_{10}(1 + 1/7) = LOG(1.14) | = | 0.058 |

LOG_{10}(1 + 1/8) = LOG(1.13) | = | 0.051 |

LOG_{10}(1 + 1/9) = LOG(1.11) | = | 0.046 |

--------- | ||

1.000 |

## 3. Second Digits, Third Digits, and Higher Order Digits

## 4. A Robust Measure of Order of Magnitude (ROM)

_{10}(Maximum/Minimum)

_{10}(P

_{99%}/P

_{1%})

**) for the minimum, and by substituting the 99th percentile (in symbols P**

_{1%}**) for the maximum.**

_{99%}## 5. Two Essential Requirements for Benford Behavior

_{10}(P

_{99%}/P

_{1%}) > 3

^{35}, 10

^{8}) or the Uniform(1, 10

^{27}) are not Benford at all, and this is so in spite of their extremely large orders of magnitude. In summary: Benford behavior in extreme generality can be found with the confluence of sufficiently large order of magnitude together with skewness of data—having a histogram falling to the right. The combination of skewness and large order of magnitude is not a guarantee of Benford behavior, but it is a strong indication of likely Benford behavior under the right conditions. Moderate (overall) quantitative skewness with a tail falling too gently to the right implies that digits are not as skewed as in the Benford configuration. Extreme (overall) quantitative skewness with a tail falling sharply to the right implies that digits are severely skewed, even more so than they are in the Benford configuration.

## 6. Three Generic Causes of the Benford Phenomenon

- (I)
- given that partition is performed on the real number basis and not exclusively on integers;
- (II)
- given that all partition acts are truly random;
- (III)
- given that partition thoroughly breaks the original quantity into numerous parts;
- (IV)
- given that each act of partition is independent and not correlated with all the previous or future acts of partitions;
- (V)
- given that no artificial or arbitrary limits or rules exist, and that partition is performed totally in free style with no limitations whatsoever.

**Uniform(min, Uniform(maxA, maxB))**, where min < maxA < maxB.

**Exponential(Exponential(Exponential(7)))**or

**Exponential(Normal(Uniform(15, 21), 3))**, among infinitely many other chaining possibilities.

^{k}(powers), usually do not respond to chaining at all, show no preference for the big, for the small, or for any size, and disobey Benford’s Law (i.e., not chain-able).

_{parameter}**∂(center)/∂(parameter) = 0**

_{→∞}## 7. The Chi-Square Test in the Context of Benford’s Law

**/Theoretical**

^{2}_{0}that the data set is authentically Benford at the p% confidence level if the chi-square statistic is larger than chi-square p% critical (threshold) value with 8 degrees of freedom for the 1st digits, or with 9 degrees of freedom for the 2nd digits.

**15.5**is less than 5%.

**16.9**is less than 5%.

^{2}## 8. The First Paradox

- (1)
- Time in seconds between the 19,451 Global Earthquakes during 2012;
- (2)
- USA Cities and Towns in 2009—19,509 Population Centers;
- (3)
- Canford PLC Price List, Oct 2013—15,194 items for sale;
- (4)
- 48,111 Star Distances from the Solar System, NASA;
- (5)
- Biological Genetic Measure—DNA V230 Bone Marrow—91,223 data points;
- (6)
- Oklahoma State 986,962 positive expenses below $1 million in 2011.

- (1)
- (2)
- (3)
- (4)
- (5)
- (6)

(1) | 29.9% | 18.8% | 13.5% | 9.3% | 7.5% | 6.2% | 5.8% | 4.8% | 4.2% |

(2) | 29.4% | 18.1% | 12.0% | 9.5% | 8.0% | 7.0% | 6.0% | 5.3% | 4.6% |

(3) | 28.8% | 17.7% | 14.2% | 9.2% | 8.1% | 7.0% | 5.3% | 5.1% | 4.6% |

(4) | 28.3% | 15.1% | 12.0% | 10.5% | 9.0% | 7.6% | 6.5% | 5.9% | 5.2% |

(5) | 29.7% | 18.2% | 12.5% | 9.9% | 7.7% | 6.6% | 5.8% | 5.0% | 4.5% |

(6) | 29.7% | 17.7% | 12.1% | 9.7% | 8.6% | 6.6% | 6.1% | 4.9% | 4.5% |

Ben | 30.1% | 17.6% | 12.5% | 9.7% | 7.9% | 6.7% | 5.8% | 5.1% | 4.6% |

- (1)
- Time in seconds between the 19,451 Global Earthquakes during 2012.For this data set we obtain the following results:OOM = LOG(Maximum/Minimum) = 6.3ROM = LOG(Percentile 99%/Percentile 1%) = 2.81st Digits: SSD = 3.1 chi-square = 53.1 > 15.5 (rejection)2nd Digits: SSD = 0.7 chi-square = 14.0
- (2)
- USA Cities and Towns in 2009—19,509 Population Centers.For this data set we obtain the following results:OOM = LOG(Maximum/Minimum) = 6.9ROM = LOG(Percentile 99%/Percentile 1%) = 3.61st Digits: SSD = 1.3 chi-square =17.4 > 15.5 (rejection)2nd Digits: SSD = 0.7 chi-square =13.1
- (3)
- Canford PLC Price List, Oct 2013—15,194 items for sale.For this data set we obtain the following results:OOM = LOG(Maximum/Minimum) = 6.1ROM = LOG(Percentile 99%/Percentile 1%) = 4.11st Digits: SSD = 5.3 chi-square = 57.7 > 15.5 (rejection)2nd Digits: SSD = 6.3 chi-square = 100.6 > 16.9 (rejection)
- (4)
- 48,111 Star Distances, NASA.For this data set we obtain the following results:OOM = LOG(Maximum/Minimum) = 4.9ROM = LOG(Percentile 99%/Percentile 1%) = 2.21st Digits: SSD = 14.1 chi-square = 538.1 > 15.5 (rejection)2nd Digits: SSD = 0.4 chi-square = 17.7 > 16.9 (rejection)
- (5)
- Biological Genetic Measure—DNA V230 Bone Marrow—91,223 data points.For this data set we obtain the following results:OOM = LOG(Maximum/Minimum) = 11.6ROM = LOG(Percentile 99%/Percentile 1%) = 5.61st Digits: SSD = 0.6 chi-square = 34.7 > 15.5 (rejection)2nd Digits: SSD = 0.3 chi-square = 27.3 > 16.9 (rejection)
- (6)
- Oklahoma State 986,962 expenses below $1 million in 2011.For this data set we obtain the following results:OOM = LOG(Maximum/Minimum) = 10.0ROM = LOG(Percentile 99%/Percentile 1%) = 4.31st Digits: SSD = 0.9 chi-square = 978.4 > 15.5 (rejection)2nd Digits: SSD = 426.4 chi-square = 406,262.8

## 9. The Second Paradox

- (1)
- List of 92 Atomic Weights in the Periodic Table, from Hydrogen to (and including) Uranium, as depicted in the table of Figure 17. Figure 18 and Figure 19 depict the graphs of the first and second digits respectively for the Atomic Weights data. See https://www.webelements.com/ (accessed on 1 April 2021) for a concise summary.For this data set we obtain the following results:OOM = LOG(Maximum/Minimum) = 2.4ROM = LOG(Percentile 99%/Percentile 1%) = 1.81st Digits: SSD = 334.9 chi-square ≈ 15.5 = 15.5 (borderline acceptance)2nd Digits: SSD = 105.7 chi-square = 9.1 < 16.9 (acceptance)
- (2)
- List of Electric Conductivity Values (in 1,000,000 Siemens/meter) for 24 Common Metals, as depicted in table of Figure 20. Figure 21 and Figure 22 depict the graphs of the first and second digits, respectively, for the Metals Electric Conductivity data. The link https://www.tibtech.com/conductivite (accessed on 1 April 2021) provides a concise source of the data.For this data set we obtain the following results:OOM = LOG(Maximum/Minimum) = 1.92ROM = LOG(Percentile 99%/Percentile 1%) = 1.871st Digits: SSD = 443.9 chi-square = 7.3 < 15.5 (acceptance)2nd Digits: SSD = 213.6 chi-square = 5.0 < 16.9 (acceptance)

## 10. The Proper Use and Context of the Chi-Square Application

**Does the data obey Benford’s Law or not?**” Or equivalently asking the simpler question: “

**Is the data set Benford or not?**” This is “yes or no”, “black or white”, “right or wrong”, type of a question.” As if there is no middle ground. Alternatively, we may ask the question: “

**Is the (tiny or mild) digital deviation from Benford for this data set due merely to chance and randomness or is it structural?**”

**How far is the data set from Benford?**” If data seems close to Benford then we might ask: “

**How close is the data set to Benford?**”One may inquire about the degree of deviation or “

**A Measure of Distance from Benford**”. And even though the definition chosen to accomplish this—no matter how reasonable—would still be ultimately arbitrary; yet it is altogether fitting and proper that we should construct it. Indeed, the Sum of Squared Deviations (SSD) measure could serve as an excellent comparison definition!

**Process**”.

**fraudulent reporting by IBM is arrived at via the chi-square test. Implicit in this whole forensic scheme is that the universe of revenue amounts, relating to all companies around the globe, obeys Benford’s Law, and which is indeed assumed to be true at least by this author (after working for many years with revenue and expense data and finding nearly all individual accounting data sets of large sizes to obey Benford’s Law). Surely, the hourly revenue of a tiny coffee shop on a street corner with a tiny clientele is not Benford, but the global aggregate—if the chief auditor at the IMF could ever obtain such enormous and confidential data—is very nearly perfectly Benford. It is only in this context that the IBM data on hand is considered a**

`/`**sample**, namely a sample from a much larger abstract population (of the universe of revenue amounts, relating to all companies around the globe); and it is only in this context too that statistical theory can lend a hand and provides us with cutoff points and threshold values, by way of indicating their exact probabilistic significance. Yet, there is a serious pitfall in such an approach, namely that the nature of the sample data should resemble the nature of the population in all its aspects, and this is rarely so, except in the mind of the eager, ambitious, and naïve auditor or statistician seeking an error-proof algorithm capable of detecting fraud where perhaps none exists. For example, the data on hand about IBM revenues is in reality the entire population, not a sample from that imaginary “universe of revenues”. This is so since each company has its own unique price list, particular clientele, unique products for sale, and belonging to some very specific sub-industry. Alternatively stated, the particular data set on hand, even if it can be thought of as a sample, was not “taken from the larger population” in a truly random fashion. It is impossible to argue that these 57,000 revenue transactions from IBM database is a random sample from the universe of that generic and global revenue data type! To take a truly random sample from such imaginary universe/population, one should take, say only, 25 random values from IMB, 12 random values from Nokia, 17 random values from GM, and so forth, in which case results are guaranteed to be nearly perfectly Benford, and the application of the chi-square test wholly justified and workable!

**population!**

`/`**population to be compared with. Certainly, the researcher cannot claim that his/her data on hand was obtained in a “truly random fashion” from that imaginary larger population of pH values! That would border on the absurd! This pH data set stands apart, proud and independent, existing in its own right. The issue of fraud does not enter here of course, and yet, one may legitimately wonder and ask “how far is digital distribution of this pH data set from Benford” (i.e., comparison). There can’t be, and there shouldn’t be of course, any talk here of probabilistic 5% or 1% significance level, or of any supposed “chances” of obtaining such a non-Benford pH digital result. Actually, this pH example is given for pedagogical purposes only; while a more realistic digit distribution for pH values would typically be perhaps as in: {0%, 3%, 21%, 13%, 37%, 19%, 7%, 0%, 0%}, where 2.0 is just about the most acidic drink, and 7.5 about the strongest alkali.**

`/`**6 value. In any case, rightly or wrongly, the chi-square test nowadays is used quite frequently whenever data set is relatively small, and it is erroneously (or rather conveniently) avoided whenever data set is deemed too large. Generally, auditors (mistakenly) consider any account with over 25,000 or 50,000 entries as “too large for the chi-square test”. This is akin to the irrational critically ill patient asking the laboratory to return his/her blood tests only if results are negative and to discard the whole thing if it brings bad news. In reality, the chi-square test should usually be avoided altogether in the context of Benford’s Law and regardless of data size, due to the often questionable basis of the underpinning statistical theory. Data size should never be the basis for deciding whether to apply the chi-square test or not, rather the correctness in the modeling of the data as a truly random sample of some larger Benford population should be the only criteria of proper application.**

`/`**6 probability value (16.6%), hence for only 10 throws the low value of 0% is quite acceptable, while for 1000 throws not even 5% is enough for establishing trust, and the unbiased-ness of the die is called into serious doubt.**

`/`**subjectively**judge a given SSD value of the data on hand to be either small enough and thus somewhat close to Benford, or too high and definitely non-Benford in nature. A more systematic way of going about it is to empirically compare SSD of the data set under consideration to the list of a large variety of SSD values of other honest and relevant data sets of the same or similar type. Such accumulated knowledge helps us in empirically deciding (in a non-statistical and non-theoretical way) on the implications of those SSD values. Yet some subjectivity is unfortunately necessary here in choosing cutoff points.

## 11. The Statistical Theory Forming the Basis for the Chi-Square Test

**sum**of N independent and identically distributed Bernoulli variables with parameter p.The above requirement for a large value of N implies that data sets with exceedingly small N value are not eligible to apply the chi-square procedure, even when data points are known to have been selected truly randomly and independently. Yet, this fact does not truly absolve the second paradox of its error.

**discrete**whole numbers distribution (Binomial) is being approximated by a

**continuous**distribution (Normal) of integrals as well as fractional values, and which is a delicate, subtle, and complex endeavor, and could be quite problematic for low values of N.

**χ**= (O − Np)/√(Npq)

**χ**

**= (O − Np)**

^{2}**/(Npq)**

^{2}**is distributed according to the chi-square distribution with 1 degree of freedom. Hence the**

^{2}**χ**

**expression above is indeed chi-square 1.**

^{2}**χ**

**can be written as:**

^{2}**χ**

**= (O − Np)**

^{2}**/(Npq)**

^{2}**χ**

**= (1)(O − Np)**

^{2}**/(Npq)**

^{2}**χ**

**= (p + (1 − p))(O − Np)**

^{2}**/(Npq)**

^{2}**χ**

**= (p + q)(O − Np)**

^{2}**/(Npq)**

^{2}**χ**

**= [p(O − Np)**

^{2}**+ q(O − Np)**

^{2}**]/(Npq)**

^{2}**χ**

**= [p(Np − O)**

^{2}**+ q(O − Np)**

^{2}**]/(Npq)**

^{2}**χ**

**= [p(Np − O + [Nq − Nq])**

^{2}**+ q(O − Np)**

^{2}**]/(Npq)**

^{2}**χ**

**= [p(Np + Nq − O − Nq)**

^{2}**+ q(O − Np)**

^{2}**]/(Npq)**

^{2}**χ**

**= [p(N(p + q) − O − Nq)**

^{2}**+ q(O − Np)**

^{2}**]/(Npq)**

^{2}**χ**

**= [p(N(1) − O − Nq)**

^{2}**+ q(O − Np)**

^{2}**]/(Npq)**

^{2}**χ**

**= [ p(N − O − Nq)**

^{2}**+ q(O − Np)**

^{2}**]/(Npq)**

^{2}**χ**

**= p(N − O − Nq)**

^{2}**/(Npq) + q(O − Np)**

^{2}**/(Npq)**

^{2}**χ**

**= (N − O − Nq)**

^{2}**/(Nq) + (O − Np)**

^{2}**/(Np)**

^{2}**= ∑(O**

^{2}_{i}−E

_{i})

^{2}**/**E

_{i}[summation index i runs from 1 to n]

**= Pearson’s cumulative test statistic, which asymptotically approaches the chi-square distribution with n Degrees of Freedom (DOF);**

^{2}_{i}= The number of observations of multinomial type I;

_{i}= Np

_{i}= The expected theoretical frequency of multinomial type i, asserted by the null hypothesis that the fraction of multinomial type i in the population is p

_{i};

**= The test statistic representing the chi-square with (9 − 1) or (10 − 1) DOF;**

^{2}_{i}= The number of values in the data set with 1st digit d, or with 2nd digit d;

_{i}= N∗Log(1 + 1/d), or N times the 2nd digit Benford distribution for d;

**main fallacy**of applying the chi-square test in the context of Benford’s Law is that the 1st digits and the 2nd digits of typical real-life data values are not distributed as Bernoulli variables, all with identical p value—even within a single data set. Moreover, this author as Benford’s Law researcher and data scientist (surely to be collaborated by other data analysts) is well-aware of the notion that the 1st digits and the 2nd digits of data values very rarely or almost never occur truly independently of each other, and are not separately selected or created out of thin air as independent and identical Bernoulli variables. Thus, the entire set of all the 1st digits and all the 2nd digits of the data values cannot be declared as constituting a truly Multinomial Distribution. Consequently, the entire edifice forming the basis for the chi-square test outlined above falls apart.

## 12. Comparison of the Chi-Square Statistic with SSD

**Theoretical**

^{2}/

^{2}**proportional format, such as 0.301, hence the factor of 100**

`/`**or simply (100) ×**

^{2}`(`100) scale in the above expression is derived from the scale conversion (0.301) × (100) = 30.1.

^{9}space, namely the square of the “distance” from the Benford point to the data digital point.

**/**N, or the sum of 9 absolute values of deviations, and so forth.

**/**(0.301) = 3.32, and the amplification of the deviation-square for digit 9 by the much bigger factor of 1

**/**(0.046) = 21.85. Hence, the chi-squared statistic overemphasizes fluctuations for high digits, such as {7, 8, 9}, and it deemphasizes fluctuations for low digits, such as {1, 2, 3}.

**33.1**, 17.6, 12.5, 9.7, 7.9, 6.7, 5.8, 5.1,

**1.6**} SSD = 18.0 chi-square =

**22.7.**

**33.1**, 17.6, 12.5, 9.7, 7.9, 6.7,

**2.8**, 5.1, 4.6} SSD = 18.0 chi-square =

**18.5.**

**33.1**, 17.6, 12.5, 9.7,

**4.9**, 6.7, 5.8, 5.1, 4.6} SSD = 18.0 chi-square =

**14.4.**

**33.1**, 17.6,

**9.5**, 9.7, 7.9, 6.7, 5.8, 5.1, 4.6} SSD = 18.0 chi-square =

**10.2.**

**33.1**,

**14.6**, 12.5, 9.7, 7.9, 6.7, 5.8, 5.1, 4.6} SSD = 18.0 chi-square =

**8.1.**

**32.1**, 17.6, 12.5, 9.7, 7.9, 6.7, 5.8, 5.1,

**2.6**} SSD = 8.0 chi-square =

**10.1.**

**31.1**,

**18.6**,

**13.5**,

**10.7**, 7.9,

**5.7**,

**4.8**,

**4.1**,

**3.6**} SSD = 8.0 chi-square =

**10.1.**

**32.1**,

**15.6**, 12.5, 9.7, 7.9, 6.7, 5.8, 5.1, 4.6} SSD = 8.0 chi-square =

**3.6.**

**7.1**,

**2.6**} SSD = 8.0 chi-square =

**16.6.**

**+**

**∆**, 0.176, 0.125, 0.097, 0.079, 0.067, 0.058, 0.051, 0.046

**−**

**∆**}.

**/**(20) or (25.17)

**/**(20), so that

**1.26**is the ratio of (chi-square)/(SSD), for all values of deviation

`∆`, regardless.

**5**Δ, 0.176 +

**3**Δ, 0.125 + Δ, 0.097, 0.079, 0.067, 0.058 −

**4**Δ, 0.051 −

**3**Δ, 0.046 −

**2**Δ}

**/**(SSD) for all values of deviation

`∆`,in many typical cases of deviations (where fixed chosen digits are affected by the same proportional deviations) suggest that there is no need to be much concerned about aiding high digits over and above low digits ostensibly in order to compensate for their milder fluctuations as opposed to the typically more dramatic fluctuations of the low digits. And this is so since SSD and chi-square here are a simply distinct scale of measurements which rise and fall together in exact proportions. Admittedly, when distinct proportional deviations occur at totally distinct digits, SSD and chi-square differ substantially and in a much more profound way than merely in a scale sense, as the chi-square statistic emphasizes deviations in high digits and deemphasizes deviations in low digits.

**/**190 or merely 5% of their weight, then no proof of drought exists. But the study of monkeys does cause an alarm, since typical weight of the monkey species there is merely 30 kg, meaning that they have lost 10

**/**30 or a whopping 33% of their weight; hence the proof of the existence of drought is decisive.

**∑|**Observed − Theoretical

**|/**(number of digits)

^{2}

^{4}

^{6}**− Theoretical**

^{3}**)**

^{3}

^{2}**|**√Observed − √Theoretical

**|**

**|**Observed

**− Theoretical**

^{2}

^{2}**|**

## 13. Testing the Lognormal via the Chi-Square Statistic

**The expected value of chi-square with 8 degrees of freedom is 8!**

**any**random distribution whatsoever for that matter!)

## 14. Testing the Fibonacci Series via the Chi-Square Statistic

**not**found via some independent and random selection processes from some imaginary Benford Universe of data-one value at a time!

## 15. Testing Partial US Population Data Sets via the Chi-Square Statistic

**tug-of-war**between the reduction in deviations from Log(1 + 1/d) proportions, and the increase in the value of N, and this “conflict” yields nearly a draw or a tie regarding resultant value of the chi-square statistic, as both factors exert nearly equal and balanced influence on it. Yet, eventually, as N reaches 19,510 city centers for the entire USA data without any corresponding further drastic reduction in deviations from the Log(1 + 1/d) proportions, chi-square finally gives in from the pressure of the high value of N, as it attains the high value of 17.5, standing well above the cutoff point of 15.5 for 5% confidence interval, and so the test declares the entire US data set to be non-Benford.

**whole**as non-Benford, while at the same time it certifies

**parts**of it as Benford!?

## 16. The Nature of Hypothetical Real-Life-Like Revenue Data

## 17. The Nature of Hypothetical Real-Life-Like Population Data

## 18. Conclusions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Kossovsky, A.E.
On the Mistaken Use of the Chi-Square Test in Benford’s Law. *Stats* **2021**, *4*, 419-453.
https://doi.org/10.3390/stats4020027

**AMA Style**

Kossovsky AE.
On the Mistaken Use of the Chi-Square Test in Benford’s Law. *Stats*. 2021; 4(2):419-453.
https://doi.org/10.3390/stats4020027

**Chicago/Turabian Style**

Kossovsky, Alex Ely.
2021. "On the Mistaken Use of the Chi-Square Test in Benford’s Law" *Stats* 4, no. 2: 419-453.
https://doi.org/10.3390/stats4020027