A general methodology is proposed for using digit distributions as an approach to examining arbitrary datasets. With the Newcomb–Benford law as a star...

7 downloads
550 Views
424KB Size

No documents

Peter N. Posch* is a PhD student in the Department of Finance, University of Ulm, Germany. He studied economics and philosophy at the University of Bonn, Germany, and graduated in 2003 with a statistical thesis on the first digit law and its use in fraud detection. His research fields cover credit risk, digital analysis and fraud detection.

Welf A. Kreiner is Professor of Physical Chemistry at the University of Ulm, Germany. He was a student at the University of Graz, Austria, and at the Secondary College of Business Administration. He has carried out research at the University of Chicago and at the National Research Council of Canada. In addition to work in other fields, such as laser spectroscopy and data processing in connection with optical phenomena, he is investigating first digit laws on statistical and scientific data. *Department of Finance. University of Ulm, 89081 Ulm, Germany Tel: +49 731 50 23596; E-mail: [email protected]

Abstract A general methodology is proposed for using digit distributions as an approach to examining arbitrary datasets. With the Newcomb–Benford law as a starting point, a more general framework for digital analysis is developed. A new measure is proposed based on this framework, namely the Digital-Fit Factor (DFF). The use of index comparison on the S&P500 and the Dow Jones Industrial Average is demonstrated. The DFF is then used to construct portfolios and measure their performance compared with that of the index. The average returns using the measure exceed the index composition by 6–14 percentage points per year by being more stable at the same time. Furthermore, these measures require only a very small proportion of the available information and are thus very efficient. Keywords: stock market index, digital analysis, Newcomb–Benford law

Introduction Stock markets are considered to have a large, if not the largest, impact on the modern world economy. For information on how major economies change, certain national stocks are gathered in an index. Naturally, there are several methods of picking stocks for inclusion in an index, and the announcement of an index change often also changes the market prices of both the included and excluded stocks. The calculations of the indices themselves are reported daily in newspapers, hourly on radio and

䉷 Palgrave Macmillan Ltd 1479-179X/06 $30.00

television and in realtime in the Internet and on news channels. Anyone who ever listened to a conversation in a bar, subway or gym would not doubt that, as well as the weather forecast, changes in the national stock market index are worth discussing extensively. Although the index companies claim that the composition of their product is based on quantitative variables such as free float (ie the number of shares that are freely available to the investing public) or trading volume (ie number of shares traded each day), there are many

Vol. 7, 1, 69–80

Journal of Asset Management

69

Posch and Kreiner

qualitative factors which have to be fulfilled in order to gain access to the index, eg quarterly balance reports or certain corporate governance rules. The index itself is seldom calculated as the arithmetic mean of the market prices of the stocks included, but is more often subject to more complicated schemes. This paper demonstrates a general methodology for digital analysis of economic measures. The generality of this approach leads to a measure that can be applied to any set of economic variables in order to compare them within time or within economic units such as country, state etc. The measure is based on an approach known as ‘digital analysis’, which is already used to detect fraud in tax sheets and company reports. The most widely used distribution in this field is known as the ‘First-Digit law’ or ‘Benford law’. This name ignores the original discovery of the law by Simon Newcomb, so this phenomenon will referred to here as the Newcomb–Benford law (NBL). The power of the approach will be demonstrated by examining the Dow Jones Industrial Average Index (DJIA) and the Standard and Poor’s Composite Index over time. It is demonstrated how this measure can be used for portfolio composition. Using the stock indices, the study shows that a portfolio maintained by a digital measure outperforms an index-based approach by 6–14 percentage points and is more stable at the same time. The remainder of this paper is organised as follows. The following section gives a short introduction to the NBL and then introduces the measures used. The third section applies these measures to stock market indices. The results given in this section are mainly descriptive, and assumptions on further utility are left to the reader. The fourth section shows the use of the DFF in

70

Journal of Asset Management

Vol. 7, 1, 69–80

portfolio composition. Conclusions are drawn in the final section.

Newcomb–Benford law Newcomb (1881) noticed that the first few pages of logarithmic tables were more worn than the later ones.1 From a statistical viewpoint, he concluded that ‘the law of probability of the occurrence of numbers is such that all mantissae of their logarithms are equally probable’. While his work remained largely unnoticed, Benford (1938) discovered the same property 50 years later and studied a wide collection of datasets. Since then, many mathematicians have tried to prove the so-called First-Digits or Benford law. Some of these proofs are quite convincing, but they all lack a proper definition of the underlying probability space. This problem was solved by Hill (1996) with the definition of an appropriate -Algebra and the derivation of a probability measure. Furthermore, Hill (1996) gives a very plausible explanation with a statistical derivation of the First-Digit phenomenon. Basics

Newcomb (1881) states that, for certain natural numbers, the mantissae of their logarithms are equally distributed.2 From this assumption, he derived his First-Digit law, which gives the probability for a first digit d 苸 {1, 2, . . ., b ⫺ 1}, where b > 2, b 苸 N gives the number base. Since most empirical datasets are given in the decimal base (b ⫽ 10), the focus will be on that number system in the remainder of this paper, but the results are also valid for arbitrary bases other than 10. The function Dk(x) denotes the kth significant digit of x 苸 ᑬ, eg D1(3.1415) ⫽ D1(31415) ⫽ D11(0.0031415) ⫽ 3

䉷 Palgrave Macmillan Ltd 1479-179X/06 $30.00

Portfolio formation and index tracking

and

D4(3.1415) ⫽ D4(31415) ⫽ D4(0.0031415) ⫽ 1

The probability of a leading digit d is given by Prob[D1(x) ⫽ d] ⫽ logb(1 ⫹ 1/d)

(1)

Recall that the logarithm to base b can be calculated using the natural logarithm (the logarithm to base e ⫽ 2.7. . .) by logb(x) ⫽ ln(x)/ln(b). For positions exceeding the first digits, this equation generalises to Prob[Dn(x) ⫽ dn] ⫽

冘

b(n–1)

logb[1 ⫹ 1/(k ⫻ b ⫹ dn)]

(2)

k=bn–2

Counting

The frequency of the occurrence of first digits is obtained from a simple counting process. Using an automatic procedure for this counting process, one has to account for the significance of the digits, eg 0.001 has a 1 as the first significant digit. To gain uniqueness the routine uses mantissae of numbers instead, ie every number x 苸 R is split into a part m 苸 [0, 10] and a part 10k for a given k 苸 N. For example consider the number x ⫽ 100 ⫽ 314.15 . . .. One can write this number as 102 (note that 苸 [1, 10]). The benefit of this notation is that the decimal point is fixed, so a counting routine based on m is unique and, furthermore, includes all the necessary information. Digital-Fit Factor

By non-linear methods, fit the density function vector of the NBL whose entries are given by Counti/N ⫽ [(di ⫹ 1)1–DFFK ⫺ di1–DFFk]/(101–DFFk ⫺ 1) (3)

䉷 Palgrave Macmillan Ltd 1479-179X/06 $30.00

where N is the number of observations, di contains the possible digits, and k is the depth of the digital analysis. For the analysis of the first significant digits, this is d ⫽ 1, 2, . . ., 9, while an examination of ongoing positions includes the zero as a possible significant digit. For a detailed deviation of Equation (3), see Kreiner (2003). The density function is thus defined as C/N ⫽ D, where C is the 1 ⫻ 10 vector of counted values for each digit 0, 1, 2, . . ., 9, N is the constant number of observations, and D is the 1 ⫻ 10 vector of the digits probability as given in equation (3) ie d0 ⫽ (0, 1, 2, . . ., 9). Analysing the first significant digits, the first row of each vector is obsolete, since 0 is not a possible first digit. In this case, the vectors reduce to 1 ⫻ 9 vectors. Note that both N and D are given a priori, ie independently of the current counted digit frequency and that the left-hand side of Equation (3) gives the percentage of first digits i in the dataset. By non-linear fit-procedures, estimators are obtained for the Digital-Fit Factor DFFk of digital depth k and its variance VAR[DFFk].3 Depending on the depth of the analysis undertaken, k varies between 1, ie the first-significant digit, and the maximum possible length given by the current dataset. The remainder of this paper will refer to DFF as the Digital-Fit-Factor for first significant digits, but it is crucial to note that the analysis described above can be performed for arbitrary digit positions k > 1. In this case, the DFF-value will be denoted by DFFk-Value. It is notable, that the information included rises with k, ie in the calculation of DFF1 only the first significant digit of each observation is included, whereas for DFF4 the first four digits are used

Vol. 7, 1, 69–80

Journal of Asset Management

71

Posch and Kreiner

Table 1 Newcomb–Benford probability (NBP) of digit d at 1st to 4th position, generated by Equation (2) and rounded to five significant digits NBP (significant digit d) d

1st

2nd

3rd

4th

0 1 2 3 4 5 6 7 8 9

– 0.30103 0.17609 0.12494 0.09691 0.07918 0.06695 0.05799 0.05115 0.04576

0.11968 0.11389 0.10882 0.10433 0.10031 0.09668 0.09337 0.09035 0.08757 0.08500

0.10178 0.10138 0.10097 0.10057 0.10018 0.09979 0.09940 0.09902 0.09864 0.09827

0.10018 0.10014 0.10010 0.10006 0.10002 0.09998 0.09994 0.09990 0.09986 0.09982

etc. For any k, the DFFk is unequal to zero by definition. Since the NBL approaches the uniform distribution with raising digit position k, eg for a first significant digit 1 the Newcomb-Benford probability (NBP) is about 30.1 per cent, while the uniform distribution is at 1/9, but the NBP for a fourth significant digit 1 is ~10.014 per cent, while the uniform probability is 1/10 (since zero is a possible fourth digit), less and less ‘new’ information is added in comparison to the uniform distribution (see Table 1 for details). Following this observation and citing an unpublished result by K. Schuerger that almost every random variable with continuous density function will show such asymptotic behaviour, this study will not focus on DFF-values for positions greater than 1. Note that this restriction is without loss of generality, since the ‘additional’ information of the NBL is greatest for DFF1 and declines afterwards. The DFF gives an indication of the skewness of the digital distribution function. If the distribution is degenerated in the sense that it has only one mass point, eg all first significant digits are equal to 3, the distribution function is not well defined and the non-linear fit is not

72

Journal of Asset Management

Vol. 7, 1, 69–80

meaningful in a statistical sense. With at least two mass points present, for example the first significant digits are either 1 or 3 but never 2, 4, 5, . . ., 9 the DFF can be calculated in most cases with sufficient accuracy. An F-test is used on the non-linear estimation of the DFF and the p-values are reported in the empirical analysis below.4 The authors recommend using a confidence level based on the purpose of the digital analysis, eg for the size of lakes in the Chicago area, a confidence level of 10 per cent might be sufficient, while in the use of fraud detection, a 1 per cent confidence level could be taken to avoid false positives. The DFF is independent of the number of total observations N as it is based on proportions rather than absolute counts. If one has, for example, a dataset with N ⫽ 500 observations of which half have a first digit of 1 (Count1 ⫽ 250) and the other half have a first digit of 3 (Count3 ⫽ 250) the corresponding DFF is around 1.68 (p-value: 0). The same results hold for N ⫽ 100 observations with Count1 ⫽ 50 and Count3 ⫽ 50. Thus the DFF is based on the proportion rather than the absolute counts. More observations, however, increase the precision of the estimate. Figure 1 shows some distributions with their DFF-values.

䉷 Palgrave Macmillan Ltd 1479-179X/06 $30.00

Portfolio formation and index tracking

DFF=0.5

first digit d DFF ⫽ 0.5 DFF ⫽ 1.0 DFF ⫽ 2.0

DFF=2.0

DFF=1.0

60.00%

50.00%

Probability

40.00%

30.00%

20.00%

10.00%

0.00% 1

2

3

4

5

6

7

8

9

1 19.16% 30.10% 55.56%

6 9.08% 6.69% 2.65%

2 14.70% 17.61% 18.52%

7 8.45% 5.80% 1.98%

3 12.39% 12.49% 9.26%

8 7.93% 5.12% 1.54%

4 10.92% 9.69% 5.56%

9 7.50% 4.58% 1.23%

5 9.87% 7.92% 3.70%

First Digit d Figure 1 Graphical representation of the digits distribution using three different values with a table of the corresponding probabilities

A DFF-value of 1 results in the digit frequency of the NBL, whereas values lower than 1 indicate a flatter distribution, and values greater than 1 a steeper distribution. Uniformly distributed digits would result in a DFF-value of zero, but cannot be estimated with the procedure described above since the values of C in Equation (3) are all the same.

Stock index analysis This section demonstrates how digital analysis in general and the DFF measure in particular can be used to examine the behaviour of stock market indices. Doubtless, these indices are a major measure of a country’s performance and have great political impact. Usually, there are two variables which are of public interest: the current level of the index and its change over a specific period of time. While the former is not

䉷 Palgrave Macmillan Ltd 1479-179X/06 $30.00

informative per se, the latter is often used to describe and compare countries’ economic wealth in terms of stock markets. Furthermore, investors use such information to evaluate investment projects. One problem which arises in using changes in the index level is the comparability over time and across economies. The usual way to incorporate information on the volatility of performance is the use of variances. This does not solve the problem, however, but rather makes it worse. Since the variance is not scale independent, it cannot be used for comparison. Using digital analysis and, in particular, the DFF leads to a more structural approach in comparing two countries and their stock index performance. This general procedure will be demonstrated with two commonly used stock indices, namely the Standard & Poor’s Composite (S&P500) Index and the Dow Jones Industrial Average (DJIA) Index.

Vol. 7, 1, 69–80

Journal of Asset Management

73

Posch and Kreiner

Table 2 Digital and market measures for S&P500 index (yearly basis): DFF and market measures; the latter are market capitalisation (MKTCAP) and price index (PI), where the latter is usually referred to as the index value Year

DFF

SD (DFF)

Market cap.

Index value

2000 2001 2002 2003

0.518 0.631 0.842 0.500

0.211 0.229 0.182 0.242

11,735,261 10,581,281 8,066,633 10,137,988

1,320 1,161 875 1,096

Table 3 DFF-values of the indices (daily basis): DFF-values of several indices on a daily basis starting at 1st January, 2003, ending at 31st December, 2003

Mean Median Stand. dev. 5th percentile 95th percentile

S&P500

DJIA

0.815 0.814 0.038 0.749 0.884

0.423 0.347 0.184 0.211 0.720

Note: DJIA, Dow Jones Industrial Average Index (USA); S&P500, Standard and Poor’s 500 Index (USA)

Table 4 Descriptive statistics of stock market indices: index-values of several indices on a daily basis starting at 1st January, 2003, ending at 31st December, 2003

Mean Median Stand. dev. 5th percentile 95th percentile

S&P500

DJIA

961.61 984.03 77.05 7,907.19 11,800.40

13,417.06 13,612.55 1,107.28 11,630.30 14,991.65

Note: DJIA, Dow Jones Industrial Average Index (USA); S&P500, Standard and Poor’s 500 Index (USA)

S&P500

The S&P500 is widely regarded as one of the best single gauges of the US equity market. It includes 500 companies of the US economy and focuses on the large-cap segments. Using the companies included, the index covers 80 per cent of the US equities and is thus an ideal proxy for the total market. The digital analysis is demonstrated using yearly data ranging from 2000 to 2003. The dataset is publicly available on the Standard and Poor’s homepage. Price data of the ultimo of each year (usually the 29th of December) were used. Table 2 shows descriptive data of the yearly S&P500, including the market capitalisation (MKTCAP) and the price index (PI). Although it seems that the

74

Journal of Asset Management

Vol. 7, 1, 69–80

index value and the DFF are positively correlated, this correlation is not significant for the yearly data. Figure 2 show the S&P500 using daily data during the year 2003, while Figure 3 gives the daily values of the DFF over the same period. In Tables 3–5 the corresponding descriptive statistics are given in the rightmost column. Using this data the correlation of the index value and the DFF is in fact negative at –0.9832 (p–value: 0). Using the return of the S&P500, this correlation is persistent but not significant for the DFF. The correlation of DFF and the daily return of the index is at –0.0801 (p-value: 0.20). This analysis shows the close connection of the index and the distribution of first digits of its stocks.

䉷 Palgrave Macmillan Ltd 1479-179X/06 $30.00

Portfolio formation and index tracking

Table 5 Returns of indices (daily basis) (%): the daily returns of several indices on daily basis starting at 1st January, 2003, ending at 31st December, 2003 S&P500 Mean Median Stand. dev. 5th percentile 95th percentile

DJIA

0.09 0.10 1.07 –1.52 1.95

0.09 0.10 1.04 –1.63 1.89

Note: DJIA, Dow Jones Industrial Average Index (USA); S&P500, Standard and Poor’s 500 Index (USA)

Keep in mind that all the information used to compute the DFF are the first digits of the stock prices. No assumptions are made on the index composition or any other measure, just pure digits. Now the questions arise: (a) ‘Is this behaviour stable?’ and (b) ‘Could it be used for investment strategies etc.?’ The second question is answered in the fourth section of this paper. The first could be answered by comparing indices across countries. This exercise is, however, beyond the scope of this paper. Dow Jones

The Dow Jones index is one of the oldest stock market indices. As the homepage of Dow Jones indicates: ‘When Charles H. Dow first unveiled his industrial stock average on May 26, 1896, the stock market was not highly regarded. Prudent investors bought bonds, which paid predictable amounts of interest and were backed by real machinery, factory buildings and other hard assets.’ Today the DJIA is maintained and reviewed by editors of the Wall Street Journal. To gain continuity, composition changes are relatively rare. The only paper known to us which uses digital analysis of stock markets is by Ley (1996), who discusses the digital distribution of daily returns of the DJIA Index. He uses the chi-squared test as the goodness-of-fit measure. Although these tests are not significant, he concludes a close relationship between

䉷 Palgrave Macmillan Ltd 1479-179X/06 $30.00

the returns and the NBL distribution. Considering the assumptions of the chi-squared test that the observation must be independent, the authors doubt his conclusion that the rejection of the test is due to a ‘weakness of the Newman–Pearson-Theory’. Furthermore Ley (1996) uses a direct comparison of the actual digits observed in the returns of the DJ and the expected ones due to the NBL. The present methodology fits a whole digit distribution and compares it with the NBL distribution. Furthermore, returns are not used. Figure 4 gives a plot of the daily DJIA values, while Table 4 gives descriptive statistics for both the Dow Jones and the S&P500 index. Figure 5 shows the DFF values for the DJIA, with the descriptive statistics given in Table 3. Note that the DFF-value contains almost the whole information of the index. Finally, Table 5 gives descriptive statistics for the return of both indices. The Dow Jones Index gives the same picture as before. The DFF and the index value are highly significant, correlated at –0.9689 (p-value: 0). Again the (daily) returns of the index show the same correlation sign, but are less significant. This shows that the direction of the correlation is stable across the two indices examined. With this result, it is possible to use standard econometric methods to predict stock index values using the digital measure of DFF. A preliminary analysis showed stable regression results using these measures.

Vol. 7, 1, 69–80

Journal of Asset Management

75

Posch and Kreiner

76

Figure 2

S&P500 stock index

Figure 3

DFF of the S&P500 stock index

Journal of Asset Management

Vol. 7, 1, 69–80

䉷 Palgrave Macmillan Ltd 1479-179X/06 $30.00

Portfolio formation and index tracking

Figure 4

Dow Jones

Figure 5

DFF of the Dow Jones

䉷 Palgrave Macmillan Ltd 1479-179X/06 $30.00

Vol. 7, 1, 69–80

Journal of Asset Management

77

Posch and Kreiner

Table 6

Descriptive statistics of the portfolio returns Portfolio return (%) using

Mean Median Stand. dev. 5th percentile 95th percentile

S&P500

DFF of S&P500

Dow Jones

DFF of Dow Jones

7.65 10.58 9.60 9.10 20.65

22.03 25.02 1.95 9.28 53.26

4.11 0.00 8.03 9.14 17.47

10.75 1.09 1.55 9.18 35.11

Since the purpose of this paper is to show a general methodology of digital analysis, these results are not reported.

Portfolio composition To demonstrate the use of the DFF index, the following portfolios are simulated. Starting with an initial portfolio value of 100, trade takes place if and only if the index change is beyond a specific threshold. Consider the following example. Day

S&P500

DFF of S&P500

PF using PF using index DFF

1 2 3 4

879.82 909.03 908.59 929.01

0.81 0.78 0.79 0.77

100 100 99.9 102.15

100 100 99.9 100.77

On day 1, the change in the S&P500 cannot be calculated since values before that date are not included in this dataset. On day 2, the change is calculated as the value of day 2 divided by the value of day 1 minus 1: 879/909 ⫺ 1 ⫽ 3.41 per cent. The value of the portfolio is 100 times the actual change, that is 103.41. At the end of day 2, the portfolio composition is revised. If the observed index change is beyond the given threshold, the portfolio is adjusted by the proportion of the index change, eg take a threshold of 2.5 per cent. Since the change observed on day 2 (3.32 per cent) is beyond the threshold, the portfolio is adjusted and, since the

78

Journal of Asset Management

Vol. 7, 1, 69–80

change is positive, 3.32 per cent of the index is bought into the portfolio. This strategy assumes that the observed trend is expected to be persistent, ie if the index rises, further increases are expected, if it falls, further decreases are expected. The level of the threshold takes account of the changes one is willing to accept without adjusting the portfolio, eg a threshold higher than the maximum observed change (eg <10 per cent as a daily change of more than 10 per cent is very unlikely) will result in not adjusting at all. In this case, all simulated portfolios have the same value. A very small threshold (eg 0.01 per cent) results in a daily adjustment. Since the choice of the threshold level influences the value of the portfolio over time, its specification is crucial. The purpose is to demonstrate the use of the DFF in portfolio composition, so a wide range of possible thresholds are simulated and the mean performance over all simulation steps is examined. The range of thresholds starts at the minimum of the daily changes observed for each index and ends at the corresponding maximum. For the DJIA the minimal (maximal) daily return was at ⫺3.61 per cent (3.59 per cent), for the S&P500 at ⫺3.52 per cent (3.54 per cent). The thresholds steps are given by the standard deviation. Robustness checks using smaller steps showed no statistical significance. Table 6 shows descriptive data of the mean of all simulations. Figures 6 and 7

䉷 Palgrave Macmillan Ltd 1479-179X/06 $30.00

Portfolio formation and index tracking

Figure 6 Dow Jones Index: portfolio comparison over time: the upper line is the portfolio value using the DFF factor, while the lower line gives the return investing in the index

Figure 7 S&P500 Index — portfolio comparison: the upper line is the portfolio value using the DFF factor, while the lower line gives the return investing in the index

䉷 Palgrave Macmillan Ltd 1479-179X/06 $30.00

Vol. 7, 1, 69–80

Journal of Asset Management

79

Posch and Kreiner

give a graphical overview of the mean performance over time.

Conclusion This paper demonstrates a general methodology for digital analysis of economic measures. The generality of this approach leads to two measures which can be applied to any set of economic variable in order to compare them within time or within economic units such as countries, states etc. The measures are based on an approach known as ‘Digital Analysis’, which is already used to detect fraud in tax returns and company reports. The most widely used distribution in this field is known as the ‘First-Digit law’ or ‘Benford law’, referred to in this study as the Newcomb–Benford law. The DFF is based on the probability of digits distribution and fitted using non-linear techniques. The power of this measure is demonstrated examining the DJIA Index and the Standard and Poor’s Composite Index over time. Using the DFF as a tool for portfolio compositions shows yields to an excess return of 6–14 percentage points compared with an index-based method. Returns on the DFF portfolios are furthermore more stable, with a standard deviation of less than two percentage points. Further research should focus on the

80

Journal of Asset Management

Vol. 7, 1, 69–80

impact of the DFF on other economic variables, such as growth rates of unemployment, cash transfers etc., in order to explain differences between national and international markets. Furthermore, the use of digital analysis in the field of index tracking and portfolio composition should be examined more closely. Notes 1. Before the invention of calculators, these tables where used to multiply numbers by adding their logarithms. 2. Note that Newcomb uses the term ‘mantissae’ in the sense of logarithmic mantissae, ie the fractional part of the logarithm of a real number. 3. An Excel program is available from the authors upon request. 4. The p-value, as usual, refers to the confidence level of an F or t test, ie for a given test p gives the probability that the test value occurred only randomly. This paper refers to a ‘high-significant’ confidence level for p < 1 per cent and ‘significant’ for p < 5 per cent.

References Benford, F. (1938) ‘The Law of Anomalous Numbers’, Proceedings of the American Philosophical Society, 78(4), 551–572. Hill, T. P. (1996) ‘A Statistical Derivation of the Significant-Digit Law’, Statistical Science, 10(4), 354–363. Kreiner, W. A. (2003) ‘On the Newcomb-Benford Law’, Z. Naturforschung, 58a, 618–622. Ley, E. (1996) ‘On the Peculiar Distribution of the US Stock Indices First Digits’, The American Statistician, 50(4), 311–314. Newcomb, S. (1881) ‘Note on the Frequency of Use of the Different Digits in Natural Numbers’, American Journal of Mathematics, 4, 39–40.

䉷 Palgrave Macmillan Ltd 1479-179X/06 $30.00

Copyright © 2020 SLIDEHEAVEN.COM. All rights reserved.

## Follow us