Last update 28-Jan-2005
Antioch Mint – Obverse Dies
1. Examined Issues
Two groups of issues of Alexander I’s tetradrachms from Antioch mint are examined:
- Undated issues:
Period: Probably early in the year 150 BC Obverse: Diademed head of Alexander I right, usually with long sideburn or with scanty beard along jawline; fillet border Reverse: ‘ΒΑΣΙΛΕΩΣ ΑΛΕΞΑΝΔΡΟΥ’ in two lines on right, ‘ΘΕΟΠΑΤΟΡΟΣ ΕΥΕΡΓΕΤΟΥ’ in two lines on left; Zeus seated left on throne, resting on sceptre and holding Nike facing right, offering wreath
- Dated issues:
Period: 150/49 - 146/5 BC (years 163 - 167 of the Seleukid Era) Obverse: Diademed head of Alexander I right, either with long sideburn or with scanty beard along jawline or clean shaven; fillet border Reverse: ‘ΒΑΣΙΛΕΩΣ ΑΛΕΞΑΝΔΡΟΥ’ in two lines on right, ‘ΘΕΟΠΑΤΟΡΟΣ ΕΥΕΡΓΕΤΟΥ’ in two lines on left; Zeus seated left on throne, resting on sceptre and holding Nike facing right, offering wreath (special issue in the year 164 SE: Athena standing left, resting hand on grounded shield and holding Nike facing right, offering wreath, spear propped in crook of Athena’s elbow); date in exergue
Observed die frequencies are presented in Table 1 and graphically in Figures 1, 2 and 3 below. The die frequencies are the numbers of dies represented in the sample exactly once, twice, three times etc. For example, 6 obverse dies used for the undated tetradrachms are each represented by exactly 1 coin in the die study, 1 obverse die used for the undated tetradrachms is represented by exactly 2 coins in the die study etc.
|Number of Coins||Number of Dies|
|Undated Issues||Dated Issues||All Issues|
Die frequencies of the dated issues with respect to individual years are presented in Table 2 below. The first column of this table shows the number of dies which are represented by the numbers of coins given in the next columns. For example, there is 1 obverse die that is represented by exactly 10 undated coins and by exactly 4 coins from the year 163 of the Seleukid Era. Similarly, there are 2 obverse dies in the die study, which are each represented by exactly 9 undated coins and by no dated coin.
Tables 1 and 2 are the complete representation of the observed data from the point of view of this analysis. That is, all quantities, statistics and estimations in the following sections are computed from these two tables.
|Number of Dies||Number of Coins|
|Undated||163 SE||164 SE||165 SE||166 SE||167 SE|
|150/9 BC||149/8 BC||148/7 BC||147/6 BC||146/5 BC|
3. Assumption of Randomness
Considerations and estimations presented in the following sections are based on the assumption that the examined data are random. That is, each coin of the analysed issues entered the sample independently with equal likelihood. As the studied corpus consists of coins included in many collections and of coins offered on the market during many decades, we can treat the data as reasonably close to random.
It is necessary to note that the data are not really random in the sense that data from a scientitic experiment can be. Coins from the same die were grouped together as they were originally made, and we do not know that they were thoroughly separated in the mixing process. So, a few dies may be over-represented by a batch of coins in a non-random manner. That should not affect averages much, but it means that we cannot expect an excellent “fit” to some theoretical model.
4. Basic Statistics
Table 3 shows the basic statistics of the examined data. By Table 2, there are two obverse dies used both for the undated and for the dated issues. The basic statistics of the dated issues with respect to individual years are presented in Table 4. Note that the sum of the numbers of different dies in the individual years (49) is greater by 6 than the total number of different dies of the dated issues presented in Table 3 (43) because 1 die was used in all five years and 2 dies were used in two subsequent years (see Table 2).
|Undated Issues||Dated Issues||All Issues|
|Number of different dies||18||43||59|
|Number of coins||110||262||372|
|Mean number of coins per die||6.11||6.09||6.31|
|Median of the number of coins per die||6||3||4|
|Std dev. of the number of coins per die||5.07||8.21||7.57|
|Maximum number of coins per die||18||32||32|
|163 SE||164 SE||165 SE||166 SE||167 SE|
|150/9 BC||149/8 BC||148/7 BC||147/6 BC||146/5 BC|
|Number of different dies||5||16||5||21||2|
|Number of coins||45||88||59||67||3|
|Mean number of coins per die||9.00||5.50||11.80||3.19||1.50|
|Median of the number of coins per die||4||2.5||5||2||1.5|
|Std dev. of the number of coins per die||11.60||7.94||11.28||3.28||0.71|
|Maximum number of coins per die||29||32||26||12||2|
As we can see in Table 3, the median of the number of observed coins per die is much lower for the dated issues than for the undated issues. However, the mean numbers of observed coins per die are nearly the same for both issues because we observe several high multiplicities in the sample of dated issues (23, 29 and twice 32 coins per die, see Table 1). A one-tailed Wilcoxon rank-sum test1 was used to test the hypothesis that the distribution of the number of coins per die is the same in both samples against the alternative that the numbers of coins per die for the undated issues tend to yield larger values than for the dated issues. The test does not reject the hypothesis that the distribution is the same in both samples (the p-value: 0.219, the rank-sum statistic: 606).
Both the undated and the dated issues are examined together in the following sections. The initial undated coinage started sometime in the year 162 SE after Alexander’s victory over Demetrios I. This short undated coinage was followed by the dated coinage in 163 SE. As we can see in Table 2, some obverse dies were used both in the year 162 SE and in the year 163 SE. Thus, there was probably no boundary between dies used for the undated and the dated issues.
5. Estimation of the Coverage
The coverage of a sample of coins of a given type is the fraction of all produced coins of the given type that are from dies represented in the sample.2 In other words, the coverage of a sample of coins of a given type is the probability that a new coin of that type will be from a die already observed in the sample. It means that 1 minus the coverage is the probability that a new coin would yield a new die. Note that the coverage is a property of the sample, not of the coinage issue.
The Good’s coverage estimator and the Esty’s formula for 95% confidence intervals were used.3 The coverage of our data sample is very high, see Table 5.
|95% confidence interval||91.8% - 98.0%|
6. Estimation of the Number of Dies
6.1. Probability Distribution of the Observed Numbers of Coins per Die
It is usually supposed that the number of coins produced by a random die has a negative binomial distribution with parameters r and p (r>0, 0<p<1). That is, that the probability that exactly n coins was produced by a random die is equal to
Γ(r+n) Γ(r)-1 Γ(n+1)-1pr (1-p)n , n = 0, 1, 2, ... ,
where Γ is the gamma function and r and p are fixed (but unknown) parameters. The negative binomial family is a two-parameter family, but standard estimates of the number of dies depends only on the shape (variability) parameter r.
It can be shown that if the numbers of coins produced by individual dies have a negative binomial distribution with parameters r and p, then the observed numbers of coins per die in a random sample have a zero-truncated negative binomial distribution with parameters r and q, i.e. with the same first parameter (the second parameter depends on the survival rate).4 It is recommended to put r = 2 or slightly less (but not less than 1) for estimating the original number of dies, see Esty, Estimating the size of a coinage: A survey and comparison of methods, Esty and Carter, The distribution of the numbers of coins struck by dies, Esty, Statistics in Numismatics, and Esty, Statistics in Numismatics, 1995-2001. Figure 4 shows the empirical cumulative distribution function and negative binomial cumulative distribution functions with the first parameter r equal to 1, 1.5 and 2. The second parameter q was estimated by the minimum distance method, i.e. it minimizes the Kolmogorov-Smirnov distance from the empirical cumulative distribution function.5 Note that the best possible fitting is achieved for r = 0.081 (the corresponding minimal Kolmogorov-Smirnov distance is equal to 0.044), see the green curve in Figure 4. Figure 5 shows the corresponding observed and expected die frequencies.
As we can see, all these three theoretical cumulative distribution functions fit the empirical cumulative distribution function badly. The “optimal” value r = 0.081 is unacceptably small for our purposes. The greater the parameter r, the greater the Kolmogorov-Smirnov distance. The chi-square goodness-of-fit test was therefore used to verify the assumption that the observed numbers of coins per die have a zero-truncated negative binomial distribution with the first parameter equal to 1 (the smallest acceptable value). The following four intervals for the number of observed coins per die were chosen for the test: 1 - 2, 3 - 4, 5 - 7, 8 and more. The unknown parameter q was estimated by the minimum chi-square method.6 On the 95% confidence level, we reject this hypothesis, see Table 6. It means that the negative binomial model does not fit our data well.
|1 - 2||27||20.5|
|3 - 4||7||13.4|
|5 - 7||9||11.9|
|8 and more||16||13.3|
|estimation of q||0.192|
(2 degrees of freedom)
|95% critical value
(2 degrees of freedom)
6.2. Number of Dies
The basic method described in Esty, Estimating the size of a coinage: A survey and comparison of methods, is based on the assumption that the number of coins produced by a random die has a negative binomial distribution. However, we showed above that the negative binomial model does not fit our data well. By Figure 4, the best possible fitting within the recommended range of the first parameter r is achieved for r = 1. Table 7 shows the corresponding estimation of the original number of dies.7
|Observed number of coins||372|
|Observed number of dies||59|
|Estimated number of dies||82|
|95% confidence interval||77 - 88|
The unknown number of all used dies is equal to the sum of the number of observed dies and of the number of unobserved dies. By Table 7, the number of unobserved dies is equal to 23 with a 95% confidence interval between 18 and 29 dies (the number of observed dies was subtracted from the estimated number of all dies and from the lower and upper confidence limit). It is important to note that the number of unobserved dies can be also roughly estimated graphically from Figure 2 as the missing die frequency for zero observed coins. This can be done by eye by fitting a smooth curve through the plotted frequencies. This curve can be either monotonic or unimodal. This graphical method indicates that the number of unobserved dies might be in the interval from about 10 to 40 dies, see Figure 2. The estimate given by Table 7 is consistent with this observation.
The high number of singletons (i.e. dies represented only by one coin in our sample) in comparison with the number of doubletons probably indicates that there were a lot of dies that broke after producing only a small number of coins.8 It means that the total number of produced dies might be much higher than Table 7 shows. Nevertheless, it is probably not possible to reliably estimate the fraction of dies which were defective and broke early. Moreover, the number of defective dies is not important for an appraisal of the size of Alexander I’s coinage. From this point of view, the estimate given by Table 7 seems to be reasonable.
- Two groups of AR tetradrachms of Alexander I from Antioch mint were examined: the undated issues (110 coins, 18 obverse dies) and the dated issues (262 coins, 43 obverse dies). In total, 372 coins and 59 obverse dies. We can suppose that the data are a fairly random sample from the population of all tetradrachms of Alexander I issued by Antioch mint.
- It seems that obverse dies were used equally intensively both for the undated issues and for the dated issues.
- The data are a highly representative sample of the population of Alexander I’s tetradrachms from Antioch mint. The 59 observed obverse dies produced an estimated 94.9% of the undated issues with a 95% confidence interval between 91.8% and 98.0%. In other words, the probability that a new coin would yield a new die not included in the die study is 5.1% with a 95% confidence interval from 2.0% to 8.2%.
- The negative binomial model does not fit our data well.
- The estimated number of all dies used for Alexander I’s coinage in Antioch is about 82 with a 95% confidence interval between 77 and 88. Nevertheless, this estimate and the corresponding confidence interval should be taken with great caution.
1 See, e.g., Conover, Practical Nonparametric Statistics, Third Edition, pp. 272-275.
2 That is, the coverage is the fraction M/N where M is the number of all coins originally struck by the dies that are observed in the sample and N is the number of all coins originally struck by all the dies used in the coinage issue.
3 See Esty, Estimating the size of a coinage: A survey and comparison of methods, Appendix 2.J, p. 208, formulae J2 and J3.
4 Let X be the number of coins produced by a random die. Consider a random sample from the population of coins produced by all dies. Let Y be the number of coins produced by the given die which are included into the sample (0≤Y≤X). For each i = 1, 2, ... , X, put Ai = 1 if the ith coin has survived and it has been included into the random sample, and put Ai = 0 if the sample does not contain the ith coin. The number of observed coins produced by the given die is then given by
Y = A1 + A2 + ... + AX .
If the random variable X has a negative binomial distribution with parameters r and p and if the sample is really random (i.e. if the zero-one random variables A1, A2, ... , AX are independent and identically distributed) then the random variable Y has a negative binomial distribution with parameters r and q = p/(p+π-pπ) where π is the probability that Ai = 1 (the survival rate).
However, we observe the random variable Y only if the die is represented in the sample, i.e. if Y ≥ 1. It means that the number of coins produced by the given die which are observed in the sample have a zero-truncated negative binomial distribution with the parameters r and q. Since the probability that Y ≥ 1 is equal to 1-qr, the probability that a random die is represented by n coins in the sample, n ≥ 1, is equal to
Γ(r+n) Γ(r)-1 Γ(n+1)-1qr (1-q)n (1-qr)-1.
6 See, e.g., Conover, Practical Nonparametric Statistics, Third Edition, pp. 243-245.
7 See Esty, Estimating the size of a coinage: A survey and comparison of methods, Appendixes 1.C, 2.H and 2.K. The formula H5 (p. 205) was used where the equal-output estimate k' was computed by the formula K1 (p. 209). The formula C2 (p. 201) was used for computation of confidence intervals.
8 The number of singletons may be also greater in a sample which is a corpus, because museums and collectors often avoid die duplicates, see Esty and Carter, The distribution of the numbers of coins struck by dies. I suppose that it is not the case of our data sample.
- Conover, W. J.:Practical Nonparametric Statistics, Third Edition. John Wiley & Sons, Inc., New York - Chichester - Weinheim - Brisbane - Singapore - Toronto, 1999.
- Esty, Warren W.:Estimating the size of a coinage: A survey and comparison of methods. Numismatic Chronicle, 146 (1986) pp. 185-215, the journal of the Royal Numismatic Society.
- Esty, Warren W.:Statistics in Numismatics, 1995-2001. To appear in Survey of Numismatic Research, 1997-2002, International Association of Professional Numismatists, Special Publication (2003).
- Esty, Warren W.:Statistics in Numismatics. Survey of Numismatic Research, 1990-1996, International Association of Professional Numismatists, Special Publication 13, Berlin (1997) 817-823.
- Esty, Warren W.; Carter, Giles F.:The distribution of the numbers of coins struck by dies. American Journal of Numismatics, the publication of the American Numismatic Society, Second Series 3-4 (1991-1992), pp. 165-186 .