Submitted by Petr Vesely on
Founded 28-Nov-2004
Last update 28-Jan-2005
Antioch Mint - Obverse Dies References
Antioch Mint – Obverse Dies
1. Examined Issues
Two groups of issues of Demetrios I’s tetradrachms from Antioch mint are examined:
- Undated issues:
Period: 162/1 - 155/4 BC Obverse: Diademed head of Demetrios right, either clean shaven or light beard; laurel wreath border Reverse: ‘ΒΑΣΙΛΕΩΣ’ on right, ‘ΔΗΜΗΤΡΙΟΥ’ on left, either without the royal epithet ‘ΣΩΤΗΡΟΣ’ or this epithet on right or this epithet in exergue; Tyche, either nude to waist or fully clothed, holding short sceptre with right hand and cornucopiae with left arm, seated left either on cippus decorated with tritonesses or on backless throne with winged lion’s leg support or on backless throne with winged tritoness support - Dated issues:
Period: 155/4 - 151/0 BC (years 158 - 162 of the Seleukid Era) Obverse: Diademed head of Demetrios right, clean shaven; laurel wreath border Reverse: ‘ΒΑΣΙΛΕΩΣ’ on right, ‘ΔΗΜΗΤΡΙΟΥ ΣΩΤΗΡΟΣ’ on left; Tyche, fully clothed, holding short sceptre with right hand and cornucopiae with left arm, seated left on backless throne with winged tritoness support; date in exergue
2. Data
Observed die frequencies are presented in Table 1 and graphically in Figure 1 below. The die frequencies are the numbers of dies represented in the sample exactly once, twice, three times etc. For example, 34 obverse dies used for the undated tetradrachms are each represented by exactly 1 coin in the die study, 20 obverse dies used for the undated tetradrachms are each represented by exactly 2 coins in the die study etc.
Number of Coins | Number of Dies | |
---|---|---|
Undated Issues | Dated Issues | |
1 | 34 | 4 |
2 | 20 | 7 |
3 | 18 | 8 |
4 | 11 | 4 |
5 | 6 | 2 |
6 | 11 | 4 |
7 | 5 | 3 |
8 | 3 | 2 |
9 | 1 | 1 |
10 | 3 | 3 |
11 | 2 | |
12 | 1 | 4 |
13 | 2 | |
14 | 1 | 1 |
15 | 1 | |
16 | 1 | 1 |
17 | 1 | 1 |
18 | 1 | |
19 | 2 | |
20 | ||
21 | 1 | |
22 | ||
23 | 1 | 1 |
24 | ||
25 | 1 | |
26 | 1 | |
27 | ||
28 | 1 |
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 13 coins from the year 158 of the Seleukid Era and by exactly 2 coins from the year 159 of the Seleukid Era. Similarly, there are 2 obverse dies in the die study, which are each represented by exactly 6 coins from the year 158 of the Seleukid Era and by no coins from other years.
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 | ||||
---|---|---|---|---|---|
158 SE | 159 SE | 160 SE | 161 SE | 162 SE | |
155/4 BC | 154/3 BC | 153/2 BC | 152/1 BC | 151/0 BC | |
1 | 17 | ||||
1 | 13 | 2 | |||
1 | 12 | ||||
1 | 10 | ||||
1 | 9 | ||||
2 | 6 | ||||
3 | 3 | ||||
1 | 2 | 19 | |||
2 | 2 | ||||
2 | 1 | ||||
1 | 19 | ||||
1 | 11 | ||||
1 | 7 | ||||
1 | 6 | ||||
1 | 3 | ||||
1 | 2 | ||||
1 | 1 | ||||
1 | 23 | ||||
1 | 13 | ||||
1 | 12 | ||||
1 | 8 | ||||
1 | 5 | ||||
1 | 4 | ||||
2 | 3 | ||||
1 | 1 | 4 | |||
1 | 1 | ||||
1 | 25 | ||||
1 | 14 | ||||
1 | 12 | ||||
1 | 11 | ||||
1 | 10 | ||||
2 | 7 | ||||
1 | 6 | 13 | |||
2 | 4 | ||||
1 | 3 | 25 | |||
1 | 3 | 10 | |||
1 | 3 | 9 | |||
1 | 2 | 4 | |||
2 | 2 | ||||
1 | 1 | 9 | |||
1 | 1 | 3 | |||
1 | 1 | 2 | |||
1 | 1 | 1 | |||
1 | 16 | ||||
1 | 8 | ||||
1 | 3 |
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. Note that there is no observed obverse die 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 (68) is greater by 12 than the total number of different dies of the dated issues presented in Table 3 (56) because 12 dies were used in two subsequent years (see Table 2).
Undated Issues | Dated Issues | |
---|---|---|
Number of different dies | 119 | 56 |
Number of coins | 492 | 461 |
Mean number of coins per die | 4.13 | 8.23 |
Median of the number of coins per die | 3 | 6 |
Std dev. of the number of coins per die | 4.31 | 6.67 |
Maximum number of coins per die | 26 | 28 |
158 SE | 159 SE | 160 SE | 161 SE | 162 SE | |
---|---|---|---|---|---|
155/4 BC | 154/3 BC | 153/2 BC | 152/1 BC | 151/0 BC | |
Number of different dies | 15 | 9 | 11 | 21 | 12 |
Number of coins | 90 | 70 | 75 | 123 | 103 |
Mean number of coins per die | 6.00 | 7.78 | 6.82 | 5.86 | 8.58 |
Median of the number of coins per die | 3 | 6 | 4 | 4 | 8.5 |
Std dev. of the number of coins per die | 5.04 | 7.08 | 6.78 | 5.88 | 6.95 |
Maximum number of coins per die | 17 | 19 | 23 | 25 | 25 |
As we can see in Table 3, there is a clear difference in the mean number of observed coins per die and in the median of the number of observed coins per die between the undated and the dated issues. A two-tailed Wilcoxon rank-sum test rejects the hypothesis that the distribution of the number of observed coins per die is the same in both samples (the p-value is 4 × 10^{-6}, i.e. nearly zero). Moreover, by Table 4, not only is the mean number of observed coins per die of all dated issues greater than the mean number of observed coins per die of the undated issues, but it is so also in each dated year. It seems that obverse dies for the dated issues (i.e. in the period 155/4 BC - 151/0 BC) were used more heavily than obverse dies for the undated issues (i.e. in the period 162/1 BC - 155/4 BC).
The maximum mean number and median of coins per die is observed in the year 162 of the Seleukid Era (151/0 BC), i.e. in the last year of Demetrios I’s reign. However, it is not possible to examine the year 162 SE separately, because many dies were used in both years 161 and 162 SE, see Table 2. The time period of the dated issues was therefore divided into two periods 158 - 160 SE (155/4 - 153/2 BC) and 161 - 162 SE (152/1 - 151/0 BC). The basic statistics of these two periods are presented in Table 5. Note that one die was used in both periods, but this fact can be disregarded in the following considerations.
158 - 160 SE | 161 - 162 SE | |
---|---|---|
155/4 - 153/2 BC | 152/1 - 151/0 BC | |
Number of different dies | 33 | 24 |
Number of coins | 235 | 226 |
Mean number of coins per die | 7.12 | 9.42 |
Median of the number of coins per die | 5 | 7.5 |
Std dev. of the number of coins per die | 6.29 | 7.11 |
Maximum number of coins per die | 23 | 28 |
We can see that the number of coins observed in the final 2-year period is nearly the same as the number of coins observed in the first 3-year period, but the mean number and median of coins per die observed in the final 2-year period is much greater than in the first 3-year period. A possible explanation might be that an extensive production of coins was quickly demanded to finance the final campaign against Alexander Balas and that there was less time to prepare new dies. A two-tailed Wilcoxon rank-sum test was therefore used to test the hypothesis that the distribution of the number of observed coins per die is the same in both samples. The p-value is equal to 0.120 so it is not possible to reject the hypothesis at the 95% level of significance. Thus the difference between these two periods can be a random deviation only.
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.1 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.2 Results are given in Table 6. Both samples have a very high coverage. In particular, the coverage of the sample of the dated issues is nearly 100%. It means that the probability that a new coin would yield a die not included in the die study is very low (0.9%). Nevertheless, this estimate and the corresponding confidence interval should be taken with caution because it is not clear how they are accurate in such case (in fact, the upper bound of the confidence interval is equal to 101.0%, i.e. more than 100%).3
Undated Issues | Dated Issues | |
---|---|---|
Coverage | 93.1% | 99.1% |
95% confidence interval | 89.6% - 96.5% | 97.3% - 100.0% |
6. Estimation of the Number of Dies
6.1. Probability Distribution of the Observed Numbers of Coins per Die
We will suppose 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). It means 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)^{-1}p^{r} (1-p)^{n} , n = 0, 1, 2, ... ,
where Γ is the gamma function and r and p are fixed (but unknown) parameters. This assumption is general enough and it is very useful for the estimation of the number of dies.4 Note that the negative binomial distribution is used, for example, to model time-to-failure in reliability theory.
The negative binomial family is a two-parameter family. 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).5 Thus the first parameter, r, is a characteristic of the population of dies (it describes the shape of the distribution), whereas the second parameter varies to fit the sample size.
It is recommended to put r = 2 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. 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 2. The following four intervals for the number of observed coins per die were chosen for the test: 1 - 2, 3 - 4, 5 - 9, 10 and more. The unknown parameter q was estimated by the minimum chi-square method.6 The results are as follows:
Undated Issues | Dated Issues | |||
---|---|---|---|---|
estimation of q | 0.367 | 0.204 | ||
Interval | observed frequency |
expected frequency |
observed frequency |
expected frequency |
1 - 2 | 54 | 45.7 | 11 | 8.5 |
3 - 4 | 29 | 33.7 | 12 | 9.8 |
5 - 9 | 26 | 33.0 | 12 | 19.6 |
10 and more | 10 | 6.6 | 21 | 18.1 |
chi-square statistic | 5.337 | 4.624 | ||
p-value (2 degrees of freedom) |
6.9% | 9.9% | ||
95% critical value (2 degrees of freedom) |
5.992 | 5.992 |
The 95% quantile from the chi-square distribution with 4 - 1 - 1 = 2 degrees of freedom is 5.992. Thus, on the 95% confidence level, it is not possible to reject the hypothesis that, for both the undated and the dated issues, the observed numbers of coins per die have a zero-truncated negative binomial distribution with the first parameter r equal to 2.
Although it is not possible to reject the assumption that the observed numbers of coins per die have a zero-truncated negative binomial distribution with the first parameter r = 2, it is attractive to investigate the observed distribution in more detail. The Kolmogorov-Smirnov distance can be used as a measure of goodness of fit of the zero-truncated negative binomial model. Let F be an empirical cumulative distribution function of the observed data and let G_{r,q} be a cumulative distribution function of the zero-truncated negative binomial distribution with parameters r and q. The Kolmogorov-Smirnov distance of the functions F and G_{r,q} is defined as
max_{n≥1} |F(n) - G_{r,q}(n)| .
For each possible value of the first parameter r, we can find such value of the second parameter q that the Kolmogorov-Smirnov distance of the empirical cumulative distribution function F and the cumulative distribution function G_{r,q} is minimal. The results are presented in Figures 2 and 5 below. The left y-axis (blue line) shows the minimal Kolmogorov-Smirnov distance of the empirical cumulative distribution function from the family of zero-truncated negative binomial distributions with a given first parameter r (the x-axis). The right y-axis (red lines) shows values of n for which the absolute value of the difference F(n) - G_{r,q}(n) is maximal (the first parameter r is given by the x-axis and the second parameter q minimizes the Kolmogorov-Smirnov distance). Thus the right y-axis shows for which n it is difficult to fit the empirical cumulative distribution function by a negative binomial distribution.
For the undated issues, the optimal value of the first parameter r is 0.586 (the corresponding value of the second parameter q is 0.191), see Figure 2. For the dated issues, the optimal value of the first parameter r is 1.051 (the corresponding value of the second parameter q is 0.119), see Figure 5. Figures 3 and 6 show the empirical cumulative distribution functions, the cumulative distribution functions given by the optimal parameters r and q and the cumulative distribution functions with the first parameter r equal to 2 and to 1.5 (the second parameter q minimizes the Kolmogorov-Smirnov distance). Figures 4 and 7 show the corresponding observed and expected die frequencies.
Note that the maximum-likelihood estimations of the parameters r and q are as follows: r = 0.456 and q = 0.160 for the undated issues and r = 1.063 and q = 0.111 for the dated issues. Both the minimum distance method and the maximum-likelihood estimation give nearly the same results for the dated issues and quite similar results for the undated issues.
6.2. Number of Dies
The methodology described in Esty, Estimating the size of a coinage: A survey and comparison of methods, Appendixes 1.C, 2.H and 2.K, was used to estimate the original number of dies.7 Only the first parameter, r, of the negative binomial model is a characteristic of the population of dies. Generally speaking, lower values of r correspond to more dies which broke early and produced few coins. The recommended value is r = 2 or slightly less (but not less than 1), 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. It seems that the negative binomial model with r = 2 underestimates the fraction of dies that were defective and broke after producing only a few coins, see Esty, Statistics in Numismatics, 1995-2001, but low-ouput dies are not important for an examination of the size of the coinage. Nevertheless, as it seems that a lower value than 2 can be more adequate (see Table 7 and Figures 2 and 5), the value r = 1.5 should be also taken into consideration. The results are given in Table 8 below.
The parameter r can also be directly estimated from the data, but such an approach is not recommended, because it often leads to erratic and unlikely high estimates of the number of dies. However, for completeness, we also include die-number estimations based on the values of r given by the minimum distance method and by the maximum-likelihood method (note that the confidence intervals for these estimations do not include an uncertainty given by the estimation of r, so these confidence intervals should be wider than presented in Table 8).
Undated Issues | Dated Issues | |
---|---|---|
observed number of coins | 492 | 461 |
observed number of dies | 119 | 56 |
a priori choice of r | r = 2 | |
estimated number of dies 95% confidence interval |
146 136 - 157 |
59 56 - 61 |
a priori choice of r | r = 1.5 | |
estimated number of dies 95% confidence interval |
152 142 - 163 |
59 56 - 62 |
minimum distance estimation of r | r = 0.590 | r = 1.050 |
estimated number of dies 95% confidence interval |
190 175 - 205 |
60 58 - 63 |
maximum-likelihood estimation of r | r = 0.456 | r = 1.063 |
estimated number of dies 95% confidence interval |
208 191 - 226 |
60 57 - 63 |
We can see that all three estimations are quite similar for the dated issues, but they are very different for the undated issues. The results based on the values of r given by the minimum distance method and by the maximum-likelihood method seem to be too high. Perhaps the choice of r = 1.5 gives the most reasonable results (see also Figures 2 - 7), even if it probably also underestimates the number of low-ouput dies similarly as the choice of r = 2.
7. Conclusions
- Two groups of AR tetradrachms of Demetrios I from Antioch mint were examined: the undated issues (492 coins, 119 obverse dies) and the dated issues (461 coins, 56 obverse dies). We can suppose that the data are a fairly random sample from the population of all tetradrachms of Demetrios I issued by Antioch mint.
- Obverse dies for the dated issues (i.e. in the period 155/4 BC - 151/0 BC) were used more heavily than obverse dies for the undated issues (i.e. in the period 162/1 BC - 155/4 BC).
- The data also indicate that in the last two years of Demetrios I’s reign (152/1 - 151/0 BC) obverse dies were used more heavily than in the previous three years (155/4 - 153/2 BC). It can relate to the final campaign against Alexander Balas, but the difference is not statistically significant at the 95% confidence level.
- The data are a highly representative sample of the population of Demetrios I’s tetradrachms from Antioch mint. The 119 obverse dies observed on the undated coins produced an estimated 93.1% of the undated issues with a 95% confidence interval between 89.6% and 96.5%. The 56 obverse dies observed on the dated coins produced an estimated 99.1% of the dated issues with a 95% confidence interval between 97.3% and 100.0%. In other words, the probability that a new coin would yield a new die not included in the die study is 6.9% for the undated issues (with a 95% confidence interval from 3.5% to 10.4%) and 0.9% for the dated issues (with a 95% confidence interval from 0.0% to 2.7%). Nevertheless, the estimate for the dated issues and the corresponding confidence interval should be taken with great caution.
- It is not possible to reject the hypothesis that, for both the undated and the dated issues, the numbers of coins per die have a negative binomial distribution with the shape parameter equal to 2. Nevertheless, it seems that a somewhat lower value than 2 can be more adequate, perhaps 1.5.
- The estimated number of dies used for the undated issues is about 152 with a 95% confidence interval between 142 and 163. The estimated number of dies used for the dated issues is about 59 with a 95% confidence interval between 56 (i.e. no missing obverse die in the data) and 62.
1 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.
2 See Esty, Estimating the size of a coinage: A survey and comparison of methods, Appendix 2.J, p. 208, formulae J2 and J3.
3 See Esty, Estimating the size of a coinage: A survey and comparison of methods, Appendix 2.M, p. 211.
4 See Esty, Estimating the size of a coinage: A survey and comparison of methods, and Esty and Carter, The distribution of the numbers of coins struck by dies.
5 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 A_{i} = 1 if the ith coin has survived and it has been included into the random sample, and put A_{i} = 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 = A_{1} + A_{2} + ... + A_{X} .
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 A_{1}, A_{2}, ... , A_{X} 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 A_{i} = 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-q^{r}, 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)^{-1}q^{r} (1-q)^{n} (1-q^{r})^{-1}.
6 See, e.g., Conover, Practical Nonparametric Statistics, Third Edition, pp. 243-245.
7 Esty, Estimating the size of a coinage: A survey and comparison of methods: 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.
References:
- 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 .