Interpretaciones y reflexiones de las narrativas de Alba

Dozens of papers and formulæ dealing with this topic have been written and I will not attempt to review them all here. The methods have been subject to one general, and two detailed reviews including computer simulation experiments to assess their effectiveness (Metcalf 1981; Esty 1986; Lyon 1989). The most commonly employed methods are those of Lyon (1965) and Esty (1984). Esty (1986, n. 21, p. 198 & Appendices 1–2, pp. 199–209) lists proposed methods which should

not be used.

All these methods use the same type of data, i.e., the number of dies used to strike the coins in the sample under examination. Die-analysis is a simple, if time consuming operation. Coin dies in antiquity were cut by hand and thus between dies of the same type or design, there will be minor variations in the letter-form, the design and the relative positions of the elements. With a careful examination of a sample of coins, the number of dies used to strike those coins can be counted. This count, however, will be an under-estimate of the number of dies used to strike that coinage, unless the sample is very large and very complete, which is rarely the case. The task of the formula is, therefore, to give an estimate of the number of dies originally used, and preferably a confidence interval, i.e., a range within which the estimate is most likely to fall, usually a 95% confidence interval. This number of dies is then multiplied by the average number of coins struck per die to give an absolute output figure. Some methods also use the number of dies represented by only one coin in the sample, the number by two, three and so on in their estimates.

All of these methods rely on the sample of coins being a random selection of those struck. Many also rely on the numbers of coins struck per die being constant, which is known to be untrue. The method suggested by Esty (1984) does not assume equal output, and is the least effected by non- random sampling (Esty 1986). I shall briefly outline the method, without any formal mathematical proof. Esty’s method is based on the concept of the coverage of a sample originally introduced into the statistical literature by Good (1953).

The definition of the coverage of a sample is the total number of coins struck by dies identified in the sample, divided by the total number of coins struck. Thus, if in a sample of 100 coins 58 dies were identified, and these 58 dies struck 580,000 coins of a total of 1,200,000 coins, the coverage would equal 0.483. More formally,

C= T

0

T

(3.1) whereC is the coverage, T

0 is the number of coins struck by dies found in the sample, and T is

the total number of coins struck. To calculate the total number of coins struck we can rearrange the formula, T = T 0 C (3.2) In a real situation, we do not knowT

0, the numbers of coins struck by the dies in the sample, but

we can estimate this by multiplying the number of dies in the sample by a figure for the average number of coins struck per die. Thus,

T 0 = T 0 0 C (3.3) whereT 0

is an estimate of the total number of coins struck, andT 0

0is an estimate of the number of

coins struck by dies observed in the sample whereT 0 0

=dwheredis the number of observed

dies, andis the number of coins struck per die.

We now need an estimate of the coverage,C 0

, which can be estimated by

C 0 =1 N 1 n (3.4) whereN

1is the number of dies which are observed exactly once in the sample, and

nis the size of

the sample. This estimate can be very good, and is not affected by unequal die output. We need, however, some confidence limits so that the accuracy of the results can be assessed. Ifnis large,

andN 1

=nis not very near 0 or 1, the limits are given by

1 (N 1 =n)z v u u t N 1 +2N 2 n N 1 n 2 n ! (3.5)

wherezis 1.96 for the 95% confidence limit, 1.65 for the 90% or 1.0 for the 68% limits, andN 2is

the number of dies represented by two coins in the sample.

Esty provides the following worked example. In a hoard of 204 coins there were 178 distinct dies of which 156 dies where represented by a single coin, 19 by two coins, two dies by three coins and 1 die by four. Using his notation n = 204;N

1 = 156;N 2 = 19;N 3 = 2;N 4 = 1. Using equation 3.4 we get C 0 =1 N 1 n =1 156 204 =0:235:

Thus our best guess is that the 178 dies represents 23.5% of the coinage struck. Using formula 3.3, and a figure of 10,000 coins struck per die, we can estimate the size of the whole coinage as

T 0 = T 0 0 C 0 = d C 0 = 10;000178 0:235 =7;570;000:

This single, point estimate is very likely to be wrong and so we calculate the 95% confidence limits forC

3.13. Estimating the size of coin issues 95 C 0 1:96 v u u t 156+219 204 156 204 2 204 ! =0:2350:083: Using values ofC 0

of 0.152 (0:235 0:083) and 0.318 (0:235+0:083) we can state that with = 10;000, there is a 95% probability that the total number of coins struck was in the range

5,600,000–11,700,000.

Two further points must be noted. Firstly, if we are only interested in the relative sizes of issues, rather than the absolute sizes, we can work with an arbitrary value ofproviding we are happy to

accept that it is constant between the issues we are comparing. Secondly, we cannot convert the coverage estimate,C

0

, to an estimate of the total number of dies (D

0

) used to strike the issue, unless we are prepared to make an assumption as to the distribution of the number of coins struck per die. If we accept the simplest situation of equal output, we can simply divide the number of observed diesdby the coverageC

0

. In the case of the above example this would give us

D 0 = d C 0 = 178 0:235 =757:

Esty (1986) suggests that the distribution of the numbers of coins struck by dies will actually follow a negative-binomial distribution with a shape parameter (t) of 2. If this is the case, the number of

dies can be calculated by

D 0 = d C 0 + n(1 C 0 ) tC 0 : (3.6)

A revised estimate using this formula witht=2gives an estimate of the number of dies of 1089.

It is very important to note, however, that this does not affect the total numbers of coins struck as estimated above. The negative-binomial distribution allows for a large number of dies which broke quickly and produced very few coins, hence the increase in the number of dies, but no increase in the numbers struck.

Lyon (1989, p. 8), however, does not agree with the use of the negative-binomial correction to the die estimate unless there is clear evidence to suggest that the number of coins struck per die is distributed in this fashion. Esty & Carter (1991–1992) explicitly examine the distribution and found that negative-binomial distribution with a shape parameter of 1.5–2 fits the empirical data extremely well. Incidentally, they also note that the variability between reverse dies is generally greater than between obverses. This is almost certainly due to the fact that the obverse die is mounted in an anvil, and the reverse is the die struck with a hammer. Lastly, they note that it is important for die studies to record not only the sample size, and the number of observed dies, but also number of dies represented by two coins, three coins etc.