L'estimation du nombre originel de coins: en augmentant l'échantillon…（François de Callataÿ, Acta Numismàtica 21–23 (1991–1993, Essays in honour of L. Villaronga), pp. 31–48）

Claims Supported by This Source

1. p. 31: Symbols — n = number of specimens in the sample, d = number of obverse dies counted in the sample, D = original number of dies estimated statistically. Depending on the method, N1 (number of dies known from a single specimen) and N2 (number of dies known from two specimens) are also needed.
2. p. 32: The methods are called by the names of their inventors (Brown, Carcassonne, Carter, Esty, Good, Guilbaud, Lyon, MacGovern, Malkmus, Mora Màs, Müller, Schroeck). Only the denarii of Crepusius have numbered reverse dies, the only case where Carter could compare the statistical estimate with reality.
3. p. 33: Esty's 1986 computer simulations favor Good's method. After 1986, papers on estimating the original number of dies decreased sharply. Among quantitative researchers, two methods are mainly used: Carter's and Good's (revised by Esty). Carter's formula became the most popular due to its ease of use.
4. p. 34: Carter's method estimates the original number of dies, including both those used for a long time and those that broke quickly. Good's method estimates the coverage, i.e., what proportion of the total production is represented by the dies appearing in the sample. The latter can effectively ignore dies that were used only for a short time.
5. pp. 34–35: The fourth test in this paper, which examines the stability of the method's results by increasing the sample size for the same coinage (10, 20, 30, 40 coins...).
6. p. 35 (Table): Mithridates' tetradrachms — 1988 sample: n=505, d=156, n/d=3.24, N1=56, N2=27 → D/Carter=175.4 (167.8<D<183.7) · D/Good=192.9. 1992 sample: n=544, d=156, n/d=3.49, N1=61 → D/Carter=175.7 · D/Good=188.6.
7. pp. 46–47: When n/d exceeds 6, D/Good is over 97% of D/Carter; for 3–6, it is 90–97%; for 2–2.5, 70–89%; for 1.25–1.5, 60–72%. When n/d is less than 2, Good's method should be used with caution. When it exceeds 3, the difference is negligible.
8. p. 47: Carter's confidence interval is too small relative to the uncertainty, giving a false sense of precision (it is an illusion to set the upper bound of D at 125.1 for a sample of 10 coins with 9 obverse dies). Esty's 95% interval is more cautious, but its lower bound can sometimes fall below the number of counted dies (which is impossible).
9. p. 48 (Conclusion): All statistical methods underestimate the original number of dies, D, because they undervalue dies that were used only briefly. If the objective is D, there is no reason to discard Carter's method; if the objective is production volume, the Good/Esty method is the most attractive. The value of n/d affects not only the relative precision but also the absolute value, so samples with similar n/d values should be compared.

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