This web page contains preliminary comments on the weight standards estimates. The theoretical concept described is tentative and the numerical example given is an illustration only, not a proper statistical analysis.

Weight decomposition

The observed weight of each coin W_observed can be expressed as the difference between its original weight W_original when it left the mint and the weight loss L due to wear, physico-chemical processes caused by environmental influences over time, and possibly other factors, such as cutting metal off the edge (the so-called clipping).1 Formally expressed

Wobserved = Woriginal – L.

(1)

The quantities W_observed, W_original and L are random variables. The distribution of W_observed is usually unimodal2 and asymmetrically skewed to the left, that is the part to the left of the mode is longer and with a more gradual slope than the part to the right of the mode. In other words, the distribution of observed weights is negatively-skewed.

The weight standard, which we denote by w, is the mean value of original weights of coins of a given denomination minted in a given time period. Formally, the weight standard is the expected value of the random variable W_original, i.e. w = E(W_original).

For each individual coin, we can find its current weight W_observed, but both the W_original and L values are unknown and unobservable. One way to estimate the weight standard w is to choose appropriate parametric models of the distribution of the random variables W_original and L, to estimate the parameters of these distributions from the observed values of W_observed, and to calculate the value of w based on these parameters.3

The distribution of the random variable W_original can be approximated by a truncated normal distribution. The truncation is necessary because coins with weights that deviate too much from the weight standard either could not be produced at all (it is hard to imagine that the technology of preparing flans would be so imperfect that coins could be produced with, for example, a 50% deviation from the weight standard), or they were captured as part of the mint’s control processes.4 We will therefore assume that the random variable W_original has a truncated normal distribution with parameters μ and σ² on some interval [a, b], where this interval corresponds to the tolerance limits. The probability density of the random variable W_original is thus given by

f(x) = 1/σ × φ((x-μ)/σ) × (Φ((b-μ)/σ) – Φ((a-μ)/σ))-1  for  x∈[a, b],

(2)

where φ and Φ are the probability density function and the cumulative distribution function of the standard normal distribution, respectively. The weight standard can thus be expressed as

w = μ + σ × (φ((b-μ)/σ) – φ((a-μ)/σ)) × (Φ((b-μ)/σ) – Φ((a-μ)/σ))-1.

(3)

In the case of a symmetric tolerance interval, i.e. if b-μ = μ-a, we get w = μ. However, too heavy coins could be less acceptable than slightly lighter coins, in which case b-μ<μ-a, and hence w is less than μ according to the equation (3).

The choice of a suitable parametric distribution of the random variable L is more complicated. The distribution of this quantity is determined not only by the time of circulation of coins and the degradation of the metal over time, but also by the selection criteria of museums and auction houses, from whose catalogues the data are drawn. Unless they are rare coin types, better preserved specimens are included in museum collections and major auctions. Thus, the analyzed data set may contain less worn lighter specimens than it would contain in a purely random selection.

For simplicity, we will assume that L has an exponential distribution with parameter λ, i.e. its probability density is given as

g(x) = λe-λx  for  x≥0.

(4)

Based on these assumptions, the random variable W_observed has a distribution given by the convolution of the truncated normal distribution and the exponential distribution. Note that if we ignore the truncation of the normal distribution, then it would be the well-known exponentially modified Gaussian distribution.

Small preliminary numerical study

For illustration, let us consider three groups of Kelenderis staters, namely groups 1, 2 and 3A (for the definition of these groups, see the Overview of Kelenderis coins in the Coin Catalogue and the weight analysis of Kelenderis staters). For simplicity, we will assume that the mint’s tolerance around the weight standard was ±0.20 g, which corresponds roughly to the weight of one tetartemorion in either direction. This gives the parameters a and b as functions of μ and it remains to estimate the parameters μ, σ and λ.

The best estimation results were achieved by the minimum-distance method using the Anderson-Darling metric. The estimates of the parameters μ, σ and λ and the resulting estimates of the weight standards and the mean weight losses are given in Table 1. The estimates of weight standards of individual groups are equal to the values of μ, because we assume a symmetric tolerance interval. The estimated mean weight losses since leaving the mint to the present are given as E(L) = 1/λ. According to these estimates, almost the same weight standards of 10.81 g and 10.79 g were used for Groups 2 and 3A, but coins from Group 3A experienced a greater weight loss on average (0.08 g vs. 0.15 g, i.e. 0.8% vs. 1.4%).

	Group 1	Group 2	Group 3A
number of observations	13	229	223
μ	10.931	10.809	10.789
σ	0.090	0.064	0.066
λ	5.204	12.214	6.665
estimated weight standard	10.93	10.81	10.79
estimated mean weight loss	0.19	0.08	0.15
	1.76%	0.76%	1.39%
observed average weight	10.73	10.73	10.64

Table 1: Estimates of the distributions of W_original and L

Figure 1 shows relative frequency histograms of observed weights and the estimated distributions of observed weights given by the convolution of the truncated normal distribution and the exponential distribution. Thus, the bars represent the relative frequencies of observations ranging from 8.70 to 11.10 g in increments of 0.10 g and the continuous curves represent the estimated distributions of the random variables W_observed. Figure 2 again shows the estimated distributions of observed weights and also the distribution of the original coin weights, i.e. the truncated normal distributions of the random variables W_original. Figure 3 shows the P-P plots, i.e. the empirical cumulative distributions of observed weights against the estimated cumulative distributions of observed weights. The P–P plots show a relatively satisfactory agreement of the estimated distributions with the observed data, with the largest systematic deviation at the right tail of group 3A. This may be due to a number of different reasons, such as inappropriateness of the chosen parametric model or inhomogeneity of the data sample.

Figure 1: Relative frequency histograms and estimated observed distributions

Figure 2: Estimated observed and original distributions

Figure 3: P-P plots

Based on the chosen tolerance limit of ±0.20 g and the estimated values of σ, the proportion of weight non-compliant flans in Groups 1, 2 and 3A that had to be re-melted or modified is equal to 2.66%, 0.17% and 0.26%, respectively. These low values show that we do not commit a large error if we neglect to truncate the distribution of the original weights, i.e. we assume that W_original has a normal distribution and thus W_observed has an exponentially modified Gaussian distribution. The results under this simplifying assumption are shown in Table 2 and Figures 4–6. The estimates of the weight standards are the same (after rounding to two decimal places), but the estimates of the parameters σ and λ are slightly different. The proportions of coins in Groups 1, 2 and 3A deviating from the weight standards by more than 0.20 g are equal to 1.64%, 0.25% and 1.81%, respectively.

	Group 1	Group 2	Group 3A
number of observations	13	229	223
μ	10.932	10.807	10.791
σ	0.083	0.066	0.085
λ	5.146	12.666	6.886
estimated weight standard	10.93	10.81	10.79
estimated mean weight loss	0.19	0.08	0.15
	1.78%	0.73%	1.35%
observed average weight	10.73	10.73	10.64

Table 2: Estimates of the distributions of W_original and L assuming an exponentially modified Gaussian distribution of W_observed

Figure 4: Relative frequency histograms and estimated observed distributions assuming an exponentially modified Gaussian distribution of W_observed

Figure 5: Estimated observed and original distributions assuming an exponentially modified Gaussian distribution of W_observed

Figure 6: P-P plots assuming an exponentially modified Gaussian distribution of W_observed