Antioch Mint Comparison with Well-preserved Coins References

Antioch Mint

1. Examined type

2. Acceptable weight range

Each coin is a priori excluded from the data sample if its weight is lesser than the lower exclusion limit or greater than the upper exclusion limit.

3. Data

The data sample was kindly provided by Arthur Houghton who accumulated it in the course of a die study of the precious metal coinage of the mint of Antioch under Antiochos VII. The data refers only to the Antioch mint and to no other.

4. Descriptive statistics

Notes: The unbiased estimation of the variance was used for the computation of the standard deviation (i.e. the number of observations minus one was used as a divisor). The sample skewness was computed without sample corrections (i.e. the skewness was computed as the square root of the number of observations times the sum of the third powers of deviations from the mean divided by the 3/2 power of the sum of the squares of deviations from the mean). Similarly, the sample kurtosis was computed as the number of observations times the sum of the fourth powers of deviations from the mean divided by the second power of the sum of the squares of deviations from the mean.

The confidence interval for mean was computed by using the Student t-distribution. The confidence intervals for median and percentiles were computed nonparametrically by using the binomial distribution (see, e.g., Conover, Practical Nonparametric Statistics, pp. 143 - 148).

5. Estimation of proportion of coins with weights within the observed range

At the 95% level of confidence, at least 98.6% of issued coins of the examined type have a weight between the smallest observation and the largest observation, i.e. between 15.82 g and 17.09 g, and at least 97.8% of issued coins of the examined type have a weight between the second smallest observation and the second largest observation, i.e. between 15.92 g and 17.03 g.

Note: These estimations are computed as tolerance limits based on the binomial distribution. See, e.g., Conover, Practical Nonparametric Statistics, pp. 150 - 155.

6. Histogram and probability density function

Histogram of the sample is presented in Figure 1. Kernel estimations of the probability density function are shown in Figure 2 (two kernel estimations were used: Epanechnikov kernel with a bandwidth of 0.069 and Gaussian kernel with a bandwidth of 0.055). The dotted curve in Figure 2 is a probability density function of a normal distribution estimated by the maximum likelihood method.

Note: The bandwidth of the Gaussian kernel was computed as h_Gauss = 0.9 × min(σ, SIQR) × n^-1/5, where σ is the standard deviation, SIQR is the standardised interquartile range (i.e. the interquartile range of the data sample divided by the interquartile range of the standard normal density) and n is the number of observations; see Silverman, Density Estimation for Statistics and Data Analysis, p. 48, formula (3.31). The bandwidth of the Epanechnikov kernel was chosen subjectively in the range from 0.75×h_Gauss to 1.25×h_Gauss.

Fig. 1: Histogram

Fig. 2: Probability density estimations

7. Test of normality

The Lilliefors test of normality was used. The test statistic of 0.065 is greater than the cutoff value of 0.048 for a 95% level test. Thus we reject the hypothesis of normality at the 95% level of significance. Normal probability plot of the sample is presented in Figure 3.

Fig. 3: Normal probability plot

8. Control marks

	Control bellow ΔΙ	Secondary Controls		Frequency		Weight
		left	right			Mean	Std	Min	Max
1		none	none	60	17.3%	16.58	0.19	16.03	17.00
2		none		50	14.4%	16.58	0.20	15.99	17.03
3		none		40	11.5%	16.50	0.22	15.95	17.01
4		none		31	8.9%	16.55	0.30	15.82	17.09
5		none		26	7.5%	16.56	0.17	16.03	16.83
6		none		11	3.2%	16.58	0.20	16.29	16.89
7		none		11	3.2%	16.47	0.19	16.03	16.69
8		none		7	2.0%	16.52	0.23	16.14	16.83
9		none		6	1.7%	16.58	0.19	16.22	16.78
10		none		4	1.2%	16.73	0.07	16.66	16.82
11		none		4	1.2%	16.54	0.08	16.47	16.65
12		none		1	0.3%	16.87	0.00	16.87	16.87
		SUB-TOTAL		251	72.3%	16.56	0.21	15.82	17.09
13		none	none	19	5.5%	16.57	0.18	16.28	16.88
14		none	none	17	4.9%	16.58	0.23	16.19	16.90
15		none	none	15	4.3%	16.60	0.14	16.39	16.81
16		none	none	13	3.7%	16.42	0.13	16.23	16.62
17		none	none	11	3.2%	16.44	0.24	15.92	16.76
18		none	none	4	1.2%	16.57	0.20	16.30	16.73
19		none	none	3	0.9%	16.55	0.41	16.16	16.98
20		none	none	3	0.9%	16.46	0.48	15.94	16.90
21		none	none	2	0.6%	16.55	0.37	16.28	16.81
22		none	none	2	0.6%	16.52	0.19	16.39	16.66
23		none	none	2	0.6%	16.50	0.18	16.37	16.62
24		none	none	1	0.3%	16.77	0.00	16.77	16.77
25		none	none	1	0.3%	16.68	0.00	16.68	16.68
26		none	none	1	0.3%	16.66	0.00	16.66	16.66
27		none	none	1	0.3%	16.57	0.00	16.57	16.57
28		none	none	1	0.3%	16.51	0.00	16.51	16.51
TOTAL				347	100%	16.55	0.21	15.82	17.09
29		none	none	1	0.3%	–	–	–	–

Note: The issue on the row 29 is mentioned here because of completeness (one such coin is registered in the database kindly provided by Arthur Houghton) but no metrological data are available.

Comparison with the Data Sample of Well-preserved Coins

1. Examined coins

Comparison of the data sample analysed above with the data sample of well-preserved coins analysed in the section Weight Studies of Well-Preserved Seleukid Coins.

2. Basic characteristics

Basic descriptive statistics of both samples and box-percentile plots1 are presented in Figures 1 and 2. Histograms are presented in Figure 3, kernel density estimations are presented in Figures 4 and 5, and empirical cumulative distribution functions are presented in Figure 6.

Notes: The bandwidth of the Gaussian kernel was computed as h_Gauss = 0.9 × min(σ, SIQR) × n^-1/5, where σ is the standard deviation, SIQR is the standardised interquartile range (i.e. the interquartile range of the data sample divided by the interquartile range of the standard normal density) and n is the number of observations; see Silverman, Density Estimation for Statistics and Data Analysis, p. 48, formula (3.31). The bandwidth of the Epanechnikov kernel was chosen subjectively in the range from 0.75×h_Gauss to 1.25×h_Gauss.

Fig. 1: Descriptive statistics

Fig. 2: Box-percentile plots

Fig. 3: Histograms

Fig. 4: Probability density estimations - Epanechnikov kernel

Fig. 5: Probability density estimations - Gaussian kernel

Fig. 6: Empirical cumulative distribution functions

3. Test of differences

The Kolmogorov-Smirnov test was used to test the hypothesis that both distributions are the same. The test statistic of 0.090 is less than the critical value of 0.213 for a 95% level test (the asymptotic p-value is 88.4%). The Wilcoxon rank sum test2 gives the same conclusion. It could be expected that the distribution of Houghton’s data might be shifted to the left with respect to the distribution of data examined in Part Ia (collected with regard to the state of preservation). However, the Kolmogorov-Smirnov one-sided test gives similar result: the asymptotic p-value is 50.5%. Thus, at the 95% level of significance, we cannot reject the hypothesis that the weight distributions of these two data samples are the same.

References:

Conover, W. J.:Practical Nonparametric Statistics, Third Edition. John Wiley & Sons, Inc., New York - Chichester - Weinheim - Brisbane - Singapore - Toronto, 1999.

Esty, Warren W.; Banfield, Jeffrey D.: The Box-Percentile Plot. Journal of Statistical Software, Volume 8, Number 17, 2003, pp. 1-14.

Silverman, B.W.:Density Estimation for Statistics and Data Analysis. Chapman and Hall, London - New York - Tokyo - Melbourne - Madras, 1993 (reprint of the first edition published in 1986).