Antioch Mint - Dated Issues References

Antioch Mint – Dated Issues

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 Demetrios I and later rulers. 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 98.5% of issued coins of the examined type have a weight between the smallest observation and the largest observation, i.e. between 15.81 g and 17.16 g, and at least 97.6% of issued coins of the examined type have a weight between the second smallest observation and the second largest observation, i.e. between 15.82 g and 17.04 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 0.061 and Gaussian kernel with a bandwidth of 0.049). 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 0.090 is greater than the cutoff value of 0.050 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. Analysis of individual years

Descriptive statistics of series from individual years (Seleukid Era 158 - 162) are presented in Table 1 and Figure 4. Box-percentile plots1 are presented in Figure 5.

Note: The box-percentile plot is a modified version of the well-known boxplot. At any height the width of the irregular “box” is proportional to the percentile of that height, up to the 50th percentile, and above the 50th percentile the width is proportional to 100 minus the percentile. Thus, the width at any given height is proportional to the percent of observations that are more extreme in that direction. As in boxplots, the median, 25th, and 75th percentiles are marked with line segments across the box. For details see Esty and Banfield, The Box-Percentile Plot.

Tab. 1: Individual years - descriptive statistics

Fig. 4: Descriptive statistics

Fig. 5: Box-percentile plots

The Kruskal-Wallis test was used to test the hypothesis that weight distributions in individual years are the same, against the alternative that at least one of the distributions tends to yield larger observations than at least one of the other distributions. The test statistic of 2.867 is less than the critical value of 9.488 for a 95% level test (the p-value is 58.0%). Thus, at the 95% level of significance, we cannot reject the hypothesis that weight distributions in individual years are the same.

Histograms, kernel estimations of probability density functions and empirical cumulative distribution functions of series from individual years are presented in Figures 6 - 9.

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. 6: Histograms

Fig. 7: Probability density estimations - Epanechnikov kernel

Fig. 8: Probability density estimations - Gaussian kernel

Fig. 9: Empirical cumulative distribution functions

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).