Weight Analysis

Antiochos VII

Comparison of Antioch Mint and Pseudo-Antioch Mints

Contents

1. Comparison without respect to control marks
   • Basic characteristics
   • Test of differences
2. Comparison with respect to control marks
   • Basic characteristics
   • Test of differences

1. Comparison without respect to control marks

1.1. Basic characteristics

Basic descriptive statistics of both samples and box-percentile plots 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.

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.

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

1.2. Test of differences

The Kolmogorov-Smirnov test was used to test the hypothesis that both distributions are the same. The test statistic of 0.082 is less than the critical value of 0.098 for a 95% level test (the asymptotic p-value is 25.6%). Thus, at the 95% level of significance, we cannot reject the hypothesis that the weight distributions of both undated and dated issues are the same. The Wilcoxon rank sum test gives the same conclusion.

It could be expected that a lighter weight standard was used for pseudo-Antioch posthumous issues. Therefore, the Kolmogorov-Smirnov one-sided test was also used for the comparison of both data samples (i.e. it was tested that the distribution of pseudo-Antioch posthumous issues is shifted to the left with respect to the distribution of Antioch lifetime issues). This test also does not reject the hypothesis that both distributions are the same (critical value: 0.059, asymptotic p-value: 56.5%).

2. Comparison with respect to control marks

2.2. Basic characteristics

Since there are only few observations for some combinations of controls, all combinations were aggregated with the purpose to compare main series. The aggregations are defined as follows (see also Table 1):

Series	Position of Nike	Control bellow ΔΙ	Secondary Controls
A1	left		no
A2	left		yes
A3	left	other	yes
B1	right		yes
B2	right		yes

Tab. 1: Overview of aggregated series

Descriptive statistics of aggregated series are presented in Table 2 and Figure 7. Box-percentile plots of aggregated series are presented in Figure 8, histograms are presented in Figure 9, kernel density estimations are presented in Figures 10 and 11, and empirical cumulative distribution functions are presented in Figure 12.

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.

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.

Fig. 7: Descriptive statistics

Fig. 8: Box-percentile plots

Fig. 9: Histograms

Fig. 10: Probability density estimations, Epanechnikov kernel

Fig. 11: Probability density estimations, Gaussian kernel

Fig. 12: Empirical cumulative distribution functions

2.2. Test of differences

The Kruskal-Wallis test was used to test the hypothesis that these five series come from the same distribution against the alternative that at least one of the distributions is different. The test statistic of 7.730 is less than the critical value of 9.488 for a 95% level test (the p-value is 10.2%). Thus, at the 95% level of significance, we cannot reject the hypothesis that these series have the same weight distribution.

The series B1 has lesser standard deviation (variance) than the other series, see Table 2 and Figures 7-8 and 10-11. For that reason, the Kolmogorov-Smirnov both-sided test was used to test which series differ from the series B1. The multiple comparisons were done at the 95% confidence level using the Bonferroni procedure (it means that an adjusted criterion of significance α' = α/4 = 0.05/4 = 1.25% was used). Results are presented in Table 3.

We can conclude that there is no statistically significant difference, although the asymptotic p-values presented in Table 3 indicate that particularly the series A3 can really have a different distribution than the series B1. It is necessary to emphasize that the data samples are small. Larger samples could give significant results.

References:

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).