Submitted by Petr Vesely on

Founded 17-Feb-2004

Last update 12-Dec-2004

Comparison of Undated and Dated Issues References

## Comparison of Undated and Dated Issues

### 1. Examined coins

Undated and dated AR tetradrachms of Demetrios I from Antioch mint. Data samples are presented in the corresponding sections.

### 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.

**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}.

### 3. Test of differences

The Kolmogorov-Smirnov test was used to test the hypothesis that both distributions are the same. The test statistic of 0.082 is less than the critical value of 0.107 for a 95% level test (the asymptotic p-value is 21.5%). 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. Note that the one-sided Kolmogorov-Smirnov test (Figure 6 indicates that the distribution of undated issues might be shifted to the left with respect to the distribution of dated issues) also does not reject the hypothesis that both distributions are the same (the asymptotic p-value is 10.7%). The Wilcoxon rank sum test2 gives the same conclusion.

1 See Statistical Glossary, Box-percentile plot.

2 Also known as the Mann-Whitney test.

### 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).