Weight Analysis

Demetrios I

Comparison of Undated and Dated Issues of Antioch Mint

Contents

1. Basic characteristics
2. Test of differences

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

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 0.107 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 test gives the same conclusion.

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).