Sampling Distribution and Asymptotic and Bootstrap Confidence Interval of the Kvalseth’s Standard Deviation from the Mode - José Moral de la Rubia (Mathematics and Computer Science: Contemporary Developments Vol. 8, 2024)
Backgrounds: One of the best statistics for measuring variability in a qualitative variable is Kvalseth’s Standard Deviation from the Mode (SDM). It is capitalized to denote its population value (parameter) and written in lowercase to represent its sample value (statistic). The author applies the delta method to derive an asymptotic standard error and to obtain a Wald-type confidence interval. Although the calculation of sdm statistic is not complex, this measure is seldom used. Aims: To disseminate and deepen the knowledge of the sdm statistic and to facilitate its use. Objectives: 1) To determine its sampling distribution under the normality hypothesis, 2) to compare the asymptotic and bootstrap standard errors, and 3) to develop an R script for point and interval (asymptotic and bootstrap) estimation of this measure. Study Design: This is a methodological study using bootstrap simulation. Methodology: For the first objective, samples of 30 different sizes (30, 50 to 950 in increments of 50, 1000 to 5000 in increments of 500, and 10,000) were drawn from five binomial distributions and two multinomial distributions. Using bootstrap, 210 sampling distributions of sdm statistics were generated. Symmetry was tested with the D’Agostino test, mesokurtosis with the Anscombe-Glynn test, and normality with the Shapiro-Francia and Kolmogorov-Smirnov-Lilliefors tests. For the second objective, the Snedecor-Cochran F-test for variance equivalence was used. The developed script was applied to a multinomial variable representing migration motives. Results: The hypothesis that the asymptotic sampling distribution is normal was upheld. The asymptotic and bootstrap standard errors were very similar, with an average absolute difference of 0.0020 and a maximum absolute difference of 0.0199. Both standard errors showed convergence, as the correlation between their absolute difference and sample size was less than or equal to -0.8 in the seven bootstrap sampling distributions of the sdm statistic (across 7 types of distribution, with 30 paired data corresponding to 30 sample sizes). Conclusion: For samples of 1000 and above, the sampling distribution of sdm statistic is normal, and its asymptotic error is adequate. However, for smaller samples, it is better to use bootstrap to compute the standard error and the bias-corrected and accelerated percentile method to obtain the confidence interval. The developed R script allows performing all these calculations, in addition to representing the sample through a frequency table and two graphs.