Area Skewness for Random Samples Drawn from an Unknown or Specified Distribution

Abstract

Singh, Gewali, and Khatiwada proposed a skewness measure for probability distributions called Area Skewness ( AS ), which has desirable properties but has not been widely applied in practice. One reason for this may be the lack of dissemination and further exploration of this new measure. Additionally, the authors focused mainly on specific probability distributions. One of the advantages of AS is that it considers the entire shape of the distribution, rather than focusing only on moments or the linear distance between central tendency statistics or quantiles. This holistic approach makes AS particularly robust in cases where the distribution deviates from normality or contains outliers. This paper aims to generalize its use to random samples with either known or unknown distributions. The study has three objectives: 1) to develop an R script for point and interval estimation of AS ; 2) to provide interpretive norms of normality by examining normality in bootstrap sampling distributions; and 3) to compare asymptotic and bootstrap standard errors. Interval estimation is approached asymptotically and through bootstrap. The script was illustrated using two examples: one with generated data and another with real-world data. Interpretive norms of normality are derived from 40 samples of various sizes, created by inverse transform sampling to follow a standard normal distribution. Bootstrap intervals at three confidence levels (0.9, 0.95, and 0.99) were obtained using the normal method, with two exceptions: the bias-corrected and accelerated percentile method for the 60-data sample and the percentile method for the 600-data sample, as these deviated from normality. Asymptotic 95% confidence intervals are also provided. The asymptotic standard error was larger than the bootstrap one, with the difference decreasing as the sample size increased. The script is concluded to have practical and educational utility for estimating AS , whose asymptotic sampling distribution is normal.