Sampling Distribution and Asymptotic and Bootstrap Confidence Interval of the Kvalseth’s Standard Deviation from the Mode | Chapter 2 | Mathematics and Computer Science: Contemporary Developments Vol. 8

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.

 

Author (s) Details

 

José Moral de la Rubia
School of Psychology, Universidad Autónoma de Nuevo León, Monterrey, Mexico.

Please see the book here:- https://doi.org/10.9734/bpi/mcscd/v8/2812

The Beta Distribution: Applications and Insights for Social and Health Sciences | Chapter 3 | Mathematics and Computer Science: Research Updates Vol. 3

The objective of this methodological-academic chapter is to provide a detailed introduction to the Beta distribution, with examples applied to the social and health sciences using data randomly generated from Beta distributions. This continuous distribution maintains its status as a well-recognized statistical tool in mathematical statistics but practitioners in these fields rarely apply it despite its wide modeling capabilities for probabilities, proportions and bounded data. The chapter begins with a historical overview to provide context, followed by an explanation of its support, parameters, and key functions, including density, cumulative distribution, quantile, moment-generating, and characteristic functions. Descriptive measures such as central tendency, variation, and shape are also discussed. The chapter then covers parameter estimation, focusing on the method of moments and maximum likelihood. The fit of a random sample to a Beta distribution with unknown parameters is tested using the G test and quantile-quantile plot. Additionally, the chapter presents the associated limiting distributions and generalizations. Four examples illustrate applications of the Beta distribution. The first example shows how to compute probabilities and descriptive measures when the two parameters of the distribution are known. The second example estimates the two shape parameters (alpha and beta) using the method of moments, calculates descriptive measures and performs probability calculations. The third example estimates alpha and beta, along with their standard errors and confidence intervals, using likelihood methods. The fourth example tests the goodness of fit of a sample to a Beta distribution using the G-test and the QQ plot. The work provides accessibility by including appendices containing R scripts for all calculations so researchers with different statistical abilities can benefit from this resource. The text closes with conclusions and suggestions for using the Beta distribution. This chapter connects mathematical theory to practical examples along with computational tools to simplify the understanding of Beta distribution thereby advancing its adoption across social and health science data modeling and analysis. Researchers can now exploit this flexible distribution across policy evaluation, clinical trials and behavioral studies to enhance their data-driven decision-making capacity.

 

Author (s) Details

 

José Moral de la Rubia
School of Psychology, Universidad Autónoma de Nuevo León, Street. Dr. Carlos Canseco 110. Col. Mitras Centro, C.P. 64460. Monterrey, Nuevo León, México.

 

Please see the book here:- https://doi.org/10.9734/bpi/mcsru/v3/4354