Rank-Based Family of Probability Laws for Testing Homogeneity of Variable Grouping

Manuel L. Esquível (), Nadezhda P. Krasii, Célia Nunes, Kwaku Opoku-Ameyaw and Pedro P. Mota
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Manuel L. Esquível: Department of Mathematics, Nova School of Science and Technology and Nova Math, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal
Nadezhda P. Krasii: Department of Higher Mathematics, Don State Technical University, Gagarin Square 1, Rostov-on-Don 344000, Russia
Célia Nunes: Department of Mathematics, Center of Mathematics and Applications, University of Beira Interior, 6201-001 Covilhã, Portugal
Kwaku Opoku-Ameyaw: Cocoa Research Institute of Ghana, New Tafo-Akim P.O. Box 8, Ghana
Pedro P. Mota: Department of Mathematics, Nova School of Science and Technology and Nova Math, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal

Mathematics, 2025, vol. 13, issue 11, 1-26

Abstract: In order to test within-group homogeneity for numerical or ordinal variable groupings, we have introduced a family of discrete probability distributions, related to the Gini mean difference, that we now study in a deeper way. A member of such a family is the law of a statistic that operates on the ranks of the values of the random variables by considering the sums of the inter-subgroups ranks of the variable grouping. Being so, a law of the family depends on several parameters such as the cardinal of the group of variables, the number of subgroups of the grouping of variables, and the cardinals of the subgroups of the grouping. The exact distribution of a law of the family faces computational challenges even for moderate values of the cardinal of the whole set of variables. Motivated by this challenge, we show that an asymptotic result allowing approximate quantile values is not possible based on the hypothesis observed in particular cases. Consequently, we propose two methodologies to deal with finite approximations for large values of the parameters. We address, in some particular cases, the quality of the distributional approximation provided by a possible finite approximation. With the purpose of illustrating the usefulness of the grouping laws, we present an application to an example of within-group homogeneity grouping analysis to a grouping originated from a clustering technique applied to cocoa breeding experiment data. The analysis brings to light the homogeneity of production output variables in one specific type of soil.

Keywords: asymptotic distribution; finite approximations; discrete grouping distribution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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