Permutation Inference for a Class of Mixture Models

In statistical surveys, people are often asked to express evaluations on several topics or to make an ordered arrangement in a list of objects (items, services, sentences, etc.); thus, the analysis of ratings and rankings is receiving a growing interest in many fields. In this framework, we develop a testing procedure for a class of mixture models with covariates (defined as CUB models), proposed by Piccolo (2003 Piccolo , D. ( 2003 ). On the moments of a mixture of uniform and shifted binomial random variables . Quaderni di Statistica 5 : 85 – 104 . [Google Scholar]) and D'Elia and Piccolo (2005 D'Elia , A. , Piccolo , D. ( 2005 ). A mixture model for preference data analysis . Computat. Statist. Data Anal. 49 : 917 – 934 .[Crossref], [Web of Science ®] , [Google Scholar]) and generally developed in a parametric context. Instead, we propose a nonparametric solution to perform inference on CUB models, specifically on the coefficients of the covariates. A simulation study proves that this approach is more appropriate in some specific data settings, mostly for small sample sizes.     

Some Elementary Theory of Permutation Tests

In literature, permutation tests are mostly derived by means of heuristic arguments (Edgington and Onghena, 2007 Edgington, E.S., Onghena, P. (2007). Randomization Tests (4th ed.). Boca Raton: Chapman & Hall/CRC.[Crossref] , [Google Scholar]; Good, 2005 Good, P. (2005). Permutation, Parametric, and Bootstrap Tests of Hypotheses (3rd ed.). New York: Springer-Verlag. [Google Scholar]). In this paper, we derive them within the sufficiency and conditionality principles of inference. Most important of their properties are exposed without formal proofs that can be found in the book by Pesarin and Salmaso (2010).