{"title":"Mixture of shifted binomial distributions for rating data","authors":"Shaoting Li, Jiahua Chen","doi":"10.1007/s10463-023-00865-7","DOIUrl":null,"url":null,"abstract":"<div><p>Rating data are a kind of ordinal categorical data routinely collected in survey sampling. The response value in such applications is confined to a finite number of ordered categories. Due to population heterogeneity, the respondents may have several different rating styles. A finite mixture model is thus most suitable to fit datasets of this nature. In this paper, we propose a two-component mixture of shifted binomial distributions for rating data. We show that this model is identifiable and propose a numerically stable penalized likelihood approach for parameter estimation. We adapt an expectation-maximization algorithm for the penalized maximum likelihood estimation. Our simulation results show that the penalized maximum likelihood estimator is consistent and effective. We fit the proposed model and other models in the literature to some real-world datasets and find the proposed model can have much better fits.</p></div>","PeriodicalId":55511,"journal":{"name":"Annals of the Institute of Statistical Mathematics","volume":"75 5","pages":"833 - 853"},"PeriodicalIF":0.8000,"publicationDate":"2023-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of the Institute of Statistical Mathematics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10463-023-00865-7","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
Rating data are a kind of ordinal categorical data routinely collected in survey sampling. The response value in such applications is confined to a finite number of ordered categories. Due to population heterogeneity, the respondents may have several different rating styles. A finite mixture model is thus most suitable to fit datasets of this nature. In this paper, we propose a two-component mixture of shifted binomial distributions for rating data. We show that this model is identifiable and propose a numerically stable penalized likelihood approach for parameter estimation. We adapt an expectation-maximization algorithm for the penalized maximum likelihood estimation. Our simulation results show that the penalized maximum likelihood estimator is consistent and effective. We fit the proposed model and other models in the literature to some real-world datasets and find the proposed model can have much better fits.
期刊介绍:
Annals of the Institute of Statistical Mathematics (AISM) aims to provide a forum for open communication among statisticians, and to contribute to the advancement of statistics as a science to enable humans to handle information in order to cope with uncertainties. It publishes high-quality papers that shed new light on the theoretical, computational and/or methodological aspects of statistical science. Emphasis is placed on (a) development of new methodologies motivated by real data, (b) development of unifying theories, and (c) analysis and improvement of existing methodologies and theories.