{"title":"预防性损失函数下乘法区域级模型中的受限贝叶斯","authors":"Elaheh Torkashvand, Mohammad Jafari Jozani","doi":"10.1002/cjs.11809","DOIUrl":null,"url":null,"abstract":"Consider the problem of benchmarking small‐area estimates under multiplicative models with positive parameters. The goal is to propose a loss function that guarantees positive constrained estimates of small‐area parameters in this situation. The weighted precautionary loss function is introduced to solve the problem. Compared with the weighted Kullback–Leibler (KL) loss function, our proposed loss function penalizes underestimation of the small‐area parameters of interest more for small values of parameters. This property is appealing when we estimate disease rates. It tends to give larger estimates of small‐area parameters compared with those obtained under the KL loss function. The hierarchical empirical Bayes and constrained hierarchical empirical Bayes estimates of small‐area parameters and their corresponding risk functions under the new proposed loss function are obtained. The performance of the proposed methods is investigated using simulation studies and a real dataset.","PeriodicalId":501595,"journal":{"name":"The Canadian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Constrained Bayes in multiplicative area‐level models under the precautionary loss function\",\"authors\":\"Elaheh Torkashvand, Mohammad Jafari Jozani\",\"doi\":\"10.1002/cjs.11809\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Consider the problem of benchmarking small‐area estimates under multiplicative models with positive parameters. The goal is to propose a loss function that guarantees positive constrained estimates of small‐area parameters in this situation. The weighted precautionary loss function is introduced to solve the problem. Compared with the weighted Kullback–Leibler (KL) loss function, our proposed loss function penalizes underestimation of the small‐area parameters of interest more for small values of parameters. This property is appealing when we estimate disease rates. It tends to give larger estimates of small‐area parameters compared with those obtained under the KL loss function. The hierarchical empirical Bayes and constrained hierarchical empirical Bayes estimates of small‐area parameters and their corresponding risk functions under the new proposed loss function are obtained. The performance of the proposed methods is investigated using simulation studies and a real dataset.\",\"PeriodicalId\":501595,\"journal\":{\"name\":\"The Canadian Journal of Statistics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-06-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"The Canadian Journal of Statistics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1002/cjs.11809\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Canadian Journal of Statistics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1002/cjs.11809","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Constrained Bayes in multiplicative area‐level models under the precautionary loss function
Consider the problem of benchmarking small‐area estimates under multiplicative models with positive parameters. The goal is to propose a loss function that guarantees positive constrained estimates of small‐area parameters in this situation. The weighted precautionary loss function is introduced to solve the problem. Compared with the weighted Kullback–Leibler (KL) loss function, our proposed loss function penalizes underestimation of the small‐area parameters of interest more for small values of parameters. This property is appealing when we estimate disease rates. It tends to give larger estimates of small‐area parameters compared with those obtained under the KL loss function. The hierarchical empirical Bayes and constrained hierarchical empirical Bayes estimates of small‐area parameters and their corresponding risk functions under the new proposed loss function are obtained. The performance of the proposed methods is investigated using simulation studies and a real dataset.