Nonparametric estimation of a regression curve becomes crucial when the underlying dependence structure between covariates and responses is not explicit. While existing literature has addressed single change‐point estimation for regression curves, the problem of multiple change points remains unresolved. In an effort to bridge this gap, this article introduces a nonparametric estimator for multiple change points by minimizing a penalized weighted sum of squared residuals, presenting consistent results under mild conditions. Additionally, we propose a cross‐validation‐based procedure that possesses the advantage of being tuning‐free. Our simulation results showcase the competitive performance of these new procedures when compared with state‐of‐the‐art methods. As an illustration of their utility, we apply these procedures to a real dataset.
{"title":"Multiple change‐point detection for regression curves","authors":"Yunlong Wang","doi":"10.1002/cjs.11816","DOIUrl":"https://doi.org/10.1002/cjs.11816","url":null,"abstract":"Nonparametric estimation of a regression curve becomes crucial when the underlying dependence structure between covariates and responses is not explicit. While existing literature has addressed single change‐point estimation for regression curves, the problem of multiple change points remains unresolved. In an effort to bridge this gap, this article introduces a nonparametric estimator for multiple change points by minimizing a penalized weighted sum of squared residuals, presenting consistent results under mild conditions. Additionally, we propose a cross‐validation‐based procedure that possesses the advantage of being tuning‐free. Our simulation results showcase the competitive performance of these new procedures when compared with state‐of‐the‐art methods. As an illustration of their utility, we apply these procedures to a real dataset.","PeriodicalId":501595,"journal":{"name":"The Canadian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141769582","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Estimating the COVID‐19 infection fatality rate, inferring the latent incidence and predicting the future epidemic evolution are critical to public health surveillance, but often challenging due to limited data availability or quality. Recently, a Bayesian framework combining time series deconvolution of deaths with a parametric Susceptible–Infectious–Recovered (SIR) model was proposed by Irons and Raftery, 2021. We assess the parameter identifiability of the model using the profile likelihood approach and simulations, when only the time series of deaths and seroprevalence survey data are available. The robustness of the model to the more complex but also more realistic Susceptible–Exposed–Infectious–Recovered (SEIR)‐based epidemics is evaluated through simulations; the influence of potential biases in the serosurveys on the inference is also investigated. We use a stationary first‐order autoregressive prior to account for the variability of transmission rate over time. The results suggest that the model is relatively robust to SEIR‐based epidemics, especially when the reproductive number is low, given sufficient information from serosurveys or priors. However, the lack of parameter identifiability under limited data availability cannot be neglected. We apply the model to infer the COVID‐19 infections in Ontario and Quebec, Canada during the Omicron era.
{"title":"An SIR‐based Bayesian framework for COVID‐19 infection estimation","authors":"Haoyu Wu, David A. Stephens, Erica E. M. Moodie","doi":"10.1002/cjs.11817","DOIUrl":"https://doi.org/10.1002/cjs.11817","url":null,"abstract":"Estimating the COVID‐19 infection fatality rate, inferring the latent incidence and predicting the future epidemic evolution are critical to public health surveillance, but often challenging due to limited data availability or quality. Recently, a Bayesian framework combining time series deconvolution of deaths with a parametric Susceptible–Infectious–Recovered (SIR) model was proposed by Irons and Raftery, 2021. We assess the parameter identifiability of the model using the profile likelihood approach and simulations, when only the time series of deaths and seroprevalence survey data are available. The robustness of the model to the more complex but also more realistic Susceptible–Exposed–Infectious–Recovered (SEIR)‐based epidemics is evaluated through simulations; the influence of potential biases in the serosurveys on the inference is also investigated. We use a stationary first‐order autoregressive prior to account for the variability of transmission rate over time. The results suggest that the model is relatively robust to SEIR‐based epidemics, especially when the reproductive number is low, given sufficient information from serosurveys or priors. However, the lack of parameter identifiability under limited data availability cannot be neglected. We apply the model to infer the COVID‐19 infections in Ontario and Quebec, Canada during the Omicron era.","PeriodicalId":501595,"journal":{"name":"The Canadian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Samantha Morrison, Constantine Gatsonis, Issa J. Dahabreh, Bing Li, Jon A. Steingrimsson
We present methods for estimating loss‐based measures of the performance of a prediction model in a target population that differs from the source population in which the model was developed, in settings where outcome and covariate data are available from the source population but only covariate data are available on a simple random sample from the target population. Prior work adjusting for differences between the two populations has used various weighting estimators with inverse odds or density ratio weights. Here, we develop more robust estimators for the target population risk (expected loss) that can be used with data‐adaptive (e.g., machine learning‐based) estimation of nuisance parameters. We examine the large‐sample properties of the estimators and evaluate finite‐sample performance in simulations. Last, we apply the methods to data from lung cancer screening using nationally representative data from the National Health and Nutrition Examination Survey (NHANES) and extend our methods to account for the complex survey design of the NHANES.
{"title":"Robust estimation of loss‐based measures of model performance under covariate shift","authors":"Samantha Morrison, Constantine Gatsonis, Issa J. Dahabreh, Bing Li, Jon A. Steingrimsson","doi":"10.1002/cjs.11815","DOIUrl":"https://doi.org/10.1002/cjs.11815","url":null,"abstract":"We present methods for estimating loss‐based measures of the performance of a prediction model in a target population that differs from the source population in which the model was developed, in settings where outcome and covariate data are available from the source population but only covariate data are available on a simple random sample from the target population. Prior work adjusting for differences between the two populations has used various weighting estimators with inverse odds or density ratio weights. Here, we develop more robust estimators for the target population risk (expected loss) that can be used with data‐adaptive (e.g., machine learning‐based) estimation of nuisance parameters. We examine the large‐sample properties of the estimators and evaluate finite‐sample performance in simulations. Last, we apply the methods to data from lung cancer screening using nationally representative data from the National Health and Nutrition Examination Survey (NHANES) and extend our methods to account for the complex survey design of the NHANES.","PeriodicalId":501595,"journal":{"name":"The Canadian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611950","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider estimation of the mean squared prediction error (MSPE) for observed best prediction (OBP) in small area estimation with count data. The OBP method has been previously developed in this context by Chen et al. (Journal of Survey Statistics and Methodology, 3, 136–161, 2015). However, estimation of the MSPE remains a challenging problem due to potential model misspecification that is considered in this setting. The latter authors proposed a bootstrap method for estimating the MSPE, whose theoretical justification is not clear. We propose to use a Prasad–Rao‐type linearization method to estimate the MSPE. Unlike the traditional linearization approaches, our method is computationally oriented and easier to implement in the same regard. Theoretical properties and empirical performance of the proposed method are studied. A real‐data application is considered.
{"title":"Estimating the mean squared prediction error of the observed best predictor associated with small area counts: A computationally oriented approach","authors":"Thuan Nguyen, Jiming Jiang","doi":"10.1002/cjs.11810","DOIUrl":"https://doi.org/10.1002/cjs.11810","url":null,"abstract":"We consider estimation of the mean squared prediction error (MSPE) for observed best prediction (OBP) in small area estimation with count data. The OBP method has been previously developed in this context by Chen et al. (<jats:italic>Journal of Survey Statistics and Methodology</jats:italic>, 3, 136–161, 2015). However, estimation of the MSPE remains a challenging problem due to potential model misspecification that is considered in this setting. The latter authors proposed a bootstrap method for estimating the MSPE, whose theoretical justification is not clear. We propose to use a Prasad–Rao‐type linearization method to estimate the MSPE. Unlike the traditional linearization approaches, our method is computationally oriented and easier to implement in the same regard. Theoretical properties and empirical performance of the proposed method are studied. A real‐data application is considered.","PeriodicalId":501595,"journal":{"name":"The Canadian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a general mixture of Markov jump processes. The key novel feature of the proposed mixture is that the generator matrices of the Markov processes comprising the mixture are entirely unconstrained. The Markov processes are mixed with distributions that depend on the initial state of the mixture process. The maximum likelihood (ML) estimates of the mixture's parameters are obtained from continuous realizations of the mixture process and their standard errors from an explicit form of the observed Fisher information matrix, which simplifies the Louis (Journal of the Royal Statistical Society Series B, 44:226–233, 1982) general formula for the same matrix. The asymptotic properties of the ML estimators are also derived. A simulation study verifies the estimates' accuracy. The proposed mixture provides an exploratory tool for identifying the homogeneous subpopulations in a heterogeneous population. This is illustrated with an application to a medical dataset.
我们提出了马尔可夫跳跃过程的一般混合物。所提混合物的关键新特征是，构成混合物的马尔可夫过程的生成矩阵完全不受制约。马尔可夫过程的混合分布取决于混合过程的初始状态。混合物参数的最大似然法（ML）估计是从混合物过程的连续实化中获得的，其标准误差是从观察到的费雪信息矩阵的明确形式中获得的，这简化了路易斯（《皇家统计学会杂志》B 辑，44:226-233, 1982 年）关于同一矩阵的一般公式。此外，还得出了 ML 估计数的渐近特性。模拟研究验证了估计的准确性。所提出的混合物为识别异质人群中的同质子群提供了一种探索性工具。我们将通过对一个医疗数据集的应用来说明这一点。
{"title":"Estimation in a general mixture of Markov jump processes","authors":"Halina Frydman, Budhi Arta Surya","doi":"10.1002/cjs.11814","DOIUrl":"https://doi.org/10.1002/cjs.11814","url":null,"abstract":"We propose a general mixture of Markov jump processes. The key novel feature of the proposed mixture is that the generator matrices of the Markov processes comprising the mixture are entirely unconstrained. The Markov processes are mixed with distributions that depend on the initial state of the mixture process. The maximum likelihood (ML) estimates of the mixture's parameters are obtained from continuous realizations of the mixture process and their standard errors from an explicit form of the observed Fisher information matrix, which simplifies the Louis (<jats:italic>Journal of the Royal Statistical Society Series B</jats:italic>, 44:226–233, 1982) general formula for the same matrix. The asymptotic properties of the ML estimators are also derived. A simulation study verifies the estimates' accuracy. The proposed mixture provides an exploratory tool for identifying the homogeneous subpopulations in a heterogeneous population. This is illustrated with an application to a medical dataset.","PeriodicalId":501595,"journal":{"name":"The Canadian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141548249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Order‐restricted hypothesis testing problems frequently arise in practice, including studies involving regression models for longitudinal data. These tests are known to be more powerful than tests that ignore such restrictions. In this article, we consider order‐restricted tests for nonlinear mixed‐effects models with measurement errors in time‐dependent covariates. We propose to use a multiple imputation method to address measurement errors, since this approach allows us to use existing complete‐data methods for order‐restricted tests. Some theoretical results are presented. We evaluate our proposed methods via simulation studies that demonstrate they are more powerful than either a competing naive method or a two‐step approach to testing hypotheses. We illustrate the use of our proposed approach by analyzing data from an HIV/AIDS study.
{"title":"Order‐restricted hypothesis tests for nonlinear mixed‐effects models with measurement errors in covariates","authors":"Yixin Zhang, Wei Liu, Lang Wu","doi":"10.1002/cjs.11812","DOIUrl":"https://doi.org/10.1002/cjs.11812","url":null,"abstract":"Order‐restricted hypothesis testing problems frequently arise in practice, including studies involving regression models for longitudinal data. These tests are known to be more powerful than tests that ignore such restrictions. In this article, we consider order‐restricted tests for nonlinear mixed‐effects models with measurement errors in time‐dependent covariates. We propose to use a multiple imputation method to address measurement errors, since this approach allows us to use existing complete‐data methods for order‐restricted tests. Some theoretical results are presented. We evaluate our proposed methods via simulation studies that demonstrate they are more powerful than either a competing naive method or a two‐step approach to testing hypotheses. We illustrate the use of our proposed approach by analyzing data from an HIV/AIDS study.","PeriodicalId":501595,"journal":{"name":"The Canadian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141548252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the first‐order stochastic dominance (SD) test in the context of two independent random samples. We introduce several test statistics that effectively capture violations of the dominance relationship, particularly in the tail regions. Additionally, we develop a resampling procedure to compute the ‐values or critical values for these tests. The proposed tests have asymptotic type I error rates for frontal configurations equal to the nominal level . Furthermore, their powers approach 1 for any fixed alternatives. Through simulation experiments, we demonstrate that our SD tests outperform the recentring test proposed by Donald and Hsu (2016) as well as the integral‐type test presented by Linton et al. (2010) in various scenarios discussed in existing literature. We also employ the proposed tests to analyze changes in the distribution of household income in the United Kingdom over time. The proposed tests offer some insights into potential dominance relationships within this context.
我们在两个独立随机样本的背景下研究了一阶随机支配（SD）检验。我们引入了几种检验统计量，它们能有效捕捉违反支配关系的情况，尤其是在尾部区域。此外，我们还开发了一种重采样程序，用于计算这些检验的-值或临界值。所提出的检验对于正面配置的渐近 I 型错误率等于标称水平。此外，对于任何固定的替代方案，它们的幂都接近 1。通过模拟实验，我们证明在现有文献讨论的各种情况下，我们的 SD 检验优于 Donald 和 Hsu（2016 年）提出的重整检验以及 Linton 等人（2010 年）提出的积分型检验。我们还利用提出的检验分析了英国家庭收入分布随时间的变化。在此背景下，所提出的检验为潜在的支配关系提供了一些见解。
{"title":"Tests for the first‐order stochastic dominance","authors":"Weiwei Zhuang, Peiming Wang, Jiahua Chen","doi":"10.1002/cjs.11811","DOIUrl":"https://doi.org/10.1002/cjs.11811","url":null,"abstract":"We study the first‐order stochastic dominance (SD) test in the context of two independent random samples. We introduce several test statistics that effectively capture violations of the dominance relationship, particularly in the tail regions. Additionally, we develop a resampling procedure to compute the ‐values or critical values for these tests. The proposed tests have asymptotic type I error rates for frontal configurations equal to the nominal level . Furthermore, their powers approach 1 for any fixed alternatives. Through simulation experiments, we demonstrate that our SD tests outperform the recentring test proposed by Donald and Hsu (2016) as well as the integral‐type test presented by Linton et al. (2010) in various scenarios discussed in existing literature. We also employ the proposed tests to analyze changes in the distribution of household income in the United Kingdom over time. The proposed tests offer some insights into potential dominance relationships within this context.","PeriodicalId":501595,"journal":{"name":"The Canadian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141548250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Galappaththige S. R. de Silva, Pankaj K. Choudhary
A tolerance band for a functional response provides a region that is expected to contain a given fraction of observations from the sampled population at each point in the domain. This band is a functional analogue of the tolerance interval for a univariate response. Although the problem of constructing functional tolerance bands has been considered for a Gaussian response, it has not been investigated for non‐Gaussian responses, which are common in biomedical applications. We describe a methodology for constructing tolerance bands for two non‐Gaussian members of the exponential family: binomial and Poisson. The approach is to first model the data using the framework of generalized functional principal components analysis. Then, a parameter is identified in which the marginal distribution of the response is stochastically monotone. We show that the tolerance limits can be readily obtained from confidence limits for this parameter, which in turn can be computed using large‐sample theory and bootstrapping. Our proposed methodology works for both dense and sparse functional data. We report the results of simulation studies designed to evaluate its performance and get recommendations for practical applications. We illustrate our proposed method using two actual biomedical studies, and also provide computer source code that implements our method.
{"title":"Tolerance bands for exponential family functional data","authors":"Galappaththige S. R. de Silva, Pankaj K. Choudhary","doi":"10.1002/cjs.11808","DOIUrl":"https://doi.org/10.1002/cjs.11808","url":null,"abstract":"A tolerance band for a functional response provides a region that is expected to contain a given fraction of observations from the sampled population at each point in the domain. This band is a functional analogue of the tolerance interval for a univariate response. Although the problem of constructing functional tolerance bands has been considered for a Gaussian response, it has not been investigated for non‐Gaussian responses, which are common in biomedical applications. We describe a methodology for constructing tolerance bands for two non‐Gaussian members of the exponential family: binomial and Poisson. The approach is to first model the data using the framework of generalized functional principal components analysis. Then, a parameter is identified in which the marginal distribution of the response is stochastically monotone. We show that the tolerance limits can be readily obtained from confidence limits for this parameter, which in turn can be computed using large‐sample theory and bootstrapping. Our proposed methodology works for both dense and sparse functional data. We report the results of simulation studies designed to evaluate its performance and get recommendations for practical applications. We illustrate our proposed method using two actual biomedical studies, and also provide computer source code that implements our method.","PeriodicalId":501595,"journal":{"name":"The Canadian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141548251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Interval‐censored data arise when a failure process is under intermittent observation and failure status is only known at assessment times. We consider the development of predictive algorithms when training samples involve interval censoring. Using censoring unbiased transformations and pseudo‐observations, we define observed data loss functions, which are unbiased estimates of the corresponding complete data loss functions. We show that regression trees based on these loss functions can recover the tree structure and yield good predictive accuracy. An application is given to a study involving individuals with psoriatic arthritis where the aim is to identify genetic markers useful for the prediction of axial disease within 10 years of a baseline assessment.
{"title":"Regression trees for interval‐censored failure time data based on censoring unbiased transformations and pseudo‐observations","authors":"Ce Yang, Xianwei Li, Liqun Diao, Richard J. Cook","doi":"10.1002/cjs.11807","DOIUrl":"https://doi.org/10.1002/cjs.11807","url":null,"abstract":"Interval‐censored data arise when a failure process is under intermittent observation and failure status is only known at assessment times. We consider the development of predictive algorithms when training samples involve interval censoring. Using censoring unbiased transformations and pseudo‐observations, we define observed data loss functions, which are unbiased estimates of the corresponding complete data loss functions. We show that regression trees based on these loss functions can recover the tree structure and yield good predictive accuracy. An application is given to a study involving individuals with psoriatic arthritis where the aim is to identify genetic markers useful for the prediction of axial disease within 10 years of a baseline assessment.","PeriodicalId":501595,"journal":{"name":"The Canadian Journal of Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141513857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"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":"https://doi.org/10.1002/cjs.11809","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":0.0,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141513858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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