Electronic Journal of Statistics

Median confidence regions in a nonparametric model

Edsel A. Peña and Taeho Kim

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Abstract

The nonparametric measurement error model (NMEM) postulates that $X_{i}=\Delta +\epsilon _{i},i=1,2,\ldots ,n;\Delta \in \Re $ with $\epsilon _{i},i=1,2,\ldots ,n$, IID from $F(\cdot )\in\mathfrak{F}_{c,0}$, where $\mathfrak{F}_{c,0}$ is the class of all continuous distributions with median $0$, so $\Delta $ is the median parameter of $X$. This paper deals with the problem of constructing a confidence region (CR) for $\Delta $ under the NMEM. Aside from the NMEM, the problem setting also arises in a variety of situations, including inference about the median lifetime of a complex system arising in engineering, reliability, biomedical, and public health settings, as well as in the economic arena such as when dealing with household income. Current methods of constructing CRs for $\Delta $ are discussed, including the $T$-statistic based CR and the Wilcoxon signed-rank statistic based CR, arguably the two default methods in applied work when a confidence interval about the center of a distribution is desired. A ‘bottom-to-top’ approach for constructing CRs is implemented, which starts by imposing reasonable invariance or equivariance conditions on the desired CRs, and then optimizing with respect to their mean contents on subclasses of $\mathfrak{F}_{c,0}$. This contrasts with the usual approach of using a pivotal quantity constructed from test statistics and/or estimators and then ‘pivoting’ to obtain the CR. Applications to a real car mileage data set and to Proschan’s famous air-conditioning data set are illustrated. Simulation studies to compare performances of the different CR methods were performed. Results of these studies indicate that the sign-statistic based CR and the optimal CR focused on symmetric distributions satisfy the confidence level requirement, though they tended to have higher contents; while three of the bootstrap-based CR procedures and one of the newly-developed adaptive CR tended to be a tad more liberal, but with smaller contents. A critical recommendation for practitioners is that, under the NMEM, the $T$-statistic based and Wilcoxon signed-rank statistic based CRs should not be used since they either have very degraded coverage probabilities or inflated contents under some of the allowable error distributions under the NMEM.

Article information

Source
Electron. J. Statist., Volume 13, Number 2 (2019), 2348-2390.

Dates
Received: April 2019
First available in Project Euclid: 18 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1563436820

Digital Object Identifier
doi:10.1214/19-EJS1577

Subjects
Primary: 62G15: Tolerance and confidence regions
Secondary: 62G09, 62G35

Keywords
Bootstrap confidence region BCa confidence region confidence region content of a confidence region equivariant confidence region expected values of differences of order statistics invariant models nonparametric confidence region nonparametric measurement error model randomized confidence region

Rights
Creative Commons Attribution 4.0 International License.

Citation

Peña, Edsel A.; Kim, Taeho. Median confidence regions in a nonparametric model. Electron. J. Statist. 13 (2019), no. 2, 2348--2390. doi:10.1214/19-EJS1577. https://projecteuclid.org/euclid.ejs/1563436820


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