## Electronic Journal of Statistics

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

### Median confidence regions in a nonparametric model

#### 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