## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 8, Number 1 (2014), 738-761.

### Nonparametric density estimation using partially rank-ordered set samples with application in estimating the distribution of wheat yield

Sahar Nazari, Mohammad Jafari Jozani, and Mahmood Kharrati-Kopaei

#### Abstract

We study nonparametric estimation of an unknown density function $f$ based on the ranked-based observations obtained from a partially rank-ordered set (PROS) sampling design. PROS sampling design has many applications in environmental, ecological and medical studies where the exact measurement of the variable of interest is costly but a small number of sampling units can be ordered with respect to the variable of interest by any means other than actual measurements and this can be done at low cost. PROS observations involve independent order statistics which are not identically distributed and most of the commonly used nonparametric techniques are not directly applicable to them. We first develop a kernel density estimator of $f$ based on an imperfect PROS sampling procedure and study its theoretical properties. Then, we consider the problem when the underlying distribution is assumed to be symmetric and introduce some plug-in kernel density estimators of $f$. We use an EM type algorithm to estimate misplacement probabilities associated with an imperfect PROS design. Finally, we expand on various numerical illustrations of our results via several simulation studies and a case study to estimate the distribution of wheat yield using the total acreage of land which is planted in wheat as an easily obtained auxiliary information. Our results show that the PROS density estimate performs better than its SRS and RSS counterparts.

#### Article information

**Source**

Electron. J. Statist., Volume 8, Number 1 (2014), 738-761.

**Dates**

First available in Project Euclid: 21 May 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1400703412

**Digital Object Identifier**

doi:10.1214/14-EJS902

**Mathematical Reviews number (MathSciNet)**

MR3211030

**Zentralblatt MATH identifier**

1348.62127

**Keywords**

Imperfect subsetting kernel function mean integrated square error nonparametric procedure optimal bandwidth ranked set sampling

#### Citation

Nazari, Sahar; Jafari Jozani, Mohammad; Kharrati-Kopaei, Mahmood. Nonparametric density estimation using partially rank-ordered set samples with application in estimating the distribution of wheat yield. Electron. J. Statist. 8 (2014), no. 1, 738--761. doi:10.1214/14-EJS902. https://projecteuclid.org/euclid.ejs/1400703412