Nonparametric density estimation using partially rank-ordered set samples with application in estimating the distribution of wheat yield. (English) Zbl 1348.62127

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

Electron. J. Stat. 8, No. 1, 738-761 (2014).

Summary: 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.

MSC:

62G07	Density estimation
62D05	Sampling theory, sample surveys
62H30	Classification and discrimination; cluster analysis (statistical aspects)
62P20	Applications of statistics to economics

Keywords:

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

Software:

KernSmooth; Rsolnp

References:

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