半序集样本的非参数密度估计及其在小麦产量分布估计中的应用
Nonparametric Density Estimation Using Partially Rank-Ordered Set Samples With Application in Estimating the Distribution of Wheat Yield
摘要：
研究了基于部分秩序集（PROS）抽样设计的秩基观测值的未知密度函数$f$的非参数估计. PROS抽样设计在环境、生态和医学研究中有许多应用，在这些研究中，感兴趣变量的精确测量是昂贵的，但是可以通过除实际测量之外的任何方式相对于感兴趣变量订购少量抽样单元，并且这可以以低成本完成。PROS观测值涉及独立的顺序统计量，它们不是同分布的，大多数常用的非参数技术不能直接应用于它们。我们首先基于不完美的PROS采样过程开发了$f$的核密度估计，并研究了其理论性质。然后，我们考虑了当底层分布假设是对称的问题，并介绍了$f$的一些插件核密度估计。我们使用EM型算法来估计与不完美的PROS设计相关的错位概率。最后，我们通过几个模拟研究和一个案例研究来扩展我们的结果的各种数值说明，以估计小麦产量的分布，使用种植小麦的土地总面积作为一个容易获得的辅助信息。我们的结果表明，PROS密度估计性能优于SRS和RSS。
英文摘要：
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 kernel density estimates 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.
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