Probit Transformation for Kernel Density Estimation on the Unit Interval
单位区间核密度估计的Probit变换方法AI 解读 ↓
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单位区间核密度估计的Probit变换方法
- 研究问题
- 对于取值限定于单位区间的连续变量，如何缓解常规核密度估计的边界偏差，并使变换后估计与反变换实际可行？
- 方法与思路
- 作者分析简单变换法表现不佳的原因，将Probit变换与局部似然密度估计结合，并设计交叉验证带宽选择规则。
- 材料陈述的贡献
- 摘要称，该方法获得较少边界问题的密度估计，推导渐近性质，并通过多种密度形状的模拟和小样本实例展示表现。
- 阅读关注点
- 应核对反变换在端点附近的行为、局部似然修正与带宽规则，以及渐近结果和模拟比较覆盖的密度类别。
- 阅读边界
- 摘要中的整体最佳属于作者在所设比较中的陈述，不能推广至所有密度。所据索引摘要版本未注明，刊本变化未知。
原文摘要
Kernel estimation of a probability density function supported on the unit interval has proved difficult, because of the well-known boundary bias issues a conventional kernel density estimator would necessarily face in this situation. Transforming the variable of interest into a variable whose density has unconstrained support, estimating that density, and obtaining an estimate of the density of the original variable through back-transformation, seems a natural idea to easily get rid of the boundary problems. In practice, however, a simple and efficient implementation of this methodology is far from immediate, and the few attempts found in the literature have been reported not to perform well. In this article, the main reasons for this failure are identified and an easy way to correct them is suggested. It turns out that combining the transformation idea with local likelihood density estimation produces viable density estimators, mostly free from boundary issues. Their asymptotic properties are derived, and a practical cross-validation bandwidth selection rule is devised. Extensive simulations demonstrate the excellent performance of these estimators compared to their main competitors for a wide range of density shapes. In fact, they turn out to be the best choice overall. Finally, they are used to successfully estimate a density of nonstandard shape supported on [0, 1] from a small-size real data sample.