We propose a new wavelet-based method for density estimation when the data are size-biased. More specifically, we consider the power of the density of interest, where this power is some value greater than or equal to half. Warped wavelet bases are employed, where warping is attained by some continuous cumulative distribution function. This can be seen as a general framework for which the conventional orthonormal wavelet estimation is the case with the standard uniform c.d.f. We show that both linear and nonlinear wavelet estimators are consistent, with optimal and/or near-optimal rates. Monte Carlo simulations are performed to compare four special set-ups which are easy to interpret in practice. A real dataset application illustrates the method. We observe that warped bases provide more flexible and better estimates for both simulated and real data. Moreover, we can see that estimating the density power (for instance, its square root) further improves results.
翻译:当数据大小偏差时,我们建议一种新的以波子为基础的密度估计方法。 更具体地说, 我们考虑利息密度的威力, 当这种威力有一定的值大于或等于一半时。 使用扭曲的波子基, 扭曲是通过某种连续的累积分布函数实现的。 这可以被视为一个总框架, 常规的正方形波子估计在标准统一 c. d. f 中是这种情况。 我们显示线性和非线性波子估计器与最佳和/ 或接近最佳的速率一致。 蒙特卡洛 模拟是用来比较四个特别的组合的, 这些组合在实践中容易解释。 一个真实的数据集应用说明了这个方法。 我们观察到, 扭曲的基为模拟数据和真实的数据提供了更灵活和更好的估计。 此外, 我们可以看到, 估计密度的功率( 例如, 其平方根) 将进一步提高结果。