We propose a new wavelet-based method for density estimation when the data are size-biased. More specifically, we consider a power of the density of interest, where this power exceeds 1/2. Warped wavelet bases are employed, where warping is attained by some continuous cumulative distribution function. A special case is the conventional orthonormal wavelet estimation, where the warping distribution is the standard continuous uniform. 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 settings which are easy to interpret in practice. An application with a real dataset on fatal traffic accidents involving alcohol illustrates the method. We observe that warped bases provide more flexible and superior estimates for both simulated and real data. Moreover, we find that estimating the power of a density (for instance, its square root) further improves the 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, 我们考虑的是利息密度的威力, 该功率超过1/2。 使用扭曲的波子基, 以某种连续的累积分布函数实现扭曲。 一个特例是常规的正态波子估计, 扭曲的分布是标准的连续统一。 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。 进行蒙特卡洛模拟以比较在实际操作中容易解释的四个特殊设置。 使用涉及酒精的致命交通事故的真实数据集来应用该方法。 我们观察到, 扭曲的基子为模拟数据和真实数据提供了更灵活和更高的估计值。 此外, 我们发现, 估计密度( 例如, 其平方根) 的功率可以进一步改善结果 。