Density Estimation for the Case of Supersmooth Measurement Error
研究了当测量误差的特征函数指数衰减时，如何自适应地估计被污染随机变量的概率密度，并提出了渐近有效的数据驱动估计量，适用于小样本和圆形数据。
Abstract The problem is to estimate the probability density of a random variable contaminated by an independent measurement error. I explore one of the worst-case scenario when the characteristic function of this measurement error decreases exponentially and thus optimal estimators converge only with logarithmic rate. The particular example of such measurement error is any random variable contaminated by normal, Cauchy, or another stable random variable. For this setting and circular data, I suggest an asymptotically efficient data-driven estimator that is adaptive to both smoothness of estimated density and distribution of measurement error. Moreover, this estimator is universal in sense that its derivatives and integral are sharp estimators of the corresponding derivatives and the cumulative distribution function, and these estimators are sharp both globally and pointwise. For the case of small sample sizes, I suggest a modified estimator that mimics an optimal linear pseudoestimator. I explore this estimator theoretically, via intensive Monte Carlo simulations and practical examples. Key Words: Acute pancreatitisAdaptationCharacteristic functionCircular dataDeconvolutionMonte CarloPointwise and global lossesSharp optimalitySmall samples