Two-Stage Nonparametric Sequential Estimation of the Directional Density with Complete and Missing Observations

作者：Efromovich Sam^*

来源：Sequential Analysis, 2015, 34(4): 425-440.

DOI：10.1080/07474946.2015.1099934

• Adaptation
• Assigned MISE
• Minimal mean stopping time
• Minimax
• Optimal sampling

Stein (1945) proposed a two-stage sequential methodology of inference that influenced numerous areas of statistics. In this article, the Stein's methodology is used and expands upon nonparametric estimation of the directional probability density. The aim is to propose a data-driven nonparametric sequential procedure that mimics performance of an oracle that knows smoothness of an estimated density and minimizes the mean stopping time given an assigned mean integrated squared error. For such a setting, using a sequential estimator is the must because smoothness of the density is unknown. It is known that for a general random variable the stated problem has no solution because an estimator cannot perform as well as the minimax oracle. At the same time, this article shows that for the case of directional density, under a mild assumption, there exists a data-driven two-stage sequential procedure that is minimax and adapts to unknown smoothness of an underlying density. Furthermore, we are able to solve the same problem for a setting where some observations in a sample may be lost due to a stochastic missing mechanism. This is a practically important and theoretically interesting problem because for missing data even the above-mentioned oracle cannot solve it without knowing the missing mechanism.

• 出版日期2015-10-2