Moderate Deviation And Large Deviation For Recursive Density Estimators

In this article.we discuss density estimation.There are many kinds of estimations in this field,like Rosenblatt estimation,Wolverton-Wagner estimation as well as Wegman-Davies estimation.What we mainly study here are the moderate and the large deviation principles for Wegman-Davies estimation.In the first chapter.we introduce the background of the stochastic approximation and state some known results.In addition,we propose our research directions and questions.The second chapter is the important part of this paper.In this part.introduces the main results f our research.Given the definition of Wegman-Davies estimate,we use Gartner-Ellis theorem to prove its moderate deviation principle.Since it is biased,we need to change the coeflicients and modify it into an unbias estimate.In the third chapter we mainly introduce the large deviation principle for Wegman-Davies recursive density estimation.as well as a better result with weakened conditions.We use probability theory to prove them both.