Recursive Density Function Of Pairwise Nqd Sequences Of Nuclear Estimated Large Sample Properties

The estimator of probability density function is always a hot topic in mathematical statistics, it received wide attention from many scholars. For the estimator problem of density function, they have proposed various methods for estimating the density function, the usual methods are histogram, kernel density estimator, nearest neighbor density estimator and so on. With further research, some scholars based on ordinary kernel density estimator, proposed recursive kernel density estimator, this method is more convenient in the calculation, so it aroused the attention by many scholars. At first, scholars discussed the property of recursive kernel density estimator under the independent and identical distribution sequences, they obtained the corresponding results. But in many cases, the sample is not independent; they have some connections, so there are scholars discussed it in the dependent case. This paper researched the large sample properties of recursive kernel density estimator under pairwise NQD sequences.Firstly, This article briefly introduces the research background and present situation at home and abroad of the estimator of density function and pairwise NQD sequences, then described the importance to discuss the estimator of probability density function under pairwise NQD.Secondly, Based on the moment inequalities and related lemma of pairwise NQD sequences, we proved the r-order mean consistency,weak consistency,strong consistency and uniformly strong consistency of recursive kernel density function estimator under pairwise NQD sequences. In certain conditions, the convergent rate of r-order mean consistency has been gained. All the studies are promoting the corresponding results in independent and identical distribution case and other associations.Finally, under suitable conditions, we using some inequalities of pairwise NQD sequences and Lindeberg-Feller central limit theorem in order to prove the asymptotic property of recursive kernel density function estimator under pairwise NQD sequences, and further extend the corresponding results in existing articles.