Consistencies Of Density Function Estimations Under Two Kinds Of Dependent Samples

It is well known that the widely orthant dependent (WOD) sequence is a class of dependent sequence including the extended negatively dependent (END), neg-atively dependent (ND), and negatively associated (NA) sequences. And WOD and END sequences are widely used in the risk analysis, multivariate analysis,reliability theory and other fields. Therefore, it is of great importance to extend the properties of nonparametric statistical large sample from independent or some dependent sequences to END and WOD sequences.In this thesis, based on END and WOD sequences, the nearest neighbor esti-mator and kernel density estimator for unknown density function are considered.Under some suitable conditions, the r-order moment consistency, strong consis-tency, strong uniform consistency axe discussed, and its convergence rates for those estimators are also obtained, which extend and improve some results in existing literatures.The paper is divided into four chapters.In Chapter 1, the study background and methodology of the unknown density function estimate are concerned, and research status of domestic and abroad about WOD and END random variable sequences are also stated. And the main theorems of this thesis are explored.In Chapter 2, using the Bernstein inequality and Rosenthal inequality of END random variables, we study the strong consistency and r-order moment consistency for recursive kernel density estimator, under some suitable conditions.In Chapter 3, by the exponential inequality of WOD random variables, we discuss the uniformly strong consistency and mean square consistency of kernel density estimator for unknown density function, under appropriate conditions.In Chapter 4, based on the Bernstein inequality of WOD random variable se-quence, we establish the uniformly strong consistency rate of the nearest neighbor density estimator for unknown density function, under some suitable conditions.