Nonparametric estimation for dependent data

Nonparametric estimation for dependent observations has a long history in statistics. Rosenblatt [42] first studied density estimation for dependent data. Since then several authors have considered nonparametric estimation under various assumptions (notable early articles include Robinson [39] and Hart [29]). For example, Hall and Hart [25], Giraitis et al. [22], Mielniczuk [34] and Estevas and Vieu [18] consider density estimation for linear processes which have long memory, whereas Cheng and Robinson [9] consider density estimation for random variables which are nonlinear functions of a linear process. A notable result, is that they show if the observations were from a linear process and have short memory, then the usual rate of convergence, known for independent observations, also holds for dependent observations. On the other hand, for long memory processes, the rate of convergence is different. Interestingly, despite long memory influencing the rate of convergence, there is no influence of long memory on the bandwidth choice, which is same regardless of short or long memory. In other words, if the observations come from a linear process, a larger bandwidth does not improve the rate of convergence