Minimum distance regression-type estimates with rates under weak dependence

Under weak dependence, a minimum distance estimate is obtained for a smooth function and its derivatives in a regression-type framework. The upper bound of the risk depends on the Kolmogorov entropy of the underlying space and the mixing coefficient. It is shown that the proposed estimates have the same rate of convergence, in the L 1-norm sense, as in the independent case

Kaplan–Meier Estimator under Association

AbstractConsider a long term study, where a series of possibly censored failure times is observed. Suppose the failure times have a common marginal distribution functionF, but they exhibit a mode of dependence characterized by positive or negative association. Under suitable regularity conditions, it is shown that the Kaplan–Meier estimatorFnofFis uniformly strongly consistent; rates for the convergence are also provided. Similar results are established for the empirical cumulative hazard rate function involved. Furthermore, a stochastic process generated byFnis shown to be weakly convergent to an appropriate Gaussian process. Finally, an estimator of the limiting variance of the Kaplan–Meier estimator is proposed and it is shown to be weakly convergent

Asymptotic Normality of Random Fields of Positively or Negatively Associated Processes

Consider a random field of real-valued random variables with finite second moment and subject to covariance invariance and finite susceptibility. Under the basic assumption of positive or negative association, asymptotic normality is established. More specifically, it is shown that the joint distribution of suitably normalized and centered at expectation sums of random variables, over any finite number of appropriately selected rectangles, is asymptotically normal. The mean vector of the limiting distribution is zero and the covariance matrix is a specified diagonal matrix.