Efficient Estimation of a Distribution Function under Quadrant Dependence

Let X 1 , X 2 , ..., be real‐valued random variables forming a strictly stationary sequence, and satisfying the basic requirement of being either pairwise positively quadrant dependent or pairwise negatively quadrant dependent. Let F^ be the marginal distribution function of the X i p s , which is estimated by the empirical distribution function F n and also by a smooth kernel‐type estimate F n , by means of the segment X 1 , ..., X n . These estimates are compared on the basis of their mean squared errors (MSE). The main results of this paper are the following. Under certain regularity conditions, the optimal bandwidth (in the MSE sense) is determined, and is found to be the same as that in the independent identically distributed case. It is also shown that n MSE( F n ( t )) and n MSE ( F^ n ( t )) tend to the same constant, as n →∞ so that one can not discriminate be tween the two estimates on the basis of the MSE. Next, if i ( n ) = min { k ∈{1, 2, ...}; MSE ( F k ( t )) ≤ MSE ( F n ( t ))}, then it is proved that i ( n )/ n tends to 1, as n →∞. Thus, once again, one can not choose one estimate over the other in terms of their asymptotic relative efficiency. If, however, the squared bias of F^ n ( t ) tends to 0 sufficiently fast, or equivalently, the bandwidth h n satisfies the requirement that nh 3 n → 0, as n →∞, it is shown that, for a suitable choice of the kernel, ( i ( n ) − n )/( nh n ) tends to a positive number, as n →∞ It follows that the deficiency of F n ( t ) with respect to F^ n ( t ), i ( n ) − n , is substantial, and, actually, tends to ∞, as n →∞. In terms of deficiency, the smooth estimate F^ n ( t ) is preferable to the empirical distribution function F n ( t )