Efficient Estimation of a Distribution Function under Quadrant Dependence

LetX1,X2, ..., 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. LetF^ be the marginal distribution function of theXips, which is estimated by the empirical distribution functionFn and also by a smooth kernel‐type estimateFn, by means of the segmentX1, ...,Xn. 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 thatn MSE(Fn(t)) andnMSE (F^n(t)) tend to the same constant, asn→∞ so that one can not discriminate be tween the two estimates on the basis of the MSE. Next, ifi(n) = min {k∈{1, 2, ...}; MSE (Fk(t)) ≤ MSE (Fn(t))}, then it is proved thati(n)/n tends to 1, asn→∞. Thus, once again, one can not choose one estimate over the other in terms of their asymptotic relative efficiency. If, however, the squared bias ofF^n(t) tends to 0 sufficiently fast, or equivalently, the bandwidthhn satisfies the requirement thatnh3n→ 0, asn→∞, it is shown that, for a suitable choice of the kernel, (i(n) −n)/(nhn) tends to a positive number, asn→∞ It follows that the deficiency ofFn(t) with respect toF^n(t),i(n) −n, is substantial, and, actually, tends to ∞, asn→∞. In terms of deficiency, the smooth estimateF^n(t) is preferable to the empirical distribution functionFn(t)