Efficient Estimation of a Distribution Function under Quadrant Dependence
Efficient Estimation of a Distribution Function under Quadrant Dependence
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DOI：
10.1111/1467-9469.00098
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发表时间：
1998-03
影响因子：
1
通讯作者：
Z. Cai;G. Roussas
中科院分区：
文献类型：
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作者：
Z. Cai;G. Roussas
LetX1 X2,…，是构成严格平稳序列的实值随机变量，并且满足两两正象限相关或两两负象限相关的基本要求。设f ^为xips的边际分布函数，它由经验分布函数fn和光滑核型估计efn估计，由分段x1，…，Xn估计。这些估计值是根据它们的均方误差（MSE）进行比较的。本文的主要研究结果如下：在一定的正则性条件下，确定了最优带宽（在MSE意义上），并发现其与独立同分布情况下的带宽相同。还表明，nMSE （Fn(t)）和nmse （F^n(t)）趋向于同一个常数asn→∞，因此无法根据MSE区分两个估计。下一步，ifi(n) = min {k∈{1,2，…}；则证明了i(n)/n趋于1，asn→∞。因此，再一次，我们不能根据渐近相对效率选择一个估计而不是另一个估计。然而，如果偏差的平方ofF^n(t)足够快地趋向于0，或者等价地，带宽满足nh3n→0，asn→∞的要求，则表明，对于核的合适选择，(i(n)−n)/(nhn)趋向于正数，asn→∞。由此可见，ofFn(t)相对于toF^n(t),i(n)−n的不足是实质性的，并且实际上趋向于∞，asn→∞。就不足而言，平滑估计ef ^n(t)优于经验分布函数fn (t)。
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)