A simple modification of the Hill estimator with applications to robustness and bias reduction
A simple modification of the Hill estimator with applications to robustness and bias reduction
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Hill 估计器的简单修改及其鲁棒性和偏差减少的应用
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K. Knight
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作者：
K. Knight
假设X1，· · ·，Xn是独立同分布的。随机变量P（Xi > x）= xL（x），定义X（1）≥ X（2）≥ · · · ≥ X（n）为顺序统计量。尾指数α的Hill估计量（Hill，1975）是一种伪极大似然估计量，基于归一化对数间距Yj = j ln（X（j）/X（j+1））的指数近似，其中j = 1，· · ·，k。在实践中，希尔估计量可能非常依赖于k = kn的选择，并且固有地对大值Yj不鲁棒，这使希尔估计量向下偏置。在本文中，我们介绍了一个简单的鲁棒性的希尔估计，有界的影响曲线，是费舍尔一致的。该估计量计算简单，并且可以调整为具有0和1之间的指定渐近效率（相对于Hill估计量）。由此产生的家庭估计也可以用来减少渐近偏差的希尔估计。我们还考虑了基于指数回归方法的Hill估计量的修改的扩展（Feuerverger和Hall，1999; Beirlant等人，1999年）。
Suppose that X1, · · · , Xn are i.i.d. random variables with P (Xi > x) = xL(x) and define X(1) ≥ X(2) ≥ · · · ≥ X(n) to be the order statistics. The Hill estimator (Hill, 1975) of the tail index α is a pseudo-maximum likelihood estimator based on the exponential approximation of the normalized log-spacings Yj = j ln(X(j)/X(j+1)) for j = 1, · · · , k. In practice, the Hill estimator can be extremely dependent on the choice of k = kn and is inherently non-robust to large values Yj, which bias the Hill estimator downward. In this paper, we introduce a simple robustification of the Hill estimator that has a bounded influence curve and is Fisher consistent. The estimator is straightforward to compute and can be tuned to have a specified asymptotic efficiency (with respect to the Hill estimator) between 0 and 1. The resulting family of estimators can also be used to reduce the asymptotic bias of the Hill estimator. We also consider extensions to modifications of the Hill estimator based on exponential regression methods (Feuerverger and Hall, 1999; Beirlant et al., 1999).