On rank estimators in increasing dimensions
On rank estimators in increasing dimensions
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DOI：
10.1016/j.jeconom.2019.08.003
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发表时间：
2019-08
影响因子：
6.3
通讯作者：
Yanqin Fan;Fang Han;Wei Li;Xiao‐Hua Zhou
中科院分区：
文献类型：
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作者：
Yanqin Fan;Fang Han;Wei Li;Xiao‐Hua Zhou
作为一个显著的例子，秩估计量家族，包括韩氏的最大秩相关(han，1987)，已经被广泛地用于研究回归问题。对于这些估计，虽然引入了线性指标来缓解维度的影响，但很少有人研究高维对推断的影响。本文通过研究一大类M-估计量的统计性质来填补这一空白，这些M-估计量的目标函数被表示为U-过程，并且在允许模型中的参数个数p_n随样本大小n而增加的增维设置中可能是不连续的。首先，我们发现在估计中，当p_n/n_→_0时，(p_n/n)_(1/2)收敛速度是可以获得的。其次，我们建立了Bahadur型上界，并研究了正态逼近的有效性，我们发现这通常需要比pn2/n→0更强的标度要求。第三，给出了渐近协方差矩阵的数值导数估计相容的条件，并证明了实现协方差估计的步长必须相对于pn进行调整，所有的理论结果都得到了模拟研究的进一步支持。
The family of rank estimators, including Han’s maximum rank correlation (Han, 1987) as a notable example, has been widely exploited in studying regression problems. For these estimators, although the linear index is introduced for alleviating the impact of dimensionality, the effect of large dimension on inference is rarely studied. This paper fills this gap via studying the statistical properties of a larger family of M-estimators, whose objective functions are formulated as U-processes and may be discontinuous in increasing dimension set-up where the number of parameters, p n, in the model is allowed to increase with the sample size, n. First, we find that often in estimation, as p n∕ n→ 0,(p n∕ n) 1∕ 2 rate of convergence is obtainable. Second, we establish Bahadur-type bounds and study the validity of normal approximation, which we find often requires a much stronger scaling requirement than p n 2∕ n→ 0. Third, we state conditions under which the numerical derivative estimator of asymptotic covariance matrix is consistent, and show that the step size in implementing the covariance estimator has to be adjusted with respect to p n. All theoretical results are further backed up by simulation studies.