On rank estimators in increasing dimensions

TL;DR

This paper investigates the statistical properties of rank estimators in high-dimensional settings, revealing how increasing parameters affect convergence, normal approximation, and covariance estimation, supported by simulations.

Contribution

It extends the understanding of rank estimators to increasing dimensions, analyzing convergence rates, normal approximation validity, and covariance estimation consistency.

Findings

Convergence rate often at (p_n/n)^{1/2} when p_n/n→0.

Normal approximation requires stronger scaling, often p_n^2/n→0.

Covariance estimator consistency depends on step size adjustment with p_n.

Abstract

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 $p_{n}/n\rightarrow 0$, $(p_{n}/n)^{1/2}$ 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 $p_{n}^{2}/n\rightarrow 0.$ 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.