On the Bernstein-smoothed lower-tail Spearman's rho estimator

TL;DR

This paper introduces a Bernstein-based estimator for lower-tail Spearman's rho that improves accuracy by reducing variance, demonstrated through simulations showing significant MSE reductions in tail regions.

Contribution

It develops a novel Bernstein estimator for lower-tail Spearman's rho with proven consistency and asymptotic normality, outperforming classical methods in tail regions.

Findings

Significant MSE reduction (up to 70%) in tail estimates.

Estimator exhibits strong consistency and asymptotic normality.

Simulation confirms variance reduction benefits in small samples.

Abstract

This note develops a Bernstein estimator for lower-tail Spearman's rho and establishes its strong consistency and asymptotic normality under mild regularity conditions. Smoothing the empirical copula yields a strictly smaller mean squared error (MSE) in tail regions by lowering sampling variance relative to the classical Spearman's rho estimator. A Monte Carlo simulation experiment with the Farlie--Gumbel--Morgenstern copula demonstrates variance reductions that translate into lower MSE estimates (up to $\sim 70\%$ lower) at deep-tail thresholds under weak to moderate dependence and small sample sizes. To facilitate reproducibility of the findings, the R code that generated all simulation results is readily accessible online.