Bernstein-smoothed estimation and bootstrap inference for the lower-tail Spearman's rho curve

Abstract

This paper studies a Bernstein-smoothed plug-in estimator for the lower-tail Spearman's rho curve, a rank-based measure of local concordance defined through a normalized copula integral over the lower-left square $[0,p]^2$. The estimator applies the lower-tail Spearman functional to the empirical Bernstein copula and introduces a degree parameter that controls finite-sample regularization. The contribution is target-specific estimation and inference for the lower-tail Spearman's rho curve rather than a new general-purpose copula estimator. We establish uniform strong consistency on compact intervals away from zero whenever $m \to \infty$. Under a target-specific integrated Bernstein-bias condition and $\sqrt n/m \to 0$, we derive functional weak convergence with the same first-order Gaussian limit as the empirical copula-based estimator; this degree regime includes $m = \lfloor n^{2/3}\rfloor$. For inference, we justify a smoothed beta bootstrap based on sampling from the empirical beta copula and construct pointwise confidence intervals from the absolute centered bootstrap root. Monte Carlo experiments for the Farlie--Gumbel--Morgenstern, Gaussian, Clayton, and Frank copulas assess pointwise coverage over the threshold grid and show that Bernstein smoothing generally reduces integrated variance and often lowers mean integrated squared error under weak to moderate dependence. Additional common-sample experiments compare the proposed estimator with the empirical beta and empirical checkerboard Bernstein copula estimators in terms of pointwise error, integrated error, computation time, selected degrees, numerical stability, and pointwise coverage. A sensitivity analysis shows that $m = \lfloor n^{2/3}\rfloor$ is a simple theoretically admissible default that avoids the most severe oversmoothing. A descriptive application to the Loss--ALAE insurance claims data illustrates our method.