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- MakaleYalnız künye2026Scopus SJRQ2A Bernstein polynomial estimation of the moment generating function
Susam, Selim Orhun
We study Bernstein polynomial smoothing for nonparametric estimation of the moment generating function (MGF) of a distribution supported on (Formula presented.). We prove strong consistency of the Bernstein-smoothed MGF estimator and show that, when the smoothing degree m grows sufficiently fast, it shares the same (Formula presented.) asymptotic normal distribution as the empirical MGF estimator, with variance (Formula presented.). A second-order bias–variance expansion reveals that Bernstein smoothing yields a bias of order (Formula presented.) and a variance-reduction term of order (Formula presented.). This leads to the optimal smoothing rate (Formula presented.) for pointwise mean squared error (MSE) minimization. Simulation results confirm that the Bernstein estimator is especially effective in challenging boundary-dominated settings, notably for U-shaped beta models and for larger positive t values. Overall, Bernstein polynomial smoothing provides a simple, theoretically justified, and practically effective tool for nonparametric MGF estimation on compact supports.
- MakaleYalnız künye2026Scopus SJRQ2A bias-corrected partial bernstein copula approach for nonparametric regression
Susam, Selim Orhun
Multivariate Bernstein polynomials yield smooth, boundary–bias–free estimators for copulas and their derivatives. The classical partial Bernstein copula derivative of Janssen et al. (2016) suffers from an uncorrected first–order bias that can dominate the mean squared error (MSE) in moderate samples. We introduce a simple double–difference correction that removes the entire leading bias term while increasing the variance by a small fixed multiplicative factor only. When this bias–reduced partial derivative is plugged into the copula based regression integral, the resulting estimator achieves substantially smaller bias and uniformly lower MSE than the classical version. Analytic bias–variance expansions are provided, and a comprehensive Monte Carlo study shows up to 50% reductions in MSE across a range of Bernstein copula degrees. A real data application to the mcycle dataset further demonstrates the practical advantages of the method, with the smoothing parameters selected by least squares cross validation (LSCV).
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İncele - MakaleAçık Erişim2026Scopus SJRQ2Bernstein-smoothed estimation and bootstrap inference for the lower-tail Spearman's rho curve
Lyu, Guanjie; Ouimet, Frédéric; Susam, Selim O.
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 (Formula presented.). 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 (Formula presented.). Under a target-specific integrated Bernstein-bias condition and (Formula presented.), we derive functional weak convergence with the same first-order Gaussian limit as the empirical copula-based estimator; this degree regime includes (Formula presented.). For inference, we justify a smoothed beta bootstrap based on sampling from the empirical beta copula and construct pointwise confidence intervals from the absolute centred 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 (Formula presented.) is a simple theoretically admissible default that avoids the most severe oversmoothing. A descriptive application to the Loss–ALAE insurance claims data illustrates how the lower-tail Spearman's rho curve can reveal local dependence features that are not captured by the global Spearman correlation.