Smoothed Bootstrap Methods for Bivariate Data

In this paper, three smoothed bootstrap methods are introduced for bivariate data. Two of them are based on nonparametric predictive inference for bivariate data with both parametric and nonparametric copulas (Coolen-Maturi et al. in J Stat Theory Pract 10:515–538, 2016; Muhammad et al. in Stat Optim Inf Comput 6:398–408, 2018). The nonparametric predictive inference methods combined with copulas use generalizations of Hill’s A(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{(n)}$$\end{document} assumption (Hill in J Am Stat Assoc 63:677–691, 1968) for bivariate data. The third smoothed bootstrap method is based on uniform kernels. All smoothed bootstrap methods are compared to Efron’s bootstrap method for bivariate data (Efron in J Am Stat Assoc 76:312–319, 1981) through simulations. The comparison is conducted in terms of the coverage of percentile confidence intervals for the Pearson, Kendall and Spearman correlations, while also two simple functions of the bivariate observations are considered. From the study, it is found that the smoothed bootstrap methods mostly perform better than Efron’s method in case of data simulated from a symmetric distribution and in case of the correlation between the variables is low or medium, in particular for small data sets. In the case of high dependence level between the variables, Efron’s bootstrap method provides better results for the Pearson, Kendall and Spearman correlations due to its restriction in sampling from the data only, contrary to the smoothed bootstrap methods, which allow more variation in sampling.