A Smooth Simultaneous Confidence Envelope For Bivariate Cumulative Distribution Functions

A smooth simultaneous confdence envelope is proposed for bivariate Ho¨lder con-tinuous cumulative distribution function (cdf), extending the work of [10] on univariatecumulative distribution function. The proposed estimator approximates the empiricalcdf estimator at a rate of o(pn1/2), while the new estimator is smooth instead of astep function. A smooth simultaneous confdence envelope is constructed based onthe smooth distribution estimator and the percentiles of the maximal bivariate Brow-nian bridges distribution. Simulation using two diferent automatic bandwidths hasprovided evidence that supports the theoretical results.