The construction of the simultaneous confidence bands for the integrated hazard function is considered. The Nelson-Aalen estimator is used. The simultaneous confidence bands based on bootstrap methods are presented. Two methods of construction of such confidence bands are proposed. The weird bootstrap method is used for resampling. Simulations are made to compare the actual coverage probability of the bootstrap and the asymptotic simultaneous confidence bands. It is shown that the equal-tailed bootstrap confidence band has the coverage probability closest to the nominal one. We also present application of our confidence bands to the data regarding survival after heart transplant.
We deal with the problem of finding a maximum of a function from the Hölder class on a quantum computer. We show matching lower and upper bounds on the complexity of this problem. We prove upper bounds by constructing an algorithm that uses a pre-existing quantum algorithm for finding maximum of a discrete sequence. To prove lower bounds we use results for finding the logical OR of sequence of bits. We show that quantum computation yields a quadratic speed-up over deterministic and randomized algorithms.
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