Smoothed Bootstrap Methods for Right-Censored Data and Bivariate Data

Abstract

This thesis introduces a smoothed bootstrap method for univariate right-censored data and investigates this bootstrap method for the coverage probability and survival function inferences through simulations. The bootstrap method relies on the right-censoring A(n) assumption, which was proposed by Coolen and Yan [21]. This assumption allows sampling from the whole data range and avoids the complication in computation that occurs due to ties and right-censored observations which often occur in the samples created by Efron's bootstrap method [31]. The performance of the proposed bootstrap method is studied on nite and innite data ranges, and compared to the performance of Efron's bootstrap method through simulations. It is found that the smoothed bootstrap method mostly outperforms Efron's bootstrap method, in particular when the sample size is small. Also, the smoothed bootstrap method and Efron's bootstrap method are compared through simulations to compute the actual Type 1 error rates of quartiles tests and two sample medians test. For bivariate data, three smoothed bootstrap methods are introduced. Two of them are based on the generalization of Nonparametric Predictive Inference for random quantities (X; Y ) with copulas, proposed by Coolen-Maturi et al. [22] and Muhammad et al. [65]. The third one is by using uniform kernels. These smoothed bootstrap methods are compared to Efron's bootstrap method [33] through simulations. It is found that the smoothed bootstrap methods mostly outperform Efron's bootstrap method in terms of the coverage probabilities for Pearson correlation and the means of T1 = X + Y and T2 = XY^2 when the data distribution is symmetric. Also, these bootstrap methods are compared to compute the Type 1 error rates of the Pearson and Kendall correlation tests to provide insight into the methods' performances. For the Pearson correlation test, the smoothed bootstrap methods mostly perform better than Efron's method, but Efron's method provides better results for the Kendall correlation test.