Multiplier bootstrap methods for conditional distributions

Abstract

The multiplier bootstrap is a fast and easy-toimplement alternative to the standard bootstrap; it has been used successfully in many statistical contexts. In this paper, resampling methods based on multipliers are proposed in a general framework where one investigates the stochastic behavior of a random vector Y ∈ R d conditional on a covariate X ∈ R. Specifically, two versions of the multiplier bootstrap adapted to empirical conditional distributions are introduced as alternatives to the conditional bootstrap and their asymptotic validity is formally established. As the method walks hand-in-hand with the functional delta method, theory around the estimation of statistical functionals is developed accordingly; this includes the interval estimation of conditional mean and variance, conditional correlation coefficient, Kendall's dependence measure and copula. Composite inference about univariate and joint conditional distributions is also considered. The sample behavior of the new bootstrap schemes and related estimation methods are investigated via simulations and an illustration on real data is provided.

Dynamic prediction of the performance reliability of high-speed railway bearings

“…The bootstrap-least squares method is an artful fusion of the bootstrap method and the least squares method. Between them, the bootstrap method can generate many groups of data, according to a group of data obtained by equiprobable sampling with replacement times [22,23]; the least squares method can find the best match of the fitting function for a group of data by minimizing the sum of the squares for the error [24,25]. The effective integration of the two can be fit many times from a group of data, and then, the least squares solutions of multiple group data can be obtained.…”

“…The higher the confidence level, the larger the upper-lower interval, which in turn effectively guarantees that the actual value is inside these bounds), and the corresponding upper and lower bounds [a L , a U ] and [c L , c U ] are acquired according to Eqs. (23) and (24). The estimated results of the fitting coefficients for bearings A and B in period X 1 -X 5 are given in Table 3.…”

“…If a larger amount of data samples are given, there will be more accurate sample moments and a more authentic probability density. The bootstrap method can generate a large number of sample data, according to the current small amount of data, using equiprobable sampling with replacement times [22,23]. It can separate systematic errors in the dynamic evaluation process by using the nuclear concept; however, it needs to take advantage of the prior information of some rules of the data sample.…”