Bootstrap Selector for the Smoothing Parameter of Beran’s Estimator

This work proposes a resampling technique to approximate the smoothing parameter of Beran’s estimator. It is based on resampling by the smoothed bootstrap and minimising the bootstrap approximation of the mean integrated squared error to find the bootstrap bandwidth. The behaviour of this method has been tested by simulation on several models. Bootstrap confidence intervals are also addressed in this research and their performance is analysed in the simulation study.

On the Admissibility of Simultaneous Bootstrap Confidence Intervals

Simultaneous confidence intervals are commonly used in joint inference of multiple parameters. When the underlying joint distribution of the estimates is unknown, nonparametric methods can be applied to provide distribution-free simultaneous confidence intervals. In this note, we propose new one-sided and two-sided nonparametric simultaneous confidence intervals based on the percentile bootstrap approach. The admissibility of the proposed intervals is established. The numerical results demonstrate that the proposed confidence intervals maintain the correct coverage probability for both normal and non-normal distributions. For smoothed bootstrap estimates, we extend Efron’s (2014) nonparametric delta method to construct nonparametric simultaneous confidence intervals. The methods are applied to construct simultaneous confidence intervals for LASSO regression estimates.

Estimate Accuracy of the Sample Determination Coefficient R2 and R2 for the Multiple Regression Models

In this paper, we propose estimating standard errors for R2 and R2 and to construct their confidence intervals, using the usual and “smoothed bootstrap methods”, which are accurate measures. It is shown by ”Monte Carlo experiments” that the smoothed bootstrap standard errors are more accurate estimates of usual bootstrap method. It is also shown that although the usual and smoothed bootstrap 0.95 confidence intervals of R2 do not include the true value of the parent coefficient of determination in some particular cases, such a phenomenon does not occur when is used.

The Smoothed Bootstrap Fine-Tuning

The bootstrap method is a well-known method to gather a full probability distribution from the dataset of a small sample. The simple bootstrap i.e. resampling from the raw dataset often leads to a significant irregularities in a shape of resulting empirical distribution due to the discontinuity of a support. The remedy for these irregularities is the smoothed bootstrap: a small random shift of source points before each resampling. This shift is controlled by specifically selected distributions. The key issue is such parameter settings of these distributions to achieve the desired characteristics of the empirical distribution. This paper describes an example of this procedure.