|
Research Projects Supported by HKU's High Performance Computing Facilities |
|
¡@ |
| Researchers: |
| Miss K Y Cheung and Dr Stephen M S Lee, Department of Statistics and Actuarial Science |
| ¡@ |
|
Project Title: |
| Iteration on m out of n Bootstrap in Nonregular Cases |
| Project Description: |
|
The bootstrap has been proved consistent in many cases. However, it may fail to give consistent estimators in some nonregular cases. The m out of n bootstrap (m/n) is a well-known remedy to the inconsistency, but becomes less efficient in cases where the conventional n out of n bootstrap works. In regular cases, bootstrap efficiency could be improved by iterating the bootstrap procedure, which is known as the iterated bootstrap. This project combines the advantages of m/n for repairing bootstrap failure and the iterated bootstrap for enhancing efficiency, and develops iterating algorithm for m/n. The focus is on the effectiveness of this iterated method in improving the efficiency of its non-iterated counterpart in the context of confidence set construction in both regular and nonregular cases. |
| Project Duration: |
| 2 years |
| Project Significance: |
|
Due to its model-free nature and easy implementation, the bootstrap has been widely used in many statistical problems, but it may not work without the regularity conditions. In practice, it is very difficult, or even impossible, to know whether the problem at hand is a regular or nonregular one. m/n is one of the commonly used techniques for rectifying bootstrap inconsistency, but is less efficient than the conventional bootstrap in regular cases. Then, an efficient and consistent method under both regular and nonregular situations seems to be desirable. |
| Results Achieved: |
|
Assume that all the bootstrap
resample sizes are chosen optimally in the following context. Naïve iteraton
on m/n could not enhance the coverage accuracy of the non-iterated
m/n.
Instead, we have developed a special iterated algorithm, which requires two
independent batches of bootstrap resamples. The first set of resamples
involves two nested levels of bootstrapping for recalibrating nominal
coverage to achieve the required coverage and the second used for setting
confidence interval end points using the calibrated nominal level. This new
scheme of iterated m/n succeeds in improving the asymptotic coverage
accuracy over the non-iterated counterpart in two of the nonregular cases:
(1) functions of means with null first derivatives and (2) Stein estimation
problem. (1) In the case of function of means with null first derivatives, the proposed iterated m/n reduces the coverage error of the non-iterated and intuitive iterated m/n from order O(n-1/2) to O(n-2/3), where n is the sample size of the original data. (2) For Stein estimation of a d-variate normal mean vector, d ³ 4, the conventional parametric bootstrap only works when the components of the mean are not all equal, but fails otherwise. The m/n is, on the other hand, consistent in both situations. With the order of bootstrap sample sizes determined from an appropriate minimax criterion, both the non-iterated and intuitive iterated m/n incur the same order of coverage errors, of order O(n-1/4), whether or not the components of the unknown mean are all equal, while the new iterative scheme successfully reduces the coverage error to order O(n-1/3). ¡@ |
| Remarks on the Use of High Performance Computing Cluster: |
|
Simulation studies are usually conducted as a supplement to the asymptotic findings for finite sample illustration. In a simulation, the computer-intensive bootstrap procedure has to be repeated many times that are computationally too expensive to be carried out by commonly used personal computers. The HPC Cluster provides high performance computers that meet such a huge computation demand and the MPI cuts the computation time by more than ten times. Furthermore, the supporting staff of HPC Cluster provides great help in optimizing the computer programs for better performance and parallel processing. |