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On the Assessment of Bootstrap Intervals for Samples of Fixed Size

Weizhen Wang, Chongxiu Yu, Zhongzhan Zhang

Abstract

A reasonable confidence interval should have a confidence coefficient no less than the given nominal level and a small expected length to reliably and accurately estimate the parameter of interest, and the bootstrap interval is considered to be an efficient interval estimation technique. In this paper, we offer a first attempt at computing the coverage probability and expected length of a parametric or percentile bootstrap interval by exact probabilistic calculation for any fixed sample size. This method is applied to the basic bootstrap intervals for functions of binomial proportions and a normal mean. None of these intervals, however, are found to have a correct confidence coefficient, which leads to illogical conclusions including that the bootstrap interval is narrower than the z-interval when estimating a normal mean. This raises a general question of how to utilize bootstrap intervals appropriately in practice since the sample size is typically fixed.

On the Assessment of Bootstrap Intervals for Samples of Fixed Size

Abstract

A reasonable confidence interval should have a confidence coefficient no less than the given nominal level and a small expected length to reliably and accurately estimate the parameter of interest, and the bootstrap interval is considered to be an efficient interval estimation technique. In this paper, we offer a first attempt at computing the coverage probability and expected length of a parametric or percentile bootstrap interval by exact probabilistic calculation for any fixed sample size. This method is applied to the basic bootstrap intervals for functions of binomial proportions and a normal mean. None of these intervals, however, are found to have a correct confidence coefficient, which leads to illogical conclusions including that the bootstrap interval is narrower than the z-interval when estimating a normal mean. This raises a general question of how to utilize bootstrap intervals appropriately in practice since the sample size is typically fixed.
Paper Structure (17 sections, 2 theorems, 77 equations, 3 figures, 4 tables)

This paper contains 17 sections, 2 theorems, 77 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. A general approach to compute the coverage probability and expected length of a bootstrap interval
  3. The coverage probability and expected length of a $1-\alpha$ parametric bootstrap interval
  4. The coverage probability and expected length of a $1-\alpha$ percentile bootstrap interval
  5. Two bootstrap intervals for a proportion based on an i.i.d. Bernoulli sample
  6. The parametric and percentile bootstrap intervals based on $\hat{p}$
  7. The bootstrap interval for $p$ based on $\breve{p}$
  8. Interval comparisons
  9. Two bootstrap intervals for two functions of two proportions based on two i.i.d. Bernoulli samples
  10. The parametric and percentile bootstrap intervals of a normal mean $\mu$ when the variance $\sigma^2$ is known
  11. The parametric bootstrap interval for $\mu$ based on $Q$
  12. The parametric and percentile bootstrap intervals for $\mu$ based on the sample mean
  13. The parametric bootstrap intervals of a normal mean $\mu$ based on the sample median
  14. The percentile bootstrap interval for the median of a symmetric and continuous population based on $MED$
  15. The parametric bootstrap interval of a normal mean based on the sample mean when $\sigma^2$ is unknown
  16. ...and 2 more sections

Key Result

Theorem 1

Let $Cover_{C_Q}(\mu)$ and $EL_{C_Q}(\mu)$ be the coverage probability and the expected length for $C_Q$, respectively. Then, and

Figures (3)

  • Figure 1: Coverage probabilities for $C_{wa}$ or $C_{wi}$ (solid) for different $(1-\alpha, n, m)$ and a reference line at $1-\alpha$ (dashed).
  • Figure 2: Expected lengths for the five intervals with the same area under the coverage probability curve (A): $C_{wa}$ (black-solid), Wald (red), $C_{wi}$ (black-dash), Wilson (blue), and Wang (green), for different $n$ and $m$.
  • Figure 3: The 3-D plots of coverage probability for the 90% bootstrap intervals: $C_d$ and $C_{\theta}$, when $(n_1,n_2,m)=(30, 60, 1000)$. The left column contains the coverage probability for $C_d$ (blue) on $H_{D}$ (the black parallelogram) and the plane $Coverage=0.9$ (the red square) at three different angles. The right column is the coverage probability for $C_{\theta}$ (green) as a function of $\theta \in [1,100]$ and $p_2\in [0,1]$ and the plane $Coverage=0.9$ (red) at three different angles. The ICPs for the two intervals are equal to 0. Most of the coverage probability values are less than 0.9, especially for $C_{\theta}$.

Theorems & Definitions (3)

  • Definition 1
  • Theorem 1
  • Theorem 2