Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Confidence intervals for intentionally biased estimators

David M. Kaplan, Xin Liu

TL;DR

The paper develops confidence intervals anchored to intentionally biased estimators to reduce MSE, showing that CI centered at the biased estimator but using the unbiased estimator's standard error can achieve higher coverage than the standard CI at common levels. It introduces a bias-bound CI that shortens length (CI$_5$) and a convex-combination CI (CI$_6$) that can further reduce length under joint normality with known correlation. Through theory, simulations, and an empirical illustration in smoothed quantile regression contexts, the authors demonstrate favorable coverage and substantial length reductions, particularly at conventional levels (e.g., 95%). Practically, CI$_5$ is recommended as a default when an MSE-reducing biased estimator is available, with CI$_6$ offering additional gains when correlation information can be reliably incorporated.

Abstract

We propose and study three confidence intervals (CIs) centered at an estimator that is intentionally biased to reduce mean squared error. The first CI simply uses an unbiased estimator's standard error; compared to centering at the unbiased estimator, this CI has higher coverage probability for confidence levels above 91.7%, even if the biased and unbiased estimators have equal mean squared error. The second CI trades some of this "excess" coverage for shorter length. The third CI is centered at a convex combination of the two estimators to further reduce length. Practically, these CIs apply broadly and are simple to compute.

Confidence intervals for intentionally biased estimators

TL;DR

The paper develops confidence intervals anchored to intentionally biased estimators to reduce MSE, showing that CI centered at the biased estimator but using the unbiased estimator's standard error can achieve higher coverage than the standard CI at common levels. It introduces a bias-bound CI that shortens length (CI) and a convex-combination CI (CI) that can further reduce length under joint normality with known correlation. Through theory, simulations, and an empirical illustration in smoothed quantile regression contexts, the authors demonstrate favorable coverage and substantial length reductions, particularly at conventional levels (e.g., 95%). Practically, CI is recommended as a default when an MSE-reducing biased estimator is available, with CI offering additional gains when correlation information can be reliably incorporated.

Abstract

We propose and study three confidence intervals (CIs) centered at an estimator that is intentionally biased to reduce mean squared error. The first CI simply uses an unbiased estimator's standard error; compared to centering at the unbiased estimator, this CI has higher coverage probability for confidence levels above 91.7%, even if the biased and unbiased estimators have equal mean squared error. The second CI trades some of this "excess" coverage for shorter length. The third CI is centered at a convex combination of the two estimators to further reduce length. Practically, these CIs apply broadly and are simple to compute.

Paper Structure

This paper contains 16 sections, 8 theorems, 33 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation and abbreviations
  3. Setup
  4. Estimators and distributions
  5. Confidence intervals
  6. Applications
  7. Coverage probability comparison: equal MSE
  8. Coverage probability comparison: lower MSE
  9. Shorter CI based on bias bound
  10. CI definition and properties
  11. Example
  12. Shortest CI based on convex combination estimator
  13. Simulation
  14. Empirical illustration
  15. Conclusion
  16. ...and 1 more sections

Key Result

Theorem 1

Consider the setting of eqn:theta-hat-distseqn:CIeqn:equal-MSE. The coverage probability of $\textrm{CI}_1$ equals the confidence level $1-\alpha$. For $1-\alpha\ge0.917$, $\textrm{CI}_2$ has coverage probability strictly above $1-\alpha$ when $b_2\ne0$. For $1-\alpha=0.9$ and any $b_2$, $\textrm{CI

Figures (2)

  • Figure 1: Coverage probability of $\textrm{CI}_2$ at various nominal levels.
  • Figure 2: $\textrm{CI}_5$ (or $\textrm{CI}_6$) length divided by $\textrm{CI}_2$ length as function of $s_2/s_1$, $95\%$ confidence level.

Theorems & Definitions (16)

  • Theorem 1
  • proof
  • Corollary 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 6 more