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Confidence intervals for median absolute deviations

Chandima N. P. G. Arachchige, Luke A. Prendergast

Abstract

The median absolute deviation (MAD) is a robust measure of scale that is simple to implement and easy to interpret. Motivated by this, we introduce interval estimators of the MAD to make reliable inferences for dispersion for a single population and ratios and differences of MADs for comparing two populations. Our simulation results show that the coverage probabilities of the intervals are very close to the nominal coverage for a variety of distributions. We have used partial influence functions to investigate the robustness properties of the difference and ratios of independent MADs.

Confidence intervals for median absolute deviations

Abstract

The median absolute deviation (MAD) is a robust measure of scale that is simple to implement and easy to interpret. Motivated by this, we introduce interval estimators of the MAD to make reliable inferences for dispersion for a single population and ratios and differences of MADs for comparing two populations. Our simulation results show that the coverage probabilities of the intervals are very close to the nominal coverage for a variety of distributions. We have used partial influence functions to investigate the robustness properties of the difference and ratios of independent MADs.

Paper Structure

This paper contains 14 sections, 1 theorem, 22 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Notations and influence functions
  3. Influence function and partial influence functions
  4. Partial influence functions of the difference and squared ratio of MADs
  5. Partial influence functions comparison
  6. Asymptotic confidence intervals
  7. Simulations and Examples
  8. Simulations
  9. Prostate data example
  10. Summary and discussion
  11. Appendix
  12. Proof of Theorem \ref{['th:PIF_Rm']}
  13. Bonett and Seier Method
  14. R code for interval estimators

Key Result

Theorem 2.1

For $\text{PIF} (x;T,F_1,F_2)$ as defined in eq:PIF, the PIFs of $\mathcal{R}_M$ are

Figures (2)

  • Figure 1: PIF$_1$ comparisons for (A) two exponential populations both with rates 0.5, 1 and 1.5 and (B) two log-normal populations both with $\mu$=0 and $\sigma$=0.5,1,1.5.
  • Figure 2: Box plots of three interesting genes selected from the prostate data set.

Theorems & Definitions (2)

  • Theorem 2.1
  • proof