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Finite sample valid confidence sets of mode

Manit Paul, Arun Kumar Kuchibhotla

TL;DR

This work tackles finite-sample interval estimation for the mode $\theta_0$ of a unimodal distribution with density $f$, addressing a gap in interval inference for this functional. It proposes three constructive approaches—order-statistic spacings (M1), M-estimation (M2) with an adaptive variant (M2'), and Edelman’s single-observation method (M3)—and extends the framework to $d$-dimensional $\gamma$-unimodal distributions. Under mild regularity on $f$ near the mode, the width of the constructed confidence sets shrinks to a singleton at near-minimax rates, adaptively to the unknown local smoothness exponent $\beta$, while the Edelman-based method yields valid coverage but does not guarantee shrinking width. Numerical experiments corroborate the finite-sample validity and show that M1 and M2 closely track oracle performance as $n$ grows, whereas M3 remains wide, highlighting the benefits of adaptive, multi-approach uncertainty quantification for mode inference.

Abstract

Estimating the mode of a unimodal distribution is a classical problem in statistics. Although there are several approaches for point-estimation of mode in the literature, very little has been explored about the interval-estimation of mode. Our work proposes a collection of novel methods of obtaining finite sample valid confidence set of the mode of a unimodal distribution. We analyze the behaviour of the width of the proposed confidence sets under some regularity assumptions of the density about the mode and show that the width of these confidence sets shrink to zero near optimally. Simply put, we show that it is possible to build finite sample valid confidence sets for the mode that shrink to a singleton as sample size increases. We support the theoretical results by showing the performance of the proposed methods on some synthetic data-sets. We believe that our confidence sets can be improved both in construction and in terms of rate.

Finite sample valid confidence sets of mode

TL;DR

This work tackles finite-sample interval estimation for the mode of a unimodal distribution with density , addressing a gap in interval inference for this functional. It proposes three constructive approaches—order-statistic spacings (M1), M-estimation (M2) with an adaptive variant (M2'), and Edelman’s single-observation method (M3)—and extends the framework to -dimensional -unimodal distributions. Under mild regularity on near the mode, the width of the constructed confidence sets shrinks to a singleton at near-minimax rates, adaptively to the unknown local smoothness exponent , while the Edelman-based method yields valid coverage but does not guarantee shrinking width. Numerical experiments corroborate the finite-sample validity and show that M1 and M2 closely track oracle performance as grows, whereas M3 remains wide, highlighting the benefits of adaptive, multi-approach uncertainty quantification for mode inference.

Abstract

Estimating the mode of a unimodal distribution is a classical problem in statistics. Although there are several approaches for point-estimation of mode in the literature, very little has been explored about the interval-estimation of mode. Our work proposes a collection of novel methods of obtaining finite sample valid confidence set of the mode of a unimodal distribution. We analyze the behaviour of the width of the proposed confidence sets under some regularity assumptions of the density about the mode and show that the width of these confidence sets shrink to zero near optimally. Simply put, we show that it is possible to build finite sample valid confidence sets for the mode that shrink to a singleton as sample size increases. We support the theoretical results by showing the performance of the proposed methods on some synthetic data-sets. We believe that our confidence sets can be improved both in construction and in terms of rate.

Paper Structure

This paper contains 21 sections, 14 theorems, 139 equations, 1 figure, 6 algorithms.

Table of Contents

  1. Introduction and Motivation
  2. Contributions and Organization
  3. Organization
  4. Methods
  5. A nested approach of obtaining confidence interval of mode using order statistics
  6. Confidence set of mode using M-estimation
  7. An adaptive confidence set of mode using M-estimation
  8. Confidence set of mode using Edelman's result
  9. Extension to $\gamma$-Unimodal (multivariate) distributions
  10. Numerical results
  11. Conclusions and Future Directions
  12. Finite sample valid confidence set of mode of dependent data
  13. Proof of \ref{['thm:main_result_os_validity']}
  14. Proof of \ref{['thm:width_analysis_os']}
  15. Proof of \ref{['thm:m_estimation']}
  16. ...and 6 more sections

Key Result

Theorem 1

Suppose $X_1, X_2, \ldots, X_n$ are independent and identically distributed as $F$ which is a unimodal distribution with mode $\theta_0$ i.e. $F(\cdot)$ is a convex function in the interval $\left(-\infty, \theta_0 \right]$ and a concave function in the interval $\left[ \theta_0, \infty \right)$. Th

Figures (1)

  • Figure 1: Comparison of coverage and width of various confidence sets of mode of $f_{\beta}$ for different sample sizes and smoothness parameter, $\beta$.

Theorems & Definitions (24)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Remark 2
  • Theorem 4
  • Remark 3
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • ...and 14 more