Finite sample valid confidence sets of mode (2503.23711v1)

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

The paper addresses an important aspect of statistical estimation—the construction of confidence sets for the mode of a unimodal distribution, which is relatively underexplored compared to point estimation. The authors present novel methods for constructing finite sample valid confidence sets for the mode, effectively bridging a significant gap in the statistical literature.

Key Contributions

1. Collection of Methods: A range of methods are introduced for constructing finite sample valid confidence sets. These include approaches based on order statistics, M-estimation procedures, and p-value calculations derived from Edelman's result. Each of these methods is designed to provide valid confidence sets with finite sample coverage guarantees, adapting to the underlying distribution's density behavior near the mode.
2. Coverage and Shrinkage: The paper meticulously analyses the width of the proposed confidence sets, demonstrating that under certain regularity conditions, these sets shrink to zero optimally as the sample size increases. Notably, this adaptivity is achieved without requiring a priori knowledge of the density's regularity parameter, which represents a substantial advancement in the field.
3. Multivariate Extension: The methods are extended to handle $\gamma$-unimodal distributions in multivariate contexts, leveraging the properties of Minkowski functionals alongside the unimodality characterization. This extension is crucial for practical applications involving complex data structures.

Implications and Speculation

Practical Implications

The development of finite sample valid confidence sets is vital for ensuring reliable statistical inference, particularly in situations with limited data. These methods offer researchers robust tools for mode estimation in unimodal distributions, which find applications across various domains, including econometrics, signal processing, and biosciences.

Theoretical Implications

The paper lays groundwork for further theoretical exploration, especially in the domain of adaptivity in statistical estimation. It advances the understanding of mode-related functionals in statistics, encouraging the reassessment of existing methodologies in light of finite sample concerns.

Speculation on Future Developments

The paper opens up avenues for future research on refining the established methods to achieve exact minimax optimal rates without conservatism. Additionally, it calls for exploration into confidence sets for modes of multivariate distributions under diverse dependent structures, posing an intriguing challenge for theoretical statisticians.

Conclusion

This paper enriches the statistical literature with innovative techniques for constructing confidence sets for the mode of unimodal distributions. The authors prescribe a comprehensive methodology that transcends conventional asymptotic approaches, ensuring finite sample valid inferences that are crucial for practical applications. These contributions significantly enhance tools for statistical inference, offering promising directions for both research and applied statistics.