Asymptotic Breakdown Point Analysis for a General Class of Minimum Divergence Estimators

TL;DR

The paper studies the asymptotic breakdown point of minimum divergence estimators in the S-divergence family, including the minimum density power divergence estimator, under general model setups. It derives a general lower bound that is independent of data dimension, and provides three practical sufficient conditions to guarantee the bound, along with a suite of examples and empirical studies. The results show that MSDEs can achieve substantial robustness (often up to 1/2) across location, scale, and other parametric settings, including high-dimensional contexts, with the breakdown point governed by tunable parameters $\alpha$ and $\lambda$ rather than dimension. The work unifies and extends prior results for DP, SHD, and MHDE, and substantiates the use of MSDEs as robust alternatives for parametric inference in diverse families, while acknowledging limitations and directions for future extension.

Abstract

Robust inference based on the minimization of statistical divergences has proved to be a useful alternative to classical techniques based on maximum likelihood and related methods. Basu et al. (1998) introduced the density power divergence (DPD) family as a measure of discrepancy between two probability density functions and used this family for robust estimation of the parameter for independent and identically distributed data. Ghosh et al. (2017) proposed a more general class of divergence measures, namely the S-divergence family and discussed its usefulness in robust parametric estimation through several asymptotic properties and some numerical illustrations. In this paper, we develop the results concerning the asymptotic breakdown point for the minimum S-divergence estimators (in particular the minimum DPD estimator) under general model setups. The primary result of this paper provides lower bounds to the asymptotic breakdown point of these estimators which are independent of the dimension of the data, in turn corroborating their usefulness in robust inference under high dimensional data.

Figures (7)

• Figure 1: Lower bound $\widetilde{\epsilon}_{(\alpha, \lambda)}$ as in Corollary \ref{['thm:sdiv-breakdown-2']} of the asymptotic breakdown point of the MSDE for different choices of $\alpha$ and $\lambda$. The black regions indicate either $A < 0$ or $B < 0$. The dotted green line indicates the implicit curve $(B/(1+\alpha))^{1/A} = 1/2$.
• Figure 2: Behaviour of MSDE (including MDPDE) under normal location model as a function of the contamination for different choices of $\alpha$ and $\lambda$ (denoted in the title of individual plots)
• Figure 3: Behaviour of MDPD estimates under normal location-scale model as a function of the contamination proportion $\epsilon$, with the location parameter in the left panel and the scale parameter in the right panel.
• Figure 4: Behaviour of MSDE under normal location-scale model as a function of the contamination proportion $\epsilon$, with the location parameter in the left panel and the scale parameter in the right panel, for different choices of $\lambda$ (denoted in the title of the individual plots)
• Figure 5: Behaviour of MDPD estimates under exponential model as a function of the contamination proportion $\epsilon$, for $\theta_0 = 10$ (Left) and $\theta_0 = 0.01$ (Right).

Theorems & Definitions (16)

• Theorem 3.1
• Lemma 3.2
• Corollary 3.1
• Corollary 3.2
• Corollary 3.3
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Corollary 3.4