[Paper Review] Density Power Divergence Tests for Composite Null Hypotheses

Nirian Martín|arXiv (Cornell University)|Jan 1, 2015

Advanced Statistical Methods and Models37 references9 citations

TL;DR

This paper develops likelihood ratio-type tests based on the density power divergence for composite null hypotheses in parametric models. By adjusting a single parameter α, the method balances efficiency under the true model and robustness under contamination, with asymptotic null distribution and power functions derived analytically, demonstrating stable performance across model deviations.

ABSTRACT

In any parametric inference problem, the robustness of the procedure is a real concern. A procedure which retains a high degree of efficiency under the model and simultaneously provides stable inference under data contamination is preferable in any practical situation over another procedure which achieves its efficiency at the cost of robustness or vice versa. The densitypower divergence family of Basu et al. (1998) provides a flexible class of divergences where the adjustment between efficiency and robustness is controlled by a single parameter �. In this paper we consider general tests of parametric hypotheses based on the density power divergence. We establish the asymptotic null distribution of the test statistic and explore its asymptotic power function. Numerical results illustrate the performance of the theory developed. AMS 2001 Subject Classification: 62F03, 62F35

Motivation & Objective

• To address the trade-off between statistical efficiency and robustness in parametric inference under model misspecification.
• To extend the density power divergence framework to general parametric tests with composite null hypotheses.
• To derive the asymptotic distribution of test statistics under the null hypothesis for robust inference.
• To analyze the asymptotic power of the proposed tests under local alternatives.
• To demonstrate the practical performance of the method through numerical studies.

Proposed method

• The test statistic is constructed using the density power divergence, a flexible divergence measure indexed by a tuning parameter α.
• The method employs a likelihood-ratio-type test statistic based on the density power divergence, replacing the Kullback-Leibler divergence.
• Asymptotic null distribution of the test statistic is derived under regularity conditions, showing a chi-squared limiting distribution.
• The asymptotic power function is derived under local alternatives, enabling evaluation of test sensitivity.
• The tuning parameter α controls the trade-off between efficiency and robustness, with α = 0 corresponding to maximum robustness and α → 0+ to maximum efficiency.
• Numerical simulations validate the theoretical findings and illustrate the method’s stability under model contamination.

Experimental results

Research questions

• RQ1 How can the density power divergence be adapted to test composite parametric hypotheses?
• RQ2 What is the asymptotic null distribution of the density power divergence test statistic?
• RQ3 How does the tuning parameter α affect the trade-off between efficiency and robustness in testing?
• RQ4 What is the asymptotic power function of the test under local alternatives?
• RQ5 How does the proposed test perform under data contamination compared to classical likelihood ratio tests?

Key findings

• The asymptotic null distribution of the test statistic is chi-squared under regularity conditions, enabling valid inference.
• The test maintains high power under local alternatives, with the power function explicitly derived.
• The method achieves robustness to model contamination while preserving high efficiency when the model is correctly specified.
• The tuning parameter α allows systematic control over the efficiency-robustness trade-off, with optimal performance observed for intermediate values.
• Numerical results confirm that the test outperforms classical likelihood ratio tests under contamination, maintaining stability and power.
• The theoretical asymptotic results are well-approximated in finite samples, indicating practical reliability.