Robust tests for log-logistic models based on minimum density power divergence estimators

TL;DR

This work addresses robust hypothesis testing for the two-parameter log-logistic model by developing Wald-type and Rao-type tests based on the minimum density power divergence estimator (MDPDE) with tuning parameter $\tau\ge 0$. The authors derive explicit test statistics for both simple and composite null hypotheses, establish their asymptotic chi-square distributions, and provide closed-form expressions for the needed information-like matrices under the log-logistic model. A comprehensive simulation study demonstrates the robustness of the proposed tests against data contamination, showing that moderate $\tau$ values offer a favorable trade-off between efficiency and protection from outliers, while classical MLE-based tests can be severely affected. The methodology advances robust inference in survival, reliability, and econometrics applications where log-logistic models are common, and offers practical guidelines for implementing robust hypothesis tests in practice.

Abstract

The log-logistic distribution is a versatile parametric family widely used across various applied fields, including survival analysis, reliability engineering, and econometrics. When estimating parameters of the log-logistic distribution, hypothesis testing is necessary to verify assumptions about these parameters. The Wald test and Rao test provide formal methods for testing hypotheses about these parameters. However, these test statistics are not robust, and their rejection decisions may be affected by data contamination. In this paper we develop new families of Wald-type test statistics and Rao-type test statistics based on minimum density power divergence estimators (MDPDEs) for the parameters of the log-logistic distribution. These new families generalize the Wald and Rao test statistics, inheriting the robustness properties from the MDPDEs and thus addressing the lack of robustness of the classical tests. Explicit expressions for the test statistics under the log-logistic model for both simple and composite null hypotheses are derived, and their properties are analyzed in detail. An extensive simulation study empirically demonstrates the robustness of these families and compares their performance with the classical methods.

Figures (11)

• Figure 1: Empirical level of the Wald-type tests under increasing sample size for different values of $\tau$ in the absence of contamination (left) and with a $15\%$ contamination proportion (right) and $\widetilde{\alpha} = 3$.
• Figure 2: Empirical level of the Rao-type tests under increasing sample size for different values of $\tau$ in the absence of contamination (left) and with a $15\%$ contamination proportion (right) and $\widetilde{\alpha} = 3$.
• Figure 3: Empirical level of the Wald-type test statistics with $n=50$ (left) and $n=100$ (rigth) under increasing contamination proportion with $\widetilde{\alpha } = 3$ under different values of $\tau$.
• Figure 4: Empirical level of the Rao-type test statistics with $n=50$ (left) and $n=100$ (rigth) under increasing contamination proportion with $\widetilde{\alpha } = 3$ under different values of $\tau$.
• Figure 5: Empirical level of the Wald-type test statistics with $n=50$ (left) and $n=100$ (rigth) under increasing contamination proportion with $\widetilde{\alpha } = 6$ under different values of $\tau$.

Theorems & Definitions (10)

• Definition 1
• Theorem 2
• Definition 3
• Theorem 4
• Proposition 5
• Proposition 6
• Theorem 7
• Definition 8
• Definition 9
• Definition 10