Ordering results of extreme order statistics from multiple-outlier scale models with dependence

Abstract: In this paper, we focus on stochastic comparisons of extreme order statistics stemming from multiple-outlier scale models with dependence. Archimedean copula is used to model dependence structure among nonnegative random variables. Sufficient conditions are obtained for comparison of the largest order statistics in the sense of the usual stochastic, reversed hazard rate, star and Lorenz orders. The smallest order statistics are also compared with respect to the usual stochastic, hazard rate, star and Lorenz orders. To illustrate the theoretical establishments, some examples are provided.

• The paper provides sufficient conditions using majorization theory and Archimedean copulas to order the largest and smallest extreme order statistics in dependent scale models.
• It details analytic forms for joint distributions and highlights the impact of parameter heterogeneity on ordering under hazard rate, star, and Lorenz orders.
• It offers examples and counterexamples that demonstrate practical implications for reliability, risk management, and income distribution analysis.

Stochastic Orderings of Extreme Order Statistics in Multiple-Outlier Scale Models with Dependence

Introduction and Problem Statement

This paper rigorously develops comparative results for extreme order statistics (maxima and minima) derived from multiple-outlier scale models when the underlying random variables exhibit statistical dependence. The analysis is motivated by the recognition that, especially in reliability theory and related applied areas, the traditional assumption of independence among component lifetimes is frequently violated in real systems exposed to common environmental stresses or shocks. The statistical dependence is modeled here via multivariate Archimedean copulas, a flexible and tractable family of copulas capturing a wide range of dependence structures.

The main mathematical objective is to determine sufficient conditions under which the largest or smallest order statistics, arising from two different groups of dependent components (each with possible outlier structure in the scale parameters and marginal distributions), can be ordered with respect to various stochastic orderings: the usual stochastic order, hazard rate order, reversed hazard rate order, star order, and Lorenz order. The central objects of study are order statistics $X_{n:n}(n_1, n_2)$ and $Y_{n^*:n^*}(n^*_1, n^*_2)$, constructed from samples with $n_1$ (resp., $n^*_1$) observations from one scale-distributed population and $n_2$ (resp., $n^*_2$) from another, with arbitrary scale parameter vectors, possibly outliers.

Technical Approach

The dependence among the random variables in each group is modeled by an Archimedean copula with generator $\psi$, enabling tractable expressions for the joint and marginal distributions of order statistics. The authors build upon advanced tools from majorization theory (especially weak super- and submajorization) and the theory of Schur-convex/concave functions, which provide powerful means of comparing symmetric functions of vectors of scale parameters.

The work delineates explicit analytic forms for the joint distributions, survival functions, and relevant derivatives of the order statistics under these models. These forms are then leveraged to derive sufficient conditions—typically, monotonicity of hazard or reversed hazard rates, convexity/concavity or log-convexity/log-concavity requirements on the generator $\psi$, and super-additivity of relevant functional compositions—that guarantee the desired stochastic orderings under majorization relations among parameter vectors.

Main Results

Largest Order Statistics

The key theorems provide sufficient conditions for the stochastic ordering (and, in stronger cases, reversed hazard rate and star orderings) of the largest order statistics from two multiple-outlier dependent samples:

• Usual Stochastic and Reversed Hazard Rate Orders: If the scale parameter vectors satisfy weak supermajorization (with respect to $\mathcal{E}_+$ or $\mathcal{D}_+$, i.e., monotonicity constraints), and if the copula generators satisfy either log-convexity (stochastic ordering) or log-concavity and related monotonicity properties (reversed hazard rate ordering), then stochastic comparisons propagate from the parameters to the order statistics. These results demonstrate that increased heterogeneity or weight in certain scale parameters leads to stochastically greater (or smaller, depending on direction of comparison) extreme order statistics.
• Star and Lorenz Orders: Under equality of baseline distributions and additional convexity requirements, the results extend to the star order, which implies the Lorenz order. This is significant for applications in economic inequality and reliability, as Lorenz ordering is closely related to notions of variability and fairness.
• Effects of Dependence: The use of Archimedean copulas introduces nontrivial challenges and extensions over the extensive literature for the independent case. The sufficient conditions now require both majorization in parameter space and compatibility with the dependence structure (often expressed via super-additivity of composite generator functionals).

Smallest Order Statistics

Parallel results are established for the smallest order statistics (series system lifetimes):

• Stochastic and Hazard Rate Orders: Analogous majorization and copula generator conditions provide ordering results for minima; these are particularly relevant where system failure is determined by the weakest component.
• Critical Role of Monotonicity and Convexity: Monotonicity properties of the hazard rates and reversed hazard rates (and their star-shaped analogs in the parameter domain) are shown to be essential. The paper includes explicit counterexamples demonstrating the failure of these orderings if the monotonicity or convexity assumptions are violated.

Generalizations and Counterexamples

The results are carefully illustrated by both concrete examples (validating the theory numerically) and counterexamples, which precisely document the limits of the stated sufficient conditions. These serve both as validation of the mathematical analysis and as practical caution for applied researchers.

Implications and Future Directions

Practical Relevance: The results provide precise guidance for practitioners modeling reliability, risk, or income distributions in systems with dependent, heterogeneous components. For example, the results specify when increased spread or weight in certain sets of scale parameters for system lifetimes (possibly due to maintenance or upgrading strategies) will reliably lead to stochastically improved system performance, even in the presence of complex dependence.

Theoretical Contributions: The extension from the independent to the dependent case with Archimedean copulas constitutes a significant increase in mathematical difficulty and applicability. The use of majorization theory and Schur-convex analysis in this context may stimulate further research on dependence modeling and stochastic ordering in multi-parameter models.

Future Research: One potential avenue suggested is the exploration of similar ordering results for even more general classes of copulas (e.g., hierarchical Archimedean, nested or non-exchangeable copulas), or for systems with both scale and shape parameter heterogeneity. Another is the characterization of necessary and sufficient conditions for ordering, rather than the sufficient conditions provided here. There is also scope for applications to optimal design and resource allocation in reliability structures under dependence.

Conclusion

The paper presents a comprehensive framework for comparing extreme order statistics (maxima and minima) from dependent multiple-outlier scale models under a variety of stochastic orders. By leveraging majorization theory and the properties of Archimedean copulas, the analysis yields explicit sufficient conditions for these comparisons and elucidates the roles of parameter heterogeneity and dependence. The results have both theoretical significance and substantial practical impact in fields like reliability, risk, and income distribution analysis.