Asymptotic random extremal ratio and product based on generalized order statistics and its dual

ABSTRACT

The extremal ratio has been used in several fields, most notably in industrial quality control, life testing, small-area variation analysis, and the classical heterogeneity of variance situation. In many biological, agricultural, military activity problems and in some quality control problems, it is almost impossible to have a fixed sample size, because some observations are always lost for various reasons. Therefore, the sample size itself is considered frequently to be an random variable (rv). Generalized order statistics (GOS) have been introduced as a unifying theme for several models of ascendingly ordered rvs. The concept of dual generalized order statistics (DGOS) is introduced to enable a common approach to descendingly ordered rvs. In this article, the limit dfs are obtained for the extremal ratio and product with random indices under non random normalization based on GOS and DGOS. Moreover, this article considers the conditions under which the cases of random and non random indices give the same asymptotic results. Some illustrative examples are obtained, which lend further support to our theoretical results.

Keywords:

• Dual generalized order statistics
• Extremal product
• Extremal ratio
• Generalized order statistics
• Random indices
• Weak convergence

MATHEMATICS SUBJECT CLASSIFICATION:

• 62G30
• 62E20

Acknowledgments

The authors greatly appreciate helpful suggestions of the two referees that significantly improved the paper.

Funding

Qin's work was partially supported by the National Natural Science Foundation of China (Nos.11271147, 11471135, 11471136). Elsawah's work was partially supported by the UIC Grant R201409 and the Zhuhai Premier Discipline Grant. Hu's work was partially supported by the Financially supported by self-determined research funds of CCNU from the colleges' basic research and operation of MOE (CCNU14A05041).