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ScienceAsia 40 (2014): 182-191 |doi: 10.2306/scienceasia1513-1874.2014.40.182

Bounds of the normal approximation to random-sum Wilcoxon statistics

Mongkhon Tuntapthai, Nattakarn Chaidee*

ABSTRACT:     Consider sequences {Xi}i=1 and {Yj}j=1 of independent and identically distributed (i.i.d.) random variables, random variables K1, K2 ranging over of all positive integers, where the Xi\'s, Yj\'s, K1, and K2 are all independent. We obtain Berry-Esseen bounds for random-sum Wilcoxon\'s statistics in the form (WK1,K2U)/V and (WK1,K2a)/b where WK1,K2=∑i=1K1j=1K2I(Xi>Yj) and U, V are random variables, and a, b are constants. We also show that the rate of convergence is O((EK2)−1/2) provided by EK1/EK2→τ for some constant τ>0 when EK1 and EK2 tend to infinity.

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Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330 Thailand

* Corresponding author, E-mail: nattakarn.c@chula.ac.th.

Received 14 May 2013, Accepted 16 Feb 2014