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Bounds of the normal approximation to random-sum Wilcoxon statistics

Mongkhon Tuntapthai, Nattakarn Chaidee^*

 
ABSTRACT:     Consider sequences {X_i}_i=1^∞ and {Y_j}_j=1^∞ of independent and identically distributed (i.i.d.) random variables, random variables K₁, K₂ ranging over of all positive integers, where the X_i\'s, Y_j\'s, K₁, and K₂ are all independent. We obtain Berry-Esseen bounds for random-sum Wilcoxon\'s statistics in the form (W_K1,K2−U)/V and (W_K1,K2−a)/b where W_K1,K2=∑_i=1^K₁∑_j=1^K₂I(X_i>Y_j) 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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* Corresponding author, E-mail: nattakarn.c@chula.ac.th.

Received 14 May 2013, Accepted 16 Feb 2014