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2 Test statistics
Most of considered tests are obtained by modifying the symmetry tests around known location parameter. Let X1, .., Xn be an i.i.d. sample with distribution function F . The tests are applied to the sample shifted by the value of the location estimator. Typical choices of location estimators are the mean and the median. Here we take a more general approach, using α-trimmed means
−1 1 F (1−α) µ(α)= xdF (x), 0 <α< 1/2, 1 2α −1 − ZF (α) including their boundary cases µ(0) – the mean and µ(1/2) – the median. The estimator is
F −1 −α 1 n (1 ) µ(α)= xdFn(x), 0 <α< 1/2. (2) 1 2α −1 − ZFn (α) In the caseb of α = 0 and α =1/2, the estimators are the sample mean and the sample median, respectively. The modified statistics we consider in this paper are:
Modified sign test • n 1 1 S= I Xj µ(α) > 0 ; n { − }− 2 i=1 X b 2 Modified Wilcoxon test • 1 1 W= I Xi + Xj 2µ(α) > 0 ; n { − }− 2 2 ≤i