Gaussian approximation of signed rank statistics process |
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Authors: | Madan L Puri Tiee-Jian Wu |
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Institution: | Indiana University, Bloomington, IN 470405, USA;University of Houston, Houston, TX 77004, USA |
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Abstract: | We consider the signed linear rank statistics of the form where the cNi's are known real numbers, Δ∈0,1] is an unknown real parameter,RΔNi is the rank of |YΔNi| among |YΔNj|, 1≤j≤N, ø is a score generating function, sgn y=1 or -1 according as y≥0 or <0, and YΔNj, 1≤j ≤N, are independent random variables with continuous cumulative distribution functions F(y?ΔdNj), 1≤ j≤N, respectively where the dfNi's are known real numbers. Under suitable assumptions on the c's, d's, φ and F, it is proved that the random process {SΔN?S0N?ESΔN, 0≤Δ≤1}, properly normalized, converges weakly to a Gaussian process, and this result is also true if ESΔN is replaced by ΔbN, where . As an application, we derive the asymptotic distribution of the properly normalized length of a confidence interval for Δ. |
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Keywords: | Primary 62E20 Secondary 62J05 60G10 Gaussian process Tightness |
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