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A broad class of statistics Mn can be viewed as solutions to an equation of the form
For ψ bounded,nondecreasing and left continuous,asymptotic representations for these generalized M-statistics Mn are studied in Aerts, Janssen en Veraver-beke (1987). In the present paper similar results are studied for the case the ψ is not necessarily nondecreasing but has to be, in some sense, piecewise-smooth. The results extend earlier work of Carroll(1978).
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