Distribution of global measures of deviation between the empirical distribution function and its concave majorant (Q927247)

Let \(X_1,\ldots,X_n\) be independent and identically distributed random variables with values in \([0,1]\) and common decreasing smooth density \(f\) whose derivative \(f^\prime\) is bounded away from zero. Let \(F_n\) denote the empirical distribution function pertaining to \(X_1,\ldots,X_n\), and let \(\widehat{F}_n\) be the least concave majorant of \(F_n\), which is defined to be the smallest concave function that lies above \(F_n\). It is shown that \[ n^{1/6}\Big(\int_0^1 n^{2k/3}(\widehat{F}_n(t)-F_n(t))^kg(t)\,dt-\mu\Big) \] is asymptotically normal as \(n\to\infty\) for any continuous function \(g\) and \(k\geq1\), where \(\mu\) is an appropriate centering constant depending on \(f\), \(f^\prime\), \(g\) and \(k\). This implies asymptotic normality of the \(L_k\)-distance \(\| \widehat{F}_n-F_n\| _k\) and of \(\int(\widehat{F}_n-F_n)^k\,dF_n\). The assumption \(f^\prime<0\) turns out to be essential, because it is also shown that for uniformly distributed and independent \(X_1,\ldots,X_n\) the asymptotic distributional behavior of \(\| \widehat{F}_n-F_n\| _k\) and of \(\int(\widehat{F}_n-F_n)^k\,dF_n\) is completely different.