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تسجيل مستخدم جديدBi-$s^*$-Concave Distributions
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We introduce new shape-constrained classes of distribution functions on R, the bi-$s^*$-concave classes. In parallel to results of Dumbgen, Kolesnyk, and Wilke (2017) for what they called the class of bi-log-concave distribution functions, we show that every $s$-concave density $f$ has a bi-$s^*$-concave distribution function $F$ for $s^*leq s/(s+1)$. Confidence bands building on existing nonparametric bands, but accounting for the shape constraint of bi-$s^*$-concavity, are also considered. The new bands extend those developed by Dumbgen et al. (2017) for the constraint of bi-log-concavity. We also make connections between bi-$s^*$-concavity and finiteness of the CsorgH{o} - Revesz constant of $F$ which plays an important role in the theory of quantile processes.
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We introduce a new shape-constrained class of distribution functions on R, the bi-$s^*$-concave class. In parallel to results of Dumbgen, Kolesnyk, and Wilke (2017) for what they called the class of bi-log-concave distribution functions, we show that
every s-concave density f has a bi-$s^*$-concave distribution function $F$ and that every bi-$s^*$-concave distribution function satisfies $gamma (F) le 1/(1+s)$ where finiteness of $$ gamma (F) equiv sup_{x} F(x) (1-F(x)) frac{| f (x)|}{f^2 (x)}, $$ the CsorgH{o} - Revesz constant of F, plays an important role in the theory of quantile processes on $R$.
Let ${P_{theta}:theta in {mathbb R}^d}$ be a log-concave location family with $P_{theta}(dx)=e^{-V(x-theta)}dx,$ where $V:{mathbb R}^dmapsto {mathbb R}$ is a known convex function and let $X_1,dots, X_n$ be i.i.d. r.v. sampled from distribution $P_{t
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