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Let $F_n$ be the empirical distribution function (df) pertaining to independent random variables with continuous df $F$. We investigate the minimizing point ${\widehat{\tau }}_n$ of the empirical process $F_n-F_0$, where $F_0$ is another df which differs from $F$. If $F$ and $F_0$ are locally Hölder-continuous of order $\alpha$ at a point $\tau$ our main result states that $n^{1/\alpha }({\widehat{\tau }}_n-\tau )$ converges in distribution. The limit variable is the almost sure unique minimizing point of a two-sided time-transformed homogeneous Poisson-process with a drift. The time-transformation...

Let be the empirical distribution function (df) pertaining to independent random variables with continuous df . We investigate the minimizing point ${\widehat{\tau }}_n$ of the empirical process , where is another df which differs from . If and are locally Hölder-continuous of order at a point our main result states that $n^{1/\alpha }({\widehat{\tau }}_n-\tau )$ converges in distribution. The limit variable is the almost sure unique minimizing point of a two-sided time-transformed homogeneous Poisson-process...

Let $ϵ-\text{(}Z)$ be the collection of all $ϵ$-optimal solutions for a stochastic process $Z$ with locally bounded trajectories defined on a topological space. For sequences $\left(Z_n\right)$ of such stochastic processes and $\left({ϵ}_n\right)$ of nonnegative random variables we give sufficient conditions for the (closed) random sets ${ϵ}_n-\text{(}{Z}_n)$ to converge in distribution with respect to the Fell-topology and to the coarser Missing-topology.

For lower-semicontinuous and convex stochastic processes $Z_n$ and nonnegative random variables ${ϵ}_n$ we investigate the pertaining random sets $A(Z_n,{ϵ}_n)$ of all ${ϵ}_n$-approximating minimizers of $Z_n$. It is shown that, if the finite dimensional distributions of the $Z_n$ converge to some $Z$ and if the ${ϵ}_n$ converge in probability to some constant $c$, then the $A(Z_n,{ϵ}_n)$ converge in distribution to $A(Z,c)$ in the hyperspace of Vietoris. As a simple corollary we obtain an extension of several argmin-theorems in the literature. In particular, in...

We establish consistent estimators of jump positions and jump altitudes of a multi-level step function that is the best $L^2$-approximation of a probability density function $f$. If $f$ itself is a step-function the number of jumps may be unknown.