| MSC: | 62F07 | Ranking and selection |
Abstract:
Following a Bechhofer statistical selection procedure we discuss from an analytical and from a probabilistic point of view
why the real function
\[ F(x) = \int_{-\infty}^{+\infty} \Phi^{a-t}\left(z+r \sqrt{x} \right) \cdot t \left(1-\Phi(z)\right)^{t-1} \varphi(z)\; dz \]
is for $x \geq 0$ and fixed integer parameters $a > 0$, $t \in ]0,a[$ as well as real parameters $r > 0$ and $\beta \in ]0,1[$
strictly monotone increasing and bounded by $1$.
Here $\varphi$ and $\Phi$ denote the p.d.f. and c.d.f. of the standard normal distribution.
Numerical procedures are described to determine the minimal natural $n$ satisfying the inequality $F(n) \geq K$ where $0< K <1$. The dependence
of $n$ on the parameters $a$, $t$ and $r$ is investigated, too. Some simulation results are given and discussed for $t>1$.