Dieter Schott

Some Remarks on a Statistical Selection Procedure of Bechhofer for Expectations

The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 71, 57 - 67 (2018)

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 $t>1$.

Keywords:   Monotone Functions, Inequalities, Selection Procedures for Expectations, Bechhofer Selection Problem, Indifference Zone Selection
Notes:   I thank the referee of the journal supplying valuable suggestions for improvement.
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