An Optimal Selection Problem Associated with the Poisson Process

TL;DR

This paper extends the Poisson process secretary problem to the last-success scenario with a specific success profile, proving optimal strategies and deriving bounds using hypergeometric functions and inequalities.

Contribution

It generalizes the last-success problem under the Karamata-Stirling profile, establishing the optimality of the myopic strategy and deriving bounds via hypergeometric functions.

Findings

The myopic strategy is optimal in the best-choice case.

Established a connection to hypergeometric functions for analysis.

Derived bounds and asymptotics for critical roots.

Abstract

Cowan and Zabczyk (1978) introduced a continuous-time generalisation of the secretary problem, where offers arrive at epochs of a homogeneous Poisson process. We expand their work to encompass the last-success problem under the Karamata-Stirling success profile. In this setting, the $k$th trial is a success with probability $p_k=\theta/(\theta+k-1)$, where $\theta > 0$. In the best-choice setting ($\theta=1$), the myopic strategy is optimal, and the proof hinges on verifying the monotonicity of certain critical roots. We extend this crucial result to the last-success case by exploiting a connection to the sign of the derivative in the first parameter of a quotient of Kummer's hypergeometric functions. Additionally, we establish an Edmundson-Madansky inequality applicable to Poisson random variables. This result enables us to adopt a probabilistic approach to derive bounds and asymptotics of the critical roots. This strengthens and improves the findings of Ciesielski and Zabczyk (1979).