Fenton-Wilkinson Order Statistics and German Tanks: A Case Study of an Orienteering Relay Race

TL;DR

This paper demonstrates how order statistics, specifically using a lognormal distribution and Fenton-Wilkinson approximations, can effectively predict team rankings in an orienteering relay race, linking statistical theory to real-world applications.

Contribution

It introduces a novel application of order statistics and Fenton-Wilkinson approximations to ordinal regression in a real-world sports context, bridging theory and practice.

Findings

Accurate predictions of team rankings using lognormal distribution assumptions.

Effective use of Fenton-Wilkinson approximations for changeover times.

Application of German tank problem estimator to total team count.

Abstract

Ordinal regression falls between discrete-valued classification and continuous-valued regression. Ordinal target variables can be associated with ranked random variables. These random variables are known as order statistics and they are closely related to ordinal regression. However, the challenge of using order statistics for ordinal regression prediction is finding a suitable parent distribution. In this work, we provide a case study of a real-world orienteering relay race by viewing it as a random process. For this process, we show that accurate order statistical ordinal regression predictions of final team rankings, or places, can be obtained by assuming a lognormal distribution of individual leg times. Moreover, we apply Fenton-Wilkinson approximations to intermediate changeover times alongside an estimator for the total number of teams as in the notorious German tank problem. The purpose of this work is, in part, to spark interest in studying the applicability of order statistics in ordinal regression problems.