Pilot-Size Optimality in Two-Stage Allocation: A General Rate-Balance Principle and an Exponential Reliability Application

Source abstract

Pilot samples arise naturally in two-stage statistical procedures, where preliminary observations are used to estimate an unknown optimal allocation before the remaining observations are assigned. The central question is the choice of pilot size. A small pilot gives poor estimation of the optimal allocation, while a large pilot commits too many observations before that allocation is known. Under allocation MSE and regular parametric estimation, these two effects lead to a sharp balance. The pilot size grows as the two-thirds power of the total sample size. The square-root pilot is first-order adequate, whereas the two-thirds rule gives the higher-order balance between estimation and early commitment. Four pilot-size regimes are considered to show the effect of choosing the pilot below, at, or above this order. The result is general and does not depend on a particular distribution. Exponential lifetime populations are used as an application for which the optimal allocation can be obtained explicitly. Reliability and medical examples illustrate implementation of the rule.