Distributions of stopping times in some sequential estimation procedures

Alicja Jokiel-Rokita,Ryszard Magiera

doi:10.1007/s00184-013-0456-6

Alicja Jokiel-Rokita, Ryszard Magiera

Open Access

https://doi.org/10.1007/s00184-013-0456-6

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Abstract

A class of sequential estimation procedures is considered in the case when relevant data may become available only at random times. The exact distributions of the optimal stopping time and the number of observations at the moment of stopping are derived in some sequential procedures. The results obtained in an explicit form are applied to derive the expected time of observing the process, the average number of observations and the expected loss of sequential estimation procedures based on delayed observations. The use of the results is illustrated in a special model of normally distributed observations and the Weibull distributed lifetimes. The probabilistic characteristics are also derived for an adaptive sequential procedures and the behavior of the adaptive procedure is compared with the corresponding optimal sequential procedure.

Highlights

In many practical problems, for example in reliability or in clinical research, data are available only at random times
In the papers mentioned above the problems of estimation of an unknown parameter were considered in different statistical models, the optimal sequential procedures obtained are determined by the stopping times which can be given in the following form τ0 = inf t
He obtained the exact distribution of the stopping time in a problem of estimating the mean of a normal population with an unknown variance, but for a “two-item at a time” sampling plan with a recurrence relation for its probabilities

Summary

Introduction

For example in reliability or in clinical research, data are available only at random times. In the papers mentioned above the problems of estimation of an unknown parameter were considered in different statistical models, the optimal sequential procedures obtained are determined by the stopping times which can be given in the following form τ0 = inf t. In the statistical literature the problems of deriving distributions of stopping times were considered mainly in the context of estimation of the unknown parameter under so called purely sequential sampling scheme. The determination of the stopping time for sequential estimation in a frequentist context was first considered by Robbins (1959) He obtained the exact distribution of the stopping time in a problem of estimating the mean of a normal population with an unknown variance, but for a “two-item at a time” sampling plan with a recurrence relation for its probabilities. We compare the risks of the optimal strategies with the risks of an adaptive strategy in estimation problems in a special model

The model

The distribution and the expectation of the optimal stopping time

The value of the risk of the optimal sequential procedures