## Institute of Mathematical Statistics Lecture Notes - Monograph Series

- Lecture Notes--Monograph Series
- Volume 45, 2004, 228-236

### An asymptotic minimax determination of the initial sample size in a two-stage sequential procedure

#### Abstract

When estimating the mean of a normal distribution with squared error loss and a cost for each observation, the optimal (fixed) sample size depends on the variance $\sigma^2$. A two-stage sequential procedure is to first conduct a pilot study from which $\sigma^2$ can be estimated, and then estimate the desired sample size. Here an asymptotic formula for the initial sample size in a two-stage sequential estimation procedure is derived-asymptotic as the cost of a single observation becomes small compared to the loss from estimation error. The experimenter must specify only the sample size, $n_0$ say, that would be used in a fixed sample size experiment; the initial sample size of the two-stage procedure is then the least integer greater than or equal to $\sqrt{n_0/2}$. The resulting procedure is shown to minimize the maximum Bayes regret, where the maximum is taken over prior distributions that are consistent with the initial guess $n_0$; and the minimax solution is shown to provide an asymptotic lower bound for optimal Bayesian choices for a wide class of prior distributions.

#### Chapter information

**Source***A Festschrift for Herman Rubin* (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2004)

**Dates**

First available in Project Euclid: 28 November 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.lnms/1196285393

**Digital Object Identifier**

doi:10.1214/lnms/1196285393

**Mathematical Reviews number (MathSciNet)**

MR2126900

**Zentralblatt MATH identifier**

1268.62097

**Subjects**

Primary: 62L12: Sequential estimation

**Keywords**

Bayesian solutions integrated risk and regret inverted gamma priors sequential point estimation squared error loss

**Rights**

Copyright © 2004, Institute of Mathematical Statistics

#### Citation

Woodroofe, Michael. An asymptotic minimax determination of the initial sample size in a two-stage sequential procedure. A Festschrift for Herman Rubin, 228--236, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2004. doi:10.1214/lnms/1196285393. https://projecteuclid.org/euclid.lnms/1196285393