SIMULTANEOUS SEQUENTIAL CONFIDENCE INTERVALS

OF FIXED WIDTHS FOR SEVERAL MEANS

USING BONFERRONI INEQUALITY

KIMMO E. E. RAATIKAINEN
University of Helsinki, Department of Computer Science
P. O. Box 26 (Teollisuuskatu 23), FIN-00014 University of Helsinki, Finland e-mail: [email protected]

Sequentially constructed confidence intervals for a single mean are widely used to control the length of a simulation run. When the analyst is simultaneously interested in several response variables, the control of a simulation run needs sequential confidence regions. In this paper we examine asymptotic properties of sequentially constructed confidence regions based on the Bonferroni inequality. We also report empirical results when simultaneous confidence intervals for normal and transformed normal means are constructed. (Simultaneous Estimation, Sequential Confidence Intervals)

1. Introduction

Sequentially constructed confidence intervals are widely used to control the precision of a single estimated mean in simulation studies. The methods proposed (see e.g. Pawlikowski, 1990) to control the precision assume that the output sequence fXigni=1 is covariance stationary. In addition, they assume that the output sequence satisfies certain regularity conditions which are fairly general (see e.g. Anderson, 1971, p. 478) so that the estimator ^?(n) = (X1 + : : : + Xn)=n has a normal limiting distribution: pn
?^?(n) ? ?
? D
?!
n!1 N(0; ) ;

where ? = E(Xi) and = limn!1 nVar[^?(n)] ; < < 1. When s2^?(n) denotes the estimated
variance of ^?(n) with ?(n) degrees of freedom, the methods approximate the distribution of (^?(n) ? ?)ffis^?(n) by the Student t-distribution with ?(n) degrees of freedom. Therefore, the simulation is terminated as soon as
s^?(n) <= ffi =t?(n)(1 ? ff=2) (1:1)