
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), FIN00014 University of Helsinki, Finland
email: [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 tdistribution with ?(n) degrees of freedom. Therefore, the simulation is
terminated as soon as
s^?(n) <= ffi =t?(n)(1 ? ff=2) (1:1)