Journal of Statistical Planning and Inference 82 (1999) 251–262 www.elsevier.com/locate/jspi Simultaneous condence intervals for multinomial proportions Joseph Glaz a ; ∗ , Cristina P. Sison b a Department of Statistics, University of Connecticut, 196 Auditorium Road, U-120 MSB 428, Storrs, CT 06269-3120, USA b Division of Biostatistics, North Shore University Hospital, Manhasset, NY 11030, USA Received 1 January 1997; accepted 1 February 1998 Abstract In this article approximate parametric bootstrap condence intervals for functions of multinomial proportions are discussed. The interesting feature of these condence intervals is that they are obtained via an Edgeworth expansion approximation for the rectangular multino- mial probabilities rather than the resampling approach. In the rst part of the article simultaneous condence intervals for multinomial proportions are considered. The parametric bootstrap con- dence interval appears to be the most accurate procedure. The use of this parametric bootstrap condence region in the sample size determination problem is also discussed. In the second part of the article approximate parametric bootstrap equal-tailed condence intervals for the minimum and maximum multinomial cell probabilities are derived. Numerical results based on a simulation study are presented to evaluate the performance of these condence intervals. We also indicate several problems for possible future research in this area. c  1999 Elsevier Science B.V. All rights reserved. MSC: primary 62F25; secondary 62H12 Keywords: Approximations; Condence regions; Edgeworth expansion; Multinomial distribution; Order statistics; Parametric bootstrap; Simultaneous inference 1. Introduction Let X =(X 1 ;:::;X k ) be the vector of cell frequencies in a sample of n observations from a multinomial distribution with cell probabilities  0 =(p 1 ;:::;p k ), where p i ¿ 0 and ∑ k i=1 p i = 1. In this article we are interested in approximate parametric bootstrap condence intervals for functions of  0 using the methods of Hall (1992). The inter- esting feature of these bootstrap condence intervals is that they are implemented via an Edgeworth expansion approximation rather than resampling. The bootstrap methods * Corresponding author. Fax: 860-486-4113. E-mail address: glaz@uconnvm.uconn.edu (J. Glaz) 0378-3758/99/$ - see front matter c  1999 Elsevier Science B.V. All rights reserved. PII: S0378-3758(99)00047-6