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Biometrika 2002 89(1):211-224; doi:10.1093/biomet/89.1.211
© 2002 by Biometrika Trust
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Empty confidence sets for epidemics, branching processes and Brownian motion

Frank G Ball1, Tom Britton2 and Philip D O'neill3

1 School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, U.Kfrank.ball{at}nottingham.ac.uk 2 Department of Mathematics, Uppsala University, P.O.Box 480, SE-751 06 Uppsala, Sweden tomb{at}math.uu.se 3 School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, U.K. philip.o'neill{at}nottingham.ac.uk

This paper treats some examples where likelihood-based inference for certain model parameters may produce empty confidence sets. The first example concerns epidemics, and the parameter of interest is the basic reproduction number R0, which is to be estimated from the final size of an epidemic in a finite population.The second example treats estimation of the mean of the offspring distribution in a branching process, based on observing the total progeny, i.e. the total number of individuals ever born in the branching process. The final example considers estimation of the linear drift in a Brownian motion, based on observing the first hitting time of some horizontal barrier.

Key Words: Branching process; Brownian motion; Confidence set; Epidemic model; Likelihood-based inference

Received April 2001. Revised August 2001


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