The problem of low counts in a signal plus noise model

Abstract

Consider the model X = B + S, where B and S are independent Poisson random variables with means µ and ν, ν is unknown, but µ is known. The model arises in particle physics and some recent articles have suggested conditioning on the observed bound on B; that is, if X = n is observed, then the suggestion is to base inference on the conditional distribution of X given B ≤ n. This conditioning is non-standard in that it does not correspond to a partition of the sample space. It is examined here from the view point of decision theory and shown to lead to admissible formal Bayes procedures. 1. Introduction. In some problems a signal S may be superimposed on a background B, leaving an observed count X = B + S. Here we suppose that B and S are independent Poisson random variables with means µ and ν, so that X has a Poisson distribution with mean θ = µ+ν; ν is regarded as an unknown parameter, but µ is assumed to be known, as might be appropriate if there were historical data on the background. What special techniques, if any, are appropriate if the observed count X is smaller than the expected background? A problem of this nature has arisen recently in physics. The KARMEN 2 Group has been searching for a neutrino oscillation, reported earlier from an experiment at the Los Alamos Neutrino Detector. As of Summer 1998, they had expected to see about 3 ± � 1 background events and at least one signal event, based on the earlier findings (and using rounded values), but had observed nothing. See Zeitnitz et al. (1998). What inference about ν is appropriate here? Can the hypothesis H0 � ν ≥ 1 be rejected? A naive analysis suggests that it can. If µ = 3 is regarded as a known quantity and the hypothesis is rewritten as H0 � θ ≥ 4, then the p value is Pθ� X ≤ 0 �≤ e −4 for θ ≥ 4, and this is less than the usual levels of significance. But this analysis is suspect, because if X = 0, then both B and S must be zero; and P4� S ≤ 0 �= e −1 , which is not less than the usual levels of significance. The second value is obtained by conditioning on the ancillary variable B = 0 and seems right in this context. The problem becomes much more interesting when the observed count is non-zero but smaller than the expected background, since then it is no longer possible to recover the value of B. Roe and Woodroofe (1999) argue that if X = n, then it is appropriate to base inferences on the conditional distribu