is the sample size. Suppose that the a priori probability of this upper limit being no larger than κ$_{0}$ is to be at least β$_{0}$ , where β$_{0}$ would be specified. In that case, valid n ’s satisfy There are several ways of constructing an approximate (1 − α ) confidence interval for p$_{0}$ , reviewed in Brown et al. (2001). 1 We suggest what they term the (unmodified) Jeffreys confidence interval, which is simply the equi- tailed (1 − α ) credible interval for p$_{0}$ with the Jeffreys prior, with a minor modification. Using this confidence interval, Figure 1 shows the probability for various choices of n with κ$_{0}$ = 0 . 005 and p$_{0}$ = κ$_{0}$/ 2. In this case, n = 10 5 seems to be a good choice, and this number is widely used in practice. Panjer recursion. The second approach is Panjer recursion; see Ross (1996, Cor. 2.5.4) or Shevchenko (2010, sec. 5). This provides a recursive calculation for Pr( S$_{t}$ = s ) whenever each X$_{i}$ is integer-valued, so that S itself is integer-valued. This calculation would often grind to a halt if ap- plied literally, but can be used to provide an approximation if the ELT is compressed, as discussed in section 6.1. Perhaps the main difficulty with Panjer recursion, once it has been ef- ficiently encoded, is that it does not provide any assessment of the error which follows from the compression of the ELT. In this situation, a precise $^{1}$It is not possible to construct an exact confidence interval without using an auxiliary randomisation. where X ∼ Binom( n, p$_{0}$ ). Pr $^{{}$u$_{1}$$_{−}$$_{α}$ ( X ; n ) ≤ κ$_{0}$ $^{}}$≥ β$_{0}$ 11