The problem of the Nile: Conditional solution to a changepoint problem

SUMMARY Inference is considered for the point in a sequence of random variables at which the probability distribution changes. An approximation to the conditional distribution of the maximum likelihood estimator of the changepoint given the ancillary values of observations adjacent to the estimated changepoint is derived and shown to be numerically equal to a Bayesian posterior distribution for the changepoint. A hydrological example is given to show that inferences based on the conditional distribution of the maximum likelihood estimator can differ sharply from inferences based on the marginal distribution. the process governing their distribution changes abruptly, and consider the problem of inference about the unknown changepoint. Published research on this and related problems has provided changepoint estimators for a class of increasingly sophisticated models; for a recent example and references, see Ferriera (1975). The present paper turns back the clock to reconsider the simplest possible changepoint problem, one involving independent random variables whose distributions are completely specified apart from the unknown changepoint. A Bayesian solution to this problem is implicit in the work of Chernoff & Zacks (1964); a frequentist solution is given by Hinkley (1970). We consider here a third solution, based on a conditional frequentist approach. In a sense to be made precise, this third solution serves as a bridge linking the previous two. The conditional solution evolves from Hinkley's frequentist approach, which bases inferences on the asymptotic sampling distribution of the maximum likelihood estimator of the changepoint. The need for conditioning arises because the maximum likelihood estimator is not a sufficient statistic, and thus inferences based on its sampling distribution can be made more informative by conditioning on the values of appropriate ancillary statistics. It turns out that for the simple changepoint problem the resulting conditional inferences are nominally equivalent to certain Bayesian inferences in the sense that numerical differences can be made arbitrarily small. Nominal equivalence of the two solutions follows from an approximate version of a result obtained by Fisher (1934) in his conditional approach to estimating a translation parameter 9: if A is ancillary in that its distribution does not depend on 0, and if the density f(x I 0) of the data x can be factorized in the form