BAYESIAN SMOOTHING WITH OCCASIONAL JUMPS

Abstract

Consider a sequence of independent random variables X 1, X 2,…,X n observed at n equally spaced time points where X i has a probability distribution which is known apart from the values of a parameter θ i ∈ R which may change from observation to observation. We consider the problem of estimating θ = (θ1, θ2,…,θ n ) given the observed values of X 1, X 2,…,X n . The paper proposes a prior distribution for the parameters θ for which sets of parameter values exhibiting no change, or no change apart from a few sudden large changes, or lots of small changes, all have positive prior probability. Markov chain sampling may be used to calculate Bayes estimates of the parameters. We report the results of a Monte Carlo study based on Poisson distributed data which compares the Bayes estimator with estimators obtained using cubic splines and with estimators derived from the Schwarz criterion. We conclude that the Bayes method is preferable in a minimax sense since it never produces the disastrously large errors of the other methods and pays only a modest price for this degree of safety. All three methods are used to smooth mortality rates for oesophageal cancer in Irish males aged 65–69 over the period 1955 through 1994.