THE DETECTION OF A SINGLE ADDITIVE OUTLIER OF UNKNOWN POSITION

Abstract

Abstract. Kudo (On the testing of outlying observations. Sankhya 17 (1956), 67–73) has derived an optimal invariant detector of a single additive outlier of unknown position in the context of an underlying Gaussian process consisting of independent and identically distributed random variables. We show how this author's arguments can be extended to derive an invariant detector of an additive outlier of unknown position for an underlying zero‐mean Gaussian stochastic process. This invariant detector depends on the parameters of this process; its properties are analysed further for the particular case of an underlying zero‐mean Gaussian AR(p) process. It provides an upper bound on the performance of any invariant detector based solely on the data and it may be ‘bootstrapped’ to provide an invariant detector based solely on the data. A plausibility argument is presented in favour of the proposition that the bootstrapped detector is nearly optimal for sufficiently large data length n. The truth of this proposition has been confirmed by simulation results for zero‐mean Gaussian AR(1) and AR(2) processes (for certain sets of possible outlier positions). The bootstrapped detector is shown to be closely related to the detector based on the approximate likelihood ratio criteria of Fox (Outliers in time series. J. Roy. Statist. Soc. Ser. B 34 (1972), 350–63) and the leave‐one‐out diagnostic of Bruce and Martin (Leave‐k‐out diagnostics in time series. J. Roy. Statist. Soc. Ser B 51 (1989), 363–424). It is also shown how the case of an underlying Gaussian process with arbitrary mean can be reduced to the case of an underlying zero‐mean Gaussian process.