Statistics and Probability Letters 78 (2008) 2685–2691 Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro Approximate predictive pivots for autoregressive processes José M. Corcuera Universitat de Barcelona, Facultat de Matemàtiques, Gran Via de les Corts, Catalanes 585, 08007 Barcelona, Spain article info Article history: Received 7 December 2006 Received in revised form 2 February 2008 Accepted 14 March 2008 Available online 23 March 2008 MSC: 62M10 62M20 62G15 62E20 abstract In this paper the author considers an autoregressive process where the parameters of the process are unknown and try to obtain pivots for predicting future observations. If we do a probabilistic prediction with the estimated model, where the parameters are estimated by a sample of size n, we introduce an error of order n −1 in the coverage probabilities of the prediction intervals. However we can reduce the order of the error if we calibrate adequately the estimated prediction bounds. The solution obtained can be expressed in terms of an approximate predictive pivot. © 2008 Elsevier B.V. All rights reserved. 1. Introduction The general setting is of prediction of an absolutely continuous (a.c.) random variable Z based on the observation y = (y 1 , y 2 ,..., y n ) corresponding to a random vector Y = (Y 1 , Y 2 ,..., Y n ), where the laws of Y and Z depend on a common and unknown parameter θ ∈ Θ ⊂ R d . A prediction statement about Z is often given by prediction limits, i.e. real functions K α (·) such that P θ {Z ≤ K α (Y )}= α, for every θ ∈ Θ and for any fixed α ∈ (0, 1). The above probability is usually called coverage probability and it is calculated with respect to the joint density of Z and Y . Sometimes the existence of exact (predictive) pivotal quantities, that is of functions of Z and Y whose distribution does not depend on θ, permit us to find an exact solution. But this is the exception. Here we look for approximate prediction limits and predictive pivots. An approximate solution is to take K α (Y ) = q α ( ˆ θ), where q α (θ) is the α-quantile of the conditional distribution of Z given Y = y, that we also assume absolutely continuous, and ˆ θ is an efficient estimator of θ. Note that, if we denote the conditional density of Z given Y = y, g(z; θ|y), then q α ( ˆ θ) will be the α-quantile of the so-called estimative density g(z; ˆ θ|y). However these predictions limits are usually imprecise, having coverage error of order O(n −1 ), that is P θ {Z ≤ q α ( ˆ θ)}= α + O(n −1 ). This is a well known result; indeed Barndorff-Nielsen and Cox (1996) suggest a way to correct these quantiles obtaining prediction limits with a coverage error of order o(n −1 ). The solution can be expressed in terms of a predictive density whose quantiles are precisely these predictions bounds. We will apply this method to the case where Y = (Y 1 , Y 2 ,..., Y n ) is such that Y k+1 − μ = p  j=1 φ j (Y k−j+1 − μ) + ε k+1 , k ∈ Z, E-mail address: jmcorcuera@ub.edu. 0167-7152/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2008.03.010