On Prediction Intervals for Conditionally Heteroscedastic Processes

Kabaila (1999) argues that the standard 1−α prediction intervals for a broad class of conditionally heteroscedastic processes are justified by their possession of what he calls the 'relevance property'. He considers both the case that the parameters of the process are known and that these parameters are unknown. We consider the former case and ask whether these prediction intervals can, alternatively, be deduced from the requirements of both (a) unconditional coverage probability 1−α and (b) minimum unconditional expected length. We show that the answer to this question is no, by presenting a counterexample. This counterexample concerns the standard 95% one-step-ahead prediction interval in the context of a simple Markovian bilinear process.     

THE DETECTION OF A SINGLE ADDITIVE OUTLIER OF UNKNOWN POSITION

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.     

Finite element post-buckling analysis for frames

Presented in this paper is a direct method of analysis of the elastic non-linear behaviour of frames. Emphasis is given to the method's capability of tracing the post-buckling path from a bifurcation point although the method can also trace the non-linear behaviour of frames with eccentricities. The method is proposed as an alternative to the main methods currently in use, the perturbation method and the incremental method. Conditions for equilibrium and stability are developed from a variational approach to the total potential. A finite element approximation is made and an efficient solution technique for the resulting non-linear equations is developed. Results for three frames are given demonstrating good agreement with solutions generated from other approaches.     

On output-error methods for system identification

In this paper are derived consistency and asymptotic normality results for the output-error method of system identification. The output-error estimator has the advantage over the prediction-error estimator of being more easily computable. However, it is shown that the output-error estimator can never be more efficient than the prediction-error estimator. The main result of the paper provides necessary and sufficient conditions for the output-error estimator and the prediction-error estimator to have the same efficiency, irrespective of the spectral density of the noise process.     

The adjustment of prediction intervals to account for errors in parameter estimation

Abstract.  Standard approximate 1 −  α prediction intervals (PIs) need to be adjusted to take account of the error in estimating the parameters. This adjustment may be aimed at setting the (unconditional) probability that the PI includes the value being predicted equal to 1 −  α . Alternatively, this adjustment may be aimed at setting the probability that the PI includes the value being predicted equal to 1 −  α , conditional on an appropriate statistic T . For an autoregressive process of order p , it has been suggested that T consist of the last p observations. We provide a new criterion by which both forms of adjustment can be compared on an equal footing. This new criterion of performance is the closeness of the coverage probability, conditional on all of the data, of the adjusted PI and 1 −  α . In this paper, we measure this closeness by the mean square of the difference between this conditional coverage probability and 1 −  α . We illustrate the application of this new criterion to a Gaussian zero-mean autoregressive process of order 1 and one-step-ahead prediction. For this example, this comparison shows that the adjustment which is aimed at setting the coverage probability equal to 1 −  α conditional on the last observation is the better of the two adjustments.     

On the efficiency of output-error estimators

In this correspondence we consider a system with rational transfer function from input to output where the transfer function haspparameters. It is proved in [1] that when the input is a sum of sinusoids and has a two-sided line spectrum consisting ofpdistinct frequencies, then the output-error estimator and the prediction-error estimator are equally efficient. This result, which is at first sight a surprising one, is given an intuitively pleasing proof in this correspondence. This proof is based on the algorithm of [2].     

ON RISSANEN'S LOWER BOUND ON THE ACCUMULATED MEAN-SQUARE PREDICTION ERROR

Abstract. Rissanen (1984) presents a lower bound on the accumulated mean square prediction error for a Gaussian ARMA process. Rissanen proves this result using coding theory. In this paper we show that this bound is related to the Cramer-Rao bound and Fisher's bound on asymptotic variances, for the case of a Gaussian AR process. This provides some insight into Rissanen's lower bound. Analogues of Rissanen's bound, for nonaccumulated prediction error and a Gaussian AR process, are also presented and discussed.     

On computing output-error estimates

In this note, we consider a system with rational transfer function from input to output where this transfer function haspparameters. It is also supposed that the input has a spectrum consisting ofpdistinct points (equivalently, we suppose that the design index =p/2). As proved in [2], this implies that the output-error estimator has the same efficiency as the prediction-error estimator. We present a very convenient algorithm for computing the output-error estimate of the transfer function from input to output.     

On Assessing Prediction Error in Autoregressive Models

Zhang and Shaman (Assessing prediction error in autoregressive models. Trans. Am. Math. Soc. 347, (1995), 627–37) pose the problem of estimating the conditional mean square one-step-ahead prediction error (CMOPE) for a Gaussian first-order autoregressive process. They put forward a certain estimator (with small asymptotic bias) of CMOPE and propose that its effectiveness be judged by its asymptotic correlation with CMOPE. Unfortunately, the derivation of this correlation by Zhang and Shaman (1995) is incomplete. It is very difficult to complete this derivation. For this reason we use Monte Carlo simulation to gain some insight into the correlation of the estimator with CMOPE. The results of this simulation show that the estimator is extremely poor. We then propose an alternative estimator (with small asymptotic bias) of CMOPE which is shown from Monte Carlo simulation results to have higher large-sample correlation with CMOPE than the estimator of CMOPE put forward by Zhang and Shaman (1995).     

The Relevance Property For Prediction Intervals

Suppose that we have time series data which we want to use to find a prediction interval for some future value of the series. It is widely recognized by time series practitioners that, to be practically useful, a prediction interval should possess the property that it relates to what actually happened during the period that the data were collected as opposed to what might have happened during that period but did not actually happen. We call this the 'relevance property'. Despite its obvious importance, this property has hitherto not been formulated in a mathematically rigorous way. We provide a mathematically rigorous formulation of this property for a broad class of conditionally heteroscedastic processes in the practical context that the parameters of the time series model must be estimated from the data. The importance in applications of this formulation is that it provides us with the most appropriate way of measuring the finite-sample coverage performance of a time series prediction interval.