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**Modelling and residual analysis of nonlinear autoregressive time series in exponential variables.**
*(English)*
Zbl 0579.62075

The model considered is of the form \(X_ n=\beta_ 1X_{n- 1}+\epsilon_ n\) with probability \(\alpha_ 1\), \(X_ n=\beta_ 2X_{n-2}+\epsilon_ n\) with probability \(\alpha_ 2\) and \(X_ n=\epsilon_ n\) with probability \(1-\alpha_ 1-\alpha_ 2\), while \(\epsilon_ n=E_ n\) with probability \(1-p_ 2-p_ 3\), \(=b_ 2E_ n\) with probability \(p_ 2\), and \(=b_ 3E_ n\) with probability \(p_ 3\), where \(E_ n\) is a sequence of independent, exponentially distributed random variables with expectation unity. The constants \(b_ 2,b_ 3,p_ 2,p_ 3\) are functions of \(\alpha_ 1,\alpha_ 2,\beta_ 1,\beta_ 2\), which are the basic parameters of the model.

Conditions on these parameters are derived which ensure that the \(X_ n\) sequence is stationary with exponential (marginal) distribution. It is shown that \(X_ n+a_ 1X_{n-1}+a_ 2X_{n-2}=R_ n\) where the \(R_ n\) are stationary and serially uncorrelated.

As pointed out by H. Tong in the discussion \(E(R_ n| F_{n-1})=0\), where \(F_ n\) is the \(\sigma\)-algebra determined by \(R_ m\), \(m\leq n\) (or equivalently \(X_ m\), \(m\leq n)\). Thus the model is linear in the restricted sense that the best linear predictor is the best predictor. The model is introduced because a marginal exponential distribution is desired. Also time reversibility is avoided, i.e. reversing time gives a very different looking realisation, and some non-linear characteristics are reproduced.

An analysis of a very long data sequence on wind velocity (speed) is given. The estimation method is based on moment estimators of \(a_ 1=\alpha_ 1\beta_ 1\), \(a_ 2=\alpha_ 2\beta_ 2\), and no further analysis to find the \(\alpha_ j,\beta_ j\) is attempted. However an analysis of the observed residuals, \(\hat R_ n\), is effected, for example by comparing observed (serial) correlations of \(\hat R_ n\), \(\hat R^ 2_{n+\ell}\), \(\ell =0,\pm 1,...\), with theoretical values based on likely looking choices for \(\alpha_ 1,\beta_ 1,\alpha_ 2,\beta_ 2.\)

The discussion, by many discussants, criticises the model on various grounds. Included are the difficulty of maximum likelihood estimation of the \(\alpha_ j,\beta_ j\); discontinuities in the conditional density of \(X_ n\), given \(X_{n-1},X_{n-2}\); lack of truly non-linear characteristics; querying of the importance of marginal distributions and lack of plausible physical basis for the model. There is a reply to these criticisms by the authors.

Conditions on these parameters are derived which ensure that the \(X_ n\) sequence is stationary with exponential (marginal) distribution. It is shown that \(X_ n+a_ 1X_{n-1}+a_ 2X_{n-2}=R_ n\) where the \(R_ n\) are stationary and serially uncorrelated.

As pointed out by H. Tong in the discussion \(E(R_ n| F_{n-1})=0\), where \(F_ n\) is the \(\sigma\)-algebra determined by \(R_ m\), \(m\leq n\) (or equivalently \(X_ m\), \(m\leq n)\). Thus the model is linear in the restricted sense that the best linear predictor is the best predictor. The model is introduced because a marginal exponential distribution is desired. Also time reversibility is avoided, i.e. reversing time gives a very different looking realisation, and some non-linear characteristics are reproduced.

An analysis of a very long data sequence on wind velocity (speed) is given. The estimation method is based on moment estimators of \(a_ 1=\alpha_ 1\beta_ 1\), \(a_ 2=\alpha_ 2\beta_ 2\), and no further analysis to find the \(\alpha_ j,\beta_ j\) is attempted. However an analysis of the observed residuals, \(\hat R_ n\), is effected, for example by comparing observed (serial) correlations of \(\hat R_ n\), \(\hat R^ 2_{n+\ell}\), \(\ell =0,\pm 1,...\), with theoretical values based on likely looking choices for \(\alpha_ 1,\beta_ 1,\alpha_ 2,\beta_ 2.\)

The discussion, by many discussants, criticises the model on various grounds. Included are the difficulty of maximum likelihood estimation of the \(\alpha_ j,\beta_ j\); discontinuities in the conditional density of \(X_ n\), given \(X_{n-1},X_{n-2}\); lack of truly non-linear characteristics; querying of the importance of marginal distributions and lack of plausible physical basis for the model. There is a reply to these criticisms by the authors.

Reviewer: E.J.Hannan

### MSC:

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

62M09 | Non-Markovian processes: estimation |

62M15 | Inference from stochastic processes and spectral analysis |