## 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.
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