1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

This is the celebrated model of autoregression (of the second order)which was applied by many statisticians deserving complete trust.Yule’s considerations leading to model (3.8) were, however, not quiteclear. In particular, for the case of the pendulum, {δ n } is not asequence of independent random variables but is rather describable bySlutsky’s moving average. However, introducing additionalparameters of that average into the model will mean having too manyparameters and extremely complicated work in its application.Yule’s mistake certainly does not logically prove that the modelis not applicable to sunspots or some economic indicator, but of courseit is a bad omen.Descartes noted that the world can be explained in many differentmanners and the problem only is, to choose that which is really valid.Most chances to be valid certainly has that manner which is the mostnatural and harmonious and does not contain contradictions. If,however, it occurs that the creator of a theory committed a mistake atthe very outset, even if only concerning a particular case, our chancesof success in other cases will sharply diminish.As to sunspots, Yule himself did not achieve a decisive positiveresult. He was compelled to change his model (3.8) by assuming thatwe observe not the variables {ξ n } themselves, but that our observationswere corrupted by an additional random error. He had to make thischange because his model did not pass a statistical check to which hesubjected it, as was supposed to be done. The change of the modelallows to make ends meet but in statistics introducing an additionalparameter is very bad.In general, Yule’s contribution (1927) is an example of a statisticalmasterpiece which, however, provided a dubious (if not negative)result often happening exactly with masterpieces.The interest emerged in forecasting stochastic processes led anotherrepresentative of the English school, Moran (1954), to study thepossibilities of applying model (3.8) for predicting solar activity. Sinceδ n does not depend on the previous behaviour of the process, that is, onvariables ξ n−1 , ξ n−2 , ..., the best possible method of forecasting theestimate of ξ n from all the previous information is to assume thatˆξ = − ( a ξ + b ξ ).n n−1 n−2Moran did that and had showed his result to his friends among radiophysicists who told him that a forecast of such a quality could havebeen possible without any science, just by naked eye. And so it was, asproved by an experiment. That was the second failure of the model ofautoregression.That model possesses, however, an excellent property: it is easilyapplied. Its parameters are easy to estimate , the correlation function isof the kind of fading sinusoidal oscillations and is comparatively easyto be interpreted. The spectral density is also expressed in a simpleway. It made sense therefore to test it many times on differing materialand hope that cases in which it works well enough will be found. It isbest to read about the application of the autoregression model inKendall & Stuart (1968).73