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

It possesses a number of unpleasant properties. First, for an ergodicprocess the actual values of B(u) rapidly decrease with an increase ofu. However, the standard deviations of the estimates B ˆ( u ) are roughlythe same for any u and have order 1/ n − u.Thus, for u of the orderof a few dozen the magnitudes B(u) themselves are very small, onlyhundredth and thousandth parts of B(0) whereas the standarddeviations (if n is not too large), tenth parts of B(0) so that the estimateis senseless.Second, these estimates B ˆ( u ) when the values u are close to eachother are not scattered chaotically near the real values because theneighbouring estimates B ˆ( u ) , B ˆ( u + 1) , B ˆ( u + 2) , ... are correlatedwith each other. When looking at a graph of their values the eyeautomatically selects rather regular oscillations, see Fig. 3, atunreasonably large values of u where actually B(u) can not bedistinguished from zero. Therefore, when estimating the correlationfunction we can not trust our eyes and all our actions becomeuncertain.The estimation of the spectral density f(λ) is preferable. Whenestimating it at points λ = λ 1 , λ 2 , ..., λ m not too close to each other, therespective estimates f ˆ(λ i) will be almost independent randomvariables, a fact first discovered by Slutsky. For estimating the spectraldensity we apply the same periodogram only suitably normed. It ishowever very indent because its variance does not tend to vanish as thenumber of observations increases. Therefore the periodogram issmoothed, i. e. a mean value with some weight is taken 8 and we obtainan estimate not of the spectral density itself but of the functionresulting from taking its mean with the same weight. This means thatthe interval of taking the mean should be small. However, thatprocedure when a small interval is chosen will little decrease thevariance of the periodogram. Practical recommendations are here aresult of a compromise between these contradictory demands.I can not go into details of mathematical tricks and I ought to saythat textbooks on the theory of stochastic processes do not usuallydescribe the estimation of the correlation function or spectral densityin any scientific manner. As I noted above, textbooks prefer to issuefrom a stochastic process given along with its correlation function.As a very reliable source of information concerning statisticalproblems, I can cite Hannan (1960). This book is, however, veryconcise and difficult to read. Jenkins & Watts [1971 – 1972] is easierto read, but less reliable. For example, they do not say sufficientlyclearly that none of the provided formulas for the variances of theestimates of the correlation function and spectral density is at allapplicable to each stationary process; some strong conditionsmathematically expressing the property of ergodicity are necessary.Nevertheless, that book is usable although regrettably their practicalexamples should be studied very critically.I wish to warn the reader who will study the sources indicated thatthe initial material on which the methods of the theory of stochasticprocesses had been developed mostly consisted of economic data,70