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

dF(λ) = f (λ) dλ, so that B( t − s) = cosλ( t − s) f (λ) dλ.π∫−πSpectral analysis, that is, an experimental determination of thespectral density f(λ), is therefore sometimes explained as thedetermination of the variances of the separate random components ofthe process. For practically applying the correlation or spectral theoryit is necessary, first, to find out the practical conclusions possible fromthe correlation function or spectrum (spectral density); and, second, tobe able to estimate the correlation function (or spectral density) byobservations.That correlation function is normally applied in statistical problems.For example, the variance of the arithmetic mean ξ is expressedthrough the sum of paired covariations, i. e., through a correlationfunction. It can also be expressed through the spectral density.However, an estimate of a spectrum, or of a correlation function, issometimes applied as a magic remedy allegedly making it possible topenetrate the essence of the observed process. It should be clearlyimagined that the correlation theory generally deals with suchcharacteristics that are far from determining the process as a whole andoften only provides a superficial information about it. If we areinterested in some problem of its structure, we must be able toformulate it in terms of the correlation theory while bearing in mindthat usually we do not know precisely either the correlation function orthe spectral density but estimate them by observations. We should thusconsider comparatively rough characteristics determinable by issuingfrom non-precise data.For example, there exists the so-called method of canonicalexpansion whose application demands the knowledge of theeigenfunctions of an integral equation in which a correlation functionof a process is included as a series. This method ought to berecognized as practically hopeless because the inaccuracy of theequation’s kernel very essentially influences the eigenfunctions. I donot know about any practical application of that method. All so-calledapplications issue from arbitrarily given correlation functions and donot deal with statistical material.The estimation of the correlation function and spectrum is rathercomplicated. At first you should estimate and subtract the mean valuem of the process. Its estimate is the arithmetic mean mˆ = ξ . Theestimate ofB(u) = Eξ t ξ t+n – m 2will beˆ 1B u m u nn−u2( ) = ∑ ξ ξ ˆt t+u− , = 0, 1, ..., −1.n − u t=169