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

smoothing is, however, in general less than it is when calculatedaccording to the rules of statistics.It occurs because, when deciding by naked eye, we have to do witha given graph, with a result determined by random experimenting; onthe other hand, when working by statistical methods, we apply astatistical model and therefore also cover the possible scatter of theresults of random experiments themselves from one of theirrealizations to another one. However, the general problem of the realpossibilities of the naked eye methods demands wide experimentalinvestigation.3. The Theory of Stochastic ProcessesHowever beneficial (in suitable cases) is the method of leastsquares, a glance at the observational series often convinces us that themodel of trend with error can not describe the observations, since weare unable to isolate by naked eye a determinate curve withobservational points chaotically scattered around it. This is whatSlutsky (1927/1937, p. 105), a co-creator of the theory of stochasticprocesses, wrote about it:Almost all of the phenomena of economic life, like many otherprocesses, social, meteorological, and others, occur in sequences ofrising and falling movements, like waves. Just as waves following eachother on the sea do not repeat each other perfectly, so economic cyclesnever repeat earlier ones exactly either in duration or in amplitude.Nevertheless, in both cases, it is almost always possible to detect, evenin the multitude of individual peculiarities of the phenomena, marks ofcertain approximate uniformities and regularities. The eye of theobserver instinctively discovers on waves of a certain order othersmaller waves, so that the idea of harmonic analysis [...] presents itselfto the mind almost spontaneously.The idea of harmonic analysis can nevertheless attempted to beachieved by the model of trend with error. It is done by the so-calledmethod of periodogram that preceded the methods of the theory ofstochastic processes and we will briefly consider it.3.1. The periodogram method. Suppose that our observationsmade at discrete moments of time, each second, say, can be describedby the modelx t = sin(λ 0 t + φ) + δ t , t = 0, 1, ..., n (3.1)where λ 0 is some parameter (circular frequency of oscillation), φ, thephase of oscillation and δ i , random error. Suppose that λ 0 is much lessthan 2π, so that successive observations of only one component sin(λ 0 t+ φ) would provide a clearly seen sine curve each unit of time (whichis much shorter than the period of oscillation, 2π/λ 0 ). The addition ofrandom errors (suppose, for the sake of simplicity, independent) willcertainly corrupt the picture. So how to reconstruct the frequency λ 0 ?Multiply our observations x i by sin(λt) and cos(λt) where λ is avariable, and consider the sums64