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

or, for discrete observations,IT(ω) = | ∑ e x |nt=0−iωt2tat many values of ω and determining one or a few maxima of I T (ω).Actually, however, as is possible to find in Timofeev’s book 24 , thereexist a few other expressions differing from I T (ω) by the limits ofintegration (summation) and also by taking into account not only themodulus, but the argument of the complex magnitudes as well.Various graphs are thus obtained whose behaviour at different ωallows to localize the possible periods contained in the observations.Strictly speaking, Timofeev’s considerations are not stochastic since astochastic approach demands to apply the notion of an ensemble ofimagined realizations of the observations (of which we see only one)and to make estimations based on these notions of the statisticalsignificance of the isolated periods. This is possible if certainassumptions about the applied model describing the observations aremade.For example, it is very convenient if the model, in the discrete case,is of the kindnx = ∑ a sin(ω t + φ ) + ξ(2.10)t k k k tk = 1where ξ 0 , ξ 1 , ..., ξ n are independent identically distributed randomvariables. In the continuous case such a model with independence ofthe values of noise ξ(t) at any no matter how close values of t (whitenoise) is less realistic. It is possible to provide a number of physicalexamples where model (2.10) is realistic.The first example to come across I can mention, can be provided bythe observations of the brightness of a variable star if measured not toofrequently. The reasonableness of the use of the periodogram methodand the possibility of certain estimates of significance in such cases isdoubtless. However, in more complicated cases, in which we areunable to discuss a stochastic model of noise corrupting the process,stochastic statistical considerations with estimation of significance areimpossible.Basing myself chiefly on the application of the periodogram methodto series in economics, I [ii] formulated a number of scepticalcomments on the actually achieved success. It were these remarks thatprompted the only scientifically doubtless objection mentioned abovein the Introduction, and it came from Timofeev. He indicated a veryinteresting unexpected example of a practical application ofperiodograms. To repeat, in this case the study has no stochasticessence (isolation of some undeniable peaks on the periodogramwhose significance is not needed to estimate). [The author describesthat successful and important industrial case.]115