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

nnA(λ) x sin λ t, B(λ) = x cosλ t.= ∑ ∑tt= 1 t=1tIn previous times this calculation for various values of fairly many λwas rather tedious, but computers removed that difficulty. Calculatenow the functionC(λ) = A 2 (λ) + B 2 (λ)called periodogram. Formerly, it was imagined as a function of theperiod, 2π/λ, rather than of frequency λ, which explains the origin ofthat term.The main statement is that, given a sufficiently large number ofobservations n, the periodogram as a function of λ will take a clearlyexpressed maximal value in a small vicinity of the real frequency λ 0 . Ifthe determinate part of the observations consists of several harmonicssin(λ 0 t + φ) rather than one; that is, ifkx = ∑ A sin(λ t + φ ) + δ ,(3.2)t j j j tj=0then the periodogram will have several maximal values situated closeto λ 0 , ..., λ k . Their heights will depend on the number of observations,n, and amplitudes, A j . When not knowing beforehand the number ofharmonics and the variances σ 2 of the errors δ i , we find ourselves in arather difficult situation. The periodogram generally has very manylocal maxima and it is incomprehensible how to interpret them, eitheras really corresponding to latent periods λ j or as occurring purelyrandomly 5 .These difficulties can be somehow overcome. It is worse that as arule there is no guarantee that model (3.2) is valid. We can likely besure that it is not. For example, when studying series of observationstaken from economics, it is seen by naked eye that they rathersmoothly depend on time (they rarely change from increasing todecreasing or vice versa) which should not occur for observationsrepresented by model (3.2): they ought to be scattered around thesmooth curve, the main term at the right side of (3.2). We maycertainly assume that that curve itself badly corresponds to our idea ofa smooth curve and that its roughness compensated random scatter, butan unreasonably large number of harmonics is needed for that tohappen.The most important problem therefore consists in determining howreasonable are the results provided by the periodogram method whenthe model (3.2) is wrong and the observational series {x} is describedby some other model. The generally known English statistician M. G.Kendall [Sir Maurice Kendall] carried out such experimental studiesdescribed in his rather rare book (1946) on mathematical statistics.This small contribution is one of the most remarkable books onmathematical statistics. Its epigraph is curious:65