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

and the variances are inversely proportional to the weights2σvar δ = .iw iI have described the assumptions imposed on the randomcomponent of our observations. Now I pass to their determinatecomponent f(t) otherwise called trend.The most simple and classical case consists in that the function f(t)is of a quite definite class but depends on some unknown parametersc 1 , c 2 , ..., c k :f(t) = F(t, c 1 , c 2 , ..., c k ) (1.8)where the function F is given by a known formula or an algorithm ofcalculation. For example, in case of the motion of an object in spacethose parameters can be understood as its coordinates and velocities atany definite moment; other, more opportune parameters can also beintroduced.Then any coordinate f(t) will be uniquely determined by theparameters and the Newtonian laws of motion (if that object has noengine). The problem consists in determining estimates of theparameters c ˆigiven observations (1.7). It is solved by the Gaussianmethod of least squares: the estimates are determined in such a waythat the minimal value of the functionn∑i=1[ x − F( t ; c ,..., c ]i i 1 k2of c i will be attained at point ( cˆ1,..., c ˆk).More often, however, is the case in which the real dependence f(t) isunknown. Here also the equality (1.8) is applied but the function F ischosen more or less arbitrarily. Thus, a polynomial might be chosenand the method of least squares once more applied.Such a non-classical situation when f(t) is not known beforehanddemands a more detailed analysis, see a concrete example in the nextChapter. Here, however, we describe an absolutely different modelalso applied for statistically treating observational series.1.4. Model of a stochastic process. The main attention is turned tothe isolation of the determinate component, the trend f(t). The valuesthemselves, x i , of the observational series (1.8) are not random;random are only the additional magnitudes δ i considered as errors,noise, and generally the devil’s machinations. Another approach ispossible with randomness being considered the main property of theseries under study which we now denote byξ 1 , ξ 2 , ..., ξ n . (1.9)Here, the most simple model consists in treating that set as arealization of an n-dimensional random variable. Such a model can beuseful if the experiment providing it can be repeated many times over,53