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

It is interesting therefore to see what occurred when the mosteminent statisticians attempted to study actual data by models of astochastic process. Rather often they experienced failure, see Chapter3. We will also briefly mention the statistical theory of turbulence inwhich the notion of stochastic process has been applied with brilliantsuccess.2. The Method of Least SquaresGauss discovered and introduced it into general usage. The classicalcase which he considered consisted in that some known relationsshould be maintained between the terms of the observational seriesx 1 , x 2 , ..., x nhad not the observations been corrupted by errors. For example, in thecase of the path of an object in space 3 it would have been possible toexpress all terms of the series through a few of its first terms had thesebeen known absolutely precisely. This classical case can becomparatively easily studied within the boundaries of mathematicalstatistics. Practical applications of the method of least squares canencounter more or less essential calculational difficulties which weleave aside. Other difficulties are connected with the possible nonfulfilmentof the assumption of the model of trend with error. Thus,errors of successive measurements of distances by radar apparentlycan not be assumed independent random variables. It is in generalunclear whether they possess a statistical character so that statisticalmethods are here unreliable and moreover helpless.The observations themselves, however, are highly precise and canbe made many times, so that statistical methods are not needed there.In spite of all the merits of the classical case, its shortcoming is that itoccurs comparatively rarely. Much more often we are convinced thatour observations can be approximated by a smooth dependencex i ≈ f(t i )where t i is a variable describing the conditions of the i-th experiment.The exact form of the function f(t) is, however, unknown.Methods strongly resembling those of the classical case are appliedhere, but their study indicates that they are not mathematicallyjustified. Mathematical statistics widely applies mathematics but is notreduced to that comparatively very transparent science. Statistics israther an art and as such it has its own secrets and we will indeedbegin by studying them.2.1. The secrets of the statistical art. When wishing to apply themethod of least squares we can in most cases use a computerprogramme compiled once and for all. It is just necessary to enter thedata, wait for the calculations to be made and the printer will provide aformula for a curve fitting the observations. However, he who passesall these procedures to a machine will be wrong. It is absolutelynecessary to represent the available data in a visible way and at least toglance at the figure.55