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

is applied with P(t) being a polynomial whose coefficients should beestimated by that method. [...] It is usually said that in case anothermodelx i = f(t i ) + δ i , i = 1, 2, ..., nis valid with f(t) not being a polynomial, we may apply the Weierstrasstheorem according to which we can approximate f(t) by the polynomialP(t) as precisely as desired. However, that reference is , however,inappropriate because any continuous function can be approximated bya polynomial of a sufficiently high degree. In practice we attempt tochoose a low rather than a high degree. [...] A polynomial of a higherdegree [can be] further from reality than that of a lower degree. Then,the Weierstrass theorem is also valid for functions of several variables.However, if the earth’s surface as shown by isolines on a topographicmap is considered a typical function of two variables, and itsapproximation is attempted, the result will be usually unsatisfactory:the degree of the polynomial should be too high. The theoreticalWeierstrass theorem and practical smoothing differ.The method of smoothing by a polynomial is therefore notmathematically justified and the success of that procedure is one moremystery of the statistical art. How can we explain the rather oftensuccess here? Apparently the human eye feels well enough thebehaviour of the graphs of analytic functions, polynomials inparticular. As students are taught, only a few points ought to becalculated, − discontinuities, extrema, sometimes points of inflexion, −and functions are then reconstructed quite well. It may be supposedthat we are able to catch whether the real dependence is approximatedwhen smoothing a broken line by a polynomial well enough. Thisstatistical procedure of smoothing a function by a polynomial isprobably only applied when success is expected after having a look atthe graph by naked eye.The situation changes at once if that procedure is attempted to bewholly accomplished automatically. In that case no data will bepreliminarily estimated and the portion of successful smoothing willbe sharply reduced. The case of functions of many variables is quitecomplicated. We are unable to show either the experimental data or theresult of smoothing and can not even say whether it was successful ornot.2.2. Smoothing by a polynomial: an example. It is time to explainthe provenance of the observations represented on Fig. 1. [It was thestudy of the damage of insulation of the stators of large turbogenerators(Belova et al 1965, 1967).] The total number of failures isnaturally comprised of failures of separate generators. Rather early inour work we decided that the probability of a failure of a givengenerator is proportional to the total area of its insulation and littledepends on its constructive or operational conditions; hydrogencooling was then almost non-existing. We have therefore studied thebehaviour of a unit area of that insulation (100sq. m. corresponding inits order to an area of insulation a large machine) without allowing forany other peculiarities. The problem of aging of the insulation, i. e., of57