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

A rather subtle arsenal is developed based on the assumption thatwe know the distribution of probabilities of the random variable (thenormal law). And once more the question emerges: wherefrom indeeddo we know it? And how precisely? And, finally, what is the practicalvalue of the product itself, of the confidence interval? A small numberof trials means small amount of information, and things are bad for us.But, whether the confidence interval will be somewhat longer orshorter, is not so important the less so since the confidence probabilitywas assigned arbitrarily.From my viewpoint, this remark is still a rather mild doubt. We mayadd: Wherefrom and how precisely do we know that, given thisconcrete situation, it is proper at all to discuss distributions ofprobabilities? Suppose, however, that the distribution of probabilitiesof the unpredictable magnitudes under discussion does exist. But then(Grekova 1976), it is not necessary to think highly of the theory ofestimation. Indeed, this theory allows us to extract the maximalamount of information not from sample data in general; the postulateon the type of distribution of probabilities is also introduced. It onlyrepresents reality with some precision at whose empirical estimationthe estimation theory is not at all aimed.And the theory’s conclusions and it itself, generally speaking,changes with the change of that distribution. It would have beennecessary to calculate the vagueness of the sought estimates of theparameters caused by the expected vagueness of the postulateddistribution. Then, the estimation theory extracts the maximal amountof information according to some specific criteria whose practicalvalue is not doubtless. Finally, that theory is based on the postulate ofindependent trials with which, as we saw, not everything was in order.It ought to be stated that the treatises on mathematical statistics do notmiss the opportunity to identify the treatment of observations, thatreally not at all simple discipline, with the scientific approach instatistics. Here is Grekova (1976, p. 112) once more:Mathematical arsenals have some hypnotic property andresearchers are often apt to believe unquestionably their calculations,and the more so the more flowery are their tools [...].In any applied science, a scientific approach presumes first of all acreation of an intuitively convincing empirical foundation. Thecomplication, rigour and cost of the mathematical arsenal should becoordinated with the reliability of the foundation. This pragmatic ruleapplied from long ago is neatly called principle of equal stability of allthe elements of an [applied − Yu. A.] investigation (Grekova 1976, p.111). The theory of estimation hardly satisfies it in due measure.3.6.2. Postulate on the existence of a distribution of probabilitiesfor sample estimates. Imagining many additional samples. Theexistence and sometimes even the type of that distribution ispostulated. Suppose that an experiment according to the pattern ofmany series is carried out. We may only repeat what was said in §3.6.1 concerning the distributions of the initial random variables.129