kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

" = (2/) 20exp (– 2 ) d = 0.9953.Thus, the rule that recommends to assume that|a – x | ≤ 2/hn (62)has confidence probability " = 0.9953. In order to ascertain definitively the meaning and thepractical importance of the notion of confidence probability, let us dwell on this example.Suppose that we want to apply the rule (62) in some sequence of cases E 1 , E 2 , …, E n . Thevalues of a k , h k , and n k correspond to each of the cases E k . However, absolutelyindependently of these values, the probability of the inequality| xk– a k | ≤ 2/h k n k (62 k )in this case is " = 0.9953. If the systems of x i ’s which correspond here to the different E k ’sare independent one from another, then the events A k consisting in that the appropriateinequalities (62 k ) are valid, are also independent. Owing to the Bernoulli theorem, given thiscondition and a sufficiently large N, the frequency M/N of these inequalities being obeyed inthe sequence of cases E k will be arbitrarily close to " = 0.9953.Consequently, in anysufficiently long series of independent cases E k the rule (62) will lead to correct results inabout 99.5% of all cases, and to wrong results in approximately 0.5%. For justifying thisconclusion it is only necessary that the set of the considered cases E 1 , E 2 , …, E N bedetermined beforehand independently of the values of the x i ’s obtained by observation.Bernstein indicated a clever example of a misunderstanding that is here possible if noattention is paid to this circumstance 12 .After all this, a warning against a wide-spread mistake 13 seems almost superfluous.Namely, the equality for the unconditional probabilityP(|a – x | ≤ 2/hn) = 0.9953follows ifP(|a – x | ≤ 2/hn|a; h) = 0.9953 (63)for all possible values of a and h. However, it does not at all follow from (63) that for anyfixed values of (1)P(|a – x | ≤ 2/hn|x 1 ; x 2 ; …; x n ) = 0.9953.In concluding this section, I note that it is sometimes necessary to consider the rules forestablishing confidence limits for an estimated parameter which do not possess any definiteconfidence probability. In such cases, the part similar to that of confidence probability isplayed by the lower bound" = inf P( ≤ ≤ |, 1 , 2 , …, n )of the conditional probability for the validity of the inequalities ≤ ≤ at variouscombinations of the values of the parameters , 1 , 2 , …, n . Following Neyman, this lowerbound has been called the coefficient of confidence of the given rule 14 .