Volume 2

Collected papers of R. A. Fisher / edited by J.H. Bennett.

  • Fisher, Ronald Aylmer, Sir, 1890-1962.
Date:
1971-1974
Catalogue details

Licence: Attribution-NonCommercial 4.0 International (CC BY-NC 4.0)

Credit: Collected papers of R. A. Fisher / edited by J.H. Bennett. Source: Wellcome Collection.

    41/568 (page 37)
    This quantity remains finite as the size of the sample is increased without limit, but increases without limit as the number of classes is increased. Consequently, as one might expect, the method of minimising yf breaks down for fine grouping. For example, suppose we have 5 classes only, in a population distributed according to the binomial distribution (P + ?) 4 - Let p be calculated from numbers observed in the 5 possible classes in a large sample. Then m x = ftp 4 , m 2 = 4np 3 q, etc., and the intrinsic accuracy is ~ fm' 2 \ 4ft V m J pq‘ The loss of information due to using the minimum y 2 solution is calculated from ' 2 \ l s 2° l m m c and Z2T2 ( 3 P 2 ~ 2 M + 3 7 2 ) r 1 S m '3 4 ( ^ y y 7) (p 4 - + 18 pV - 2 P? 3 + ? 4 ) n » and is in fact i ( 3 P 2 - + 3 2 2 ) - ( V 4 - 2 P 3( ? + 18pV - 2pq 3 + g 4 ) 2 . r 7 , W P 1 This is least when p = q, and is then equal to 20 pq’ or equivalent to the loss of 5 observations. In approaching the maximum likelihood solution by successive approximations we have seen that starting with an inefficient statistic, a single process of approximation will in ordinary cases give an efficient statistic differing from the maximum likelihood solution, by a quantity which with increasing samples decreases as ft -1 . The loss of information of such efficient statistics is therefore finite for large samples, for the additional variance of dL/d6 will be L*V(T -6), and L increases proportionately to the sample. If the process of approximation be repeated a statistic is obtained differing from the maximum likelihood solution only to the order of n~%, and for such a statistic the loss of accuracy, beyond that suffered by the maximum likelihood solution will tend to zero for large samples. Delete this line. Delete the underlined expression.