FOUNDATIONS OF THEORETICAL STATISTICS. 
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Of the criteria used in problems of Estimation only the criterion of Consistency has 
hitherto been widely applied ; in Section 5 are given examples of the adequate and 
inadequate application of this criterion. The criterion of Efficiency is shown to be a 
special but important case of the criterion of Sufficiency, which latter requires that the 
whole of the relevant information supplied by a sample shall be contained in the statistics 
calculated. 
In order to make clear the nature of the general method of satisfying the criterion 
of Sufficiency, which is here put forward, it has been thought necessary to reconsider 
Bayes’ problem in the light of the more recent criticisms to which the idea of “ inverse 
probability ” has been exposed. The conclusion is drawn that two radically distinct 
concepts, both of importance in influencing our judgment, have been confused under 
the single name of probability . It is proposed to use the term likelihood to designate 
the state of our information with respect to the parameters of hypothetical populations, 
and it is shown that the quantitative measure of likelihood does not obey the mathe¬ 
matical laws of probability. 
A proof is given in Section 7 that the criterion of Sufficiency is satisfied by that set 
of values for the parameters of which the likelihood is a maximum, and that the same 
function may be used to calculate the efficiency of any other statistics, or. in other- 
words, the percentage of the total available information which is made use of by such 
statistics. 
This quantitative treatment of the information supplied by a sample is illustrated by 
an investigation of the efficiency of the method of moments in fitting the Pearsonian 
curves of Type III. 
Section 9 treats of the location and scaling of Error Curves in general, and contains 
definitions and illustrations of the intrinsic accuracy, and of the centre of location of such 
curves. 
In Section 10 the efficiency of the method of moments in fitting the general Pearsonian 
curves is tested and discussed. High efficiency is only found in the neighbourhood of 
the normal point. The two causes of failure of the method of moments in locating these 
curves are discussed and illustrated. The special cause is discovered for the high 
efficiency of the third and fourth moments in the neighbourhood of the normal point. 
It is to be understood that the low efficiency of the moments of a sample in estimating 
the form of these curves does not at all diminish the value of the notation of moments as 
a means of the comparative specification of the form of such curves as have finite moment 
coefficients. 
Section 12 illustrates the application of the method of maximum likelihood to dis¬ 
continuous distributions. The Poisson series is shown to be sufficiently fitted by the 
mean. In the case of grouped normal data, the Sheppard correction of the crude 
moments is shown to have a very high efficiency, as compared to recent attempts to 
improve such fits by making y 2 a minimum ; the reason being that x 2 is an expression 
only approximate to a true value derivable from likelihood. As a final illustration of 
