FOUNDATIONS OF THEOKETICAL STATISTICS. 
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is the best statistic for locating the curve, is the normal or gaussian curve of errors. If 
the curve is not of this form the mean is not necessarily, as we have seen, of any value 
whatever. The determination of the true curves of variation for different types of work 
is therefore of great practical importance, and this can only Ire done by different workers 
recording their data in full without rejections, however they may please to treat the 
data so recorded. Assuredly an observer need be exposed to no criticism, if after 
recording data which are not probably normal in distribution, he prefers to adopt some 
value other than the arithmetic mean. 
6. Formal Solution of Problems of Estimation. 
The form in which the criterion of sufficiency has been presented is not of direct 
assistance in the solution of problems of estimation. For it is necessary first to know 
the statistic concerned and its surface of distribution, with an infinite number of other 
statistics, before its sufficiency can be tested. For the solution of problems of 
estimation we require a method which for each particular problem will lead us 
automatically to the statistic by which the criterion of sufficiency is satisfied. Such a 
method is, I believe, provided by the Method of Maximum Likelihood, although I am 
not satisfied as to the mathematical rigour of any proof which I can put forward to 
that effect. Headers of the ensuing pages are invited to form their own opinion as 
to the possibility of the method of the maximum likelihood leading in any case to an 
insufficient statistic. For my own part 1 should gladly have withheld publication until 
a rigorously complete proof could have been formulated; but the number and variety 
of the new results which the method discloses press for publication, and at the same 
time I am not insensible of the advantage which accrues to Applied Mathematics from 
the co-operation of the Pure Mathematician, and this co-operation is not infrequently 
called forth by the very imperfections of writers on Applied Mathematics. 
If in any distribution involving unknown parameters 0,, fb, 0 3 , ... , the chance of 
an observation falling in the range dx be represented by 
f{x, 0 U 0 2 , ...)dx, 
then the chance that in a sample of n, n x fall in the range dx v 
so on, will be 
n ! 
li (v) 
ii {/(* 
p> 
0 U 0 2 ,... )dx p y 
n 2 in the range dx 2 , and 
The method of maximum likelihood consists simply in choosing that set of values 
for the parameters which makes this quantity a maximum, and since in this expression 
the parameters are only involved in the function/, we have to make 
S (log/) 
