The use of confidence or fiducial limits illustrated in the case of the binomial

CJ Clopper, ES Pearson - Biometrika, 1934 - JSTOR
CJ Clopper, ES Pearson
Biometrika, 1934JSTOR
(1) General Discussion. In facing the problem of statistical estimation it may often be
desirable to obtain from a random sample a single estimate, say a, of the value of an
unknown parameter, a, in the population sampled. It has always, however, been realised
that this single value is of little use unless associated with a measure of its reliability and the
traditional practice has been to give with a its probable error (or more recently its standard
error), in the form a+ pe (a)..................... From this inforination it was possible, if the sample …
(1) General Discussion. In facing the problem of statistical estimation it may often be desirable to obtain from a random sample a single estimate, say a, of the value of an unknown parameter, a, in the population sampled. It has always, however, been realised that this single value is of little use unless associated with a measure of its reliability and the traditional practice has been to give with a its probable error (or more recently its standard error), in the form a+ pe (a)..................... From this inforination it was possible, if the sample was not too small, to draw the conclusion that the unknown value of a lay within the limits ai= a-3xp. e (a) and a2= a+ 3 xp. e (a)...............(2) with a high degree of probability. But it was neither easy to give any precise definition of this measure of probability nor to assess the extent of error involved in estimating the value of p. e (a) from the sample. The recent work of RA Fisher introducing the conception of the fiducial interval has inade it possible under certain conditions to treat this problem of estimation in a simple yet powerful manner*. It is proposed in the present paper to illustrate on the following problem the ideas involved in this method of approach. A sample of n units is randomly drawn from a very large population in which the proportion of units bearing a certain character, A, is p. In the sample x individuals bear the character A and n-x do not. p is unknown and the problem is to obtain limits p, and P2 such that we may feel with a given degree of confidence that
Pl< P< P2...........(3). In the first place, how is this degree of confidence to be defined? The underlying conception involved in all problems of this type is extremely simple. In our statistical experience it is likely that we shall meet many values of n and of x; a rule must be laid down for determining pi and p2 given n and x. Our confidence that p lies within the interval (pI, P2) will depend upon the proportion of times that this prediction is correct in the long run of statistical experience, and this
JSTOR