By D. R. Cox, D. V. Hinkley (auth.)
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Example text
Special multinomial distribution (etd). 28 in which Y can take on the values 1, 2, ... , 6 with probabilities . 3 the parameter space being the interval [- I, I]. For this very special multinomial distribution there are six possible ancillary statistics, namely (N1 + N 4 , N2 + N s , N3 + N 6 ), ••• , (N1 + N 6 , N2 + N s , N3 + N 4 ), in an obvious notation. If we define the transformation g by g Y = Y + 3 (mod 6) the induced parameter transformation isg*O = -0. The only ancillary statistic unaffected by the transformation g is (N1 + N 4 , N2 + N s , N3 + N 6 ) and on the grounds that the inference should be unchanged by this seemingly harmless transformation there is a unique choice of ancillary statistic.
The new parameter ¢ is often called the natural parameter for the problem, for several technical and practical reasons. One of these is that the ratio of likelihood functions at ¢1 and ¢2 < ¢1 is an increasing function of the sufficient statistic. Also it will tum out that comparisons of different sets of data are most easily achieved in terms of comparisons of the natural parameter values. For example, the natural parameter for the binomial distribution is ¢ = 10g{0/O - O)}, the so-called log odds ratio.
Note that this includes very general forms of sequential sampling in which observations may be taken singly or in groups. Suppose that the data are (n, Yl, ... , Yn). e. IY,(y 2 IYl; 0) ···Pn-I(Yl, ... YnIYn-, ..... f. of Yt> ... , Yn • Therefore, if the strong likelihood principle were accepted, the conclusion to be drawn about 0 would be the same as if n were fixed. Note, however, that N is not in general an ancillary statistic and that conditioning on its value is not a consequence of the conditionality principle as formulated above.