By R. Azencott, Y. Guivarc'h, R. F. Gundy, P. L. Hennequin
Publication via Azencott, R., Guivarc'h, Y., Gundy, R. F.
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Extra info for Ecole d'Ete de Probabilites de Saint-Flour VIII
The value of U* is unaffected by the choice of origin. 5. Exact results for U*. A detailed study of exact results for Ul has been made by Stephens ,  who gives the exact first four moments, the exact distribution of U% for n = 1, 2, 3,4, some exact results for the lower tail of the distribution for any n and tables of significance points for both upper and lower tails. The algebra required for these exact results is too lengthy to be included here. For those parts of the range where exact results were not available, Pearson curves based on the first four moments were used by Stephens to obtain approximate significance points.
The orthonormal solutions of Inspection shows that h(t) = 1, /I = 0 is a solution. Let h(t) be any other eigenfunction. Since this must be orthogonal to h(t) = 1 the condition 38 J. DURBIN must be satisfied. 2) we therefore obtain where zlt z2, • • • are independent N(0,1) variables. 2) we observe the interesting fact that 4U2 has the same distribution as the sum of two independent values of W2. Putting we obtain where w t , w 2 , ••• are independent exponentially distributed variables and this has characteristic function by the infinite product for the sine function.
4. Watson's statistic U*. We now consider the statistic of Cramer-von Mises type appropriate for tests on the circle. f. F0(x) Watson  suggested the statistic on putting t = and similarly for practical work: Hence we obtain the form more suitable for As before, an origin P is chosen at an arbitrary point on where = the circle prior to the calculation of Fn(x) or Fn(i]. Although Watson introduced this statistic specifically for tests on the circle it can, like Vn, be used for tests on the line and indeed can be expected to be more powerful than W2n for tests against shifts in variance of a symmetric distribution.
Ecole d'Ete de Probabilites de Saint-Flour VIII by R. Azencott, Y. Guivarc'h, R. F. Gundy, P. L. Hennequin