Components of Cramér-Von Mises Statistics. I

摘要: Let F n (x) be the sample distribution function derived from a of independent uniform (0, 1) variables. The paper is mainly concerned with orthogonal representation Cramer-von Mises statistic W 2 in form Σ ∞ j=1 (jπ) -2 z nj where are principal components $\sqrt n\{F_n(x) - x\}$ . It shown that identically distributed for each and their significance points tabulated. Their use testing goodness fit discussed asymptotic powers compared those , Anderson Darling's A Watson's U against shifts mean variance normal distribution. residual p also given various p. analogous to Legendre polynomial introduced by Neyman as basis his "smooth" test fit. relationship Fourier series analysis x discussed. An alternative set Pyke's modification considered. Tests based on applied data coal-mining disasters.