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Asymptotic Properties of Tests Based on Linear Combinations of the Orthogonal Components of the Cramer-Von Mises Statistic
Let $X_1, X_2, \cdots, X_n$ be independent identically distributed random variables defined on the unit interval. The generalized $j$th orthogonal component is defined as $V_{nj} = n^{-\frac{1}{2}} \sum^n_{i = 1} d_j(X_i),$ where $\{1, d_1, d_2, \cdots\}$ is an orthonormal basis for $\mathscr{L}_2(\lbrack 0, 1\rbrack).$ These statistics are a generalization of …