Asymptotic Properties of Tests Based on Linear Combinations of the Orthogonal Components of the Cramer-Von Mises Statistic

Let X₁, X₂, ⋯, Xₙ be independent identically distributed random variables defined on the unit interval. The generalized jth orthogonal component is defined as Vₙⱼ = n-1/2 ∑ⁿi = 1 dⱼ(Xᵢ), where \1, d₁, d₂, ⋯\ is an orthonormal basis for L₂( 0, 1). These statistics are a generalization of the orthogonal components of the Cramer-von Mises statistic [2]. Linear combinations of the Vₙⱼ are applied to the problem of testing the null hypothesis of a uniform distribution against the alternative density pₙ(x) = 1 + h(x)/n¹/2 + kₙ(x)/n where $h(x)$ is square integrable and kₙ(x) is dominated by a square integrable function. When aⱼ = ∫ h(x) dⱼ(x) dx, tests based on ∑ᵐj = 1 aⱼ Vₙⱼ are shown to be asymptotically most powerful as min (m, n) → ∞. The asymptotic power and efficiency of these tests are computed. A procedure is developed for choosing among possible density functions when a goodness-of-fit test rejects the null hypothesis.