ABSTRACT
We treat the general theory of χ 2 tests, give applications to the treatment of contingency tables, and consider the more general problem of testing against restricted alternatives. We make the general assumptions of Section 23, and we choose Μ(θ) to be a nonsingular generalized inverse of C(θ). Therefore, our assumptions are
https://www.w3.org/1998/Math/MathML"> n ( Z n − A ( θ 0 ) ) → L N ( 0 , C ( θ 0 ) ) , Z ∈ ℝ d , θ 0 ∈ Θ open ⊂ ℝ k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136288/c12c5089-77f8-4e14-84a1-364a5e26ae30/content/eq1224.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .
A(θ) is bicontinuous; Ȧ(θ) is continuous and has full rank k ≤ d.
B(θ) is continuous nonsingular and there exists a > 0 such that Μ(θ) > α I for all θ ∈ Θ.
C(θ)M(θ)C(θ) = C(θ) and C(θ)M(θ)Ȧ(θ) = Ȧ(θ).
