Optimal asymptotic tests of composite hypotheses for continuous time stochastic processes

Abstract

Consider a stochastic process {X_t, t ≥ O} whose distributions depend on an unknown parameter (γ,θ). A locally asymptotically most powerful test, for testing the composite hypothesis H₀ : γ = γ₀ against H₁ : γ ≠ γ₀ in the presence of a nuisance parameter θ is developed following the concept of C(α)-tests introduced by Neyman. Results are illustrated by means of example of process {X(t),t ≥ 0} satisfying the linear stochastic differential equation dX(t) = (γX(t) + θ)dt+dW(t),t ≥ 0.