Asymptotic inference for a stochastic differential equation with uniformly distributed time delay

For affine stochastic differential equation with uniformly distributed time delay the local asymptotic properties of the likelihood function are studied. Local asymptotic normality, local asymptotic mixed normality, periodic local asymptotic mixed normality or local asymptotic quadraticity is proved for different values of the parameter. Applications to the asymptotic behaviour of the maximum likelihood estimator of the parameter based on continuous sample are given.