ASYMPTOTIC INFERENCE FOR SEMIMARTINGALE MODELS WITH SINGULAR PARAMETER POINTS

We consider semimartingale models which admit locally a quadratic approximation of the log-likelihood process with asymptotics as the observation time increases to infinity. A parameter point is termed singular if the model fails to be locally asymptotically mixed normal. The limiting distributions of asymptotic ML-estimators at singular and nearly singular parameter points are studied. They differ rather drastically from the normal (or mixed normal) limiting distribution at nonsingular parameters. The singularities of several models for diffusion processes, diffusions with jumps and point processes are discussed.