Difference based estimators and infill statistics

Infill statistics, that is, statistical inference based on very dense observations over a fixed domain has become of late a subject of growing importance. On the other hand, it is a known phenomenon that in many cases infill statistics do not provide optimal rates. The degree of sub-optimality is related to how much parameter-related information is lost because of dense sampling, which in turn is related to sample path regularity. In the stationary Gaussian case this is determined by the large value behaviour of the spectral density and its derivatives. Moreover, many interesting non stationary examples such as non linear functionals of stationary Gaussian processes or diffusion processes driven by a stationary increment Gaussian process can also be seen to depend on the large value behaviour of the spectral density of the underlying process. In this article we discuss several examples in a unified frequency domain approach providing a general framework relating sample path regularity to estimation rates. This includes examples such as volatility estimation for diffusions and fractional diffusions, multifractals and non-linear functions of Gaussian processes. As a final example we include the problem of estimation in the presence of an additive white noise, known as the nugget effect or micro-structure error. Copyright Springer Science+Business Media Dordrecht 2015