  We study parametric estimation of ergodic diffusions observed at high
frequency. Different from the previous studies, we suppose that sampling
stepsize is unknown, thereby making the conventional Gaussian quasi-likelihood
not directly applicable. In this situation, we construct estimators of both
model parameters and sampling stepsize in a fully explicit way, and prove that
they are jointly asymptotically normally distributed. The $L^{q}$-boundedness
of the obtained estimator is also derived. Further, we propose the Schwarz
(BIC) type statistics for model selection and show its model-selection
consistency. We conducted some numerical experiments and found that the
observed finite-sample performance well supports our theoretical findings. Also
provided is a real data example.
