a.k.a. “Fancy ARIMA”.
Classically, you estimate statistics from many i.i.d. realisations from a presumed generating process.
What if your data are realisations of sequntially dependent time series? How do you estimate parameters from a single time series realisation?
By being a flashy quant!
Bonus points: How do you do this with many time series, whose parameters themselves have a distribution you wish to estimate?
See Mark Podolskij who explains “high frequency asymptotics” well. I think that the original framework is due to Jacod. (i.e. when you don’t have an asymptotic limit in number of data points, but in how densely you sample a single time series.)
This feels contrived for me, but it is probably interesting if you are not working with a multivariate Brownian motion, but a rather general Lévy process or something with interesting jumps AND continuous movement, and can sample with arbitrary density but not arbitrarily long. AFAICT this is basically only finance.
- Barndorff-Nielsen, O. E., & Sørensen, M. (1994) A Review of Some Aspects of Asymptotic Likelihood Theory for Stochastic Processes. International Statistical Review / Revue Internationale de Statistique, 62(1), 133–165. DOI.
- Bibby, B. M., & Sørensen, M. (1995) Martingale Estimation Functions for Discretely Observed Diffusion Processes. Bernoulli, 1(1/2), 17–39. DOI.
- Duembgen, M., & Podolskij, M. (2015) High-frequency asymptotics for path-dependent functionals of Itô semimartingales. Stochastic Processes and Their Applications, 125(4), 1195–1217. DOI.
- Feigin, P. D.(1976) Maximum Likelihood Estimation for Continuous-Time Stochastic Processes. Advances in Applied Probability, 8(4), 712–736. DOI.
- Heinrich, C., & Podolskij, M. (2014) On spectral distribution of high dimensional covariation matrices. arXiv:1410.6764 [Math].
- Heyde, C. C., & Seneta, E. (2010) Estimation Theory for Growth and Immigration Rates in a Multiplicative Process. In R. Maller, I. Basawa, P. Hall, & E. Seneta (Eds.), Selected Works of C.C. Heyde (pp. 214–235). Springer New York
- Jacod, J. (1997) On continuous conditional Gaussian martingales and stable convergence in law. In J. Azéma, M. Yor, & M. Emery (Eds.), Séminaire de Probabilités XXXI (pp. 232–246). Springer Berlin Heidelberg
- Jacod, J., Podolskij, M., & Vetter, M. (2010) Limit theorems for moving averages of discretized processes plus noise. The Annals of Statistics, 38(3), 1478–1545. DOI.
- Podolskij, M., & Vetter, M. (2010) Understanding limit theorems for semimartingales: a short survey: Limit theorems for semimartingales. Statistica Neerlandica, 64(3), 329–351. DOI.