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 sequentially 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 little outside finance.

## References

*International Statistical Review / Revue Internationale de Statistique*62 (1): 133–65.

*Bernoulli*1 (1/2): 17–39.

*Stochastic Processes and Their Applications*125 (4): 1195–1217.

*Advances in Applied Probability*8 (4): 712–36.

*arXiv:1410.6764 [Math]*, October.

*Selected Works of C.C. Heyde*, edited by Ross Maller, Ishwar Basawa, Peter Hall, and Eugene Seneta, 214–35. Selected Works in Probability and Statistics. Springer New York.

*Séminaire de Probabilités XXXI*, edited by Jacques Azéma, Marc Yor, and Michel Emery, 232–46. Lecture Notes in Mathematics 1655. Springer Berlin Heidelberg.

*The Annals of Statistics*38 (3): 1478–1545.

*Statistica Neerlandica*64 (3): 329–51.

## No comments yet. Why not leave one?