# Large sample theory

Many things are similar in the eventual limit.

under construction ⚠️: I merged two notbooks here. The seams are showing.

We use asymptotic approximations all the time in statistics, most frequently in asymptotic pivots that motivate classical tests e.g. in classical hypothesis tests or an information penalty. We use the asymptotic delta method to motivate robust statistics, or infinite neural networks. There are various specialised mechanism; I am fond of the Stein methods. Also fun, Feynman-Kac formulae give us central limit theorems for all manner of weird processes.

There is much to be said on the various central limit theorems, but I will not be the one to say it right this minute, because this is a placeholder.

A convenient feature of M-estimation, and especially maximum likelihood esteimation is simple behaviour of estimators in the asymptotic large-sample-size limit, which can give you, e.g. variance estimates, or motivate information criteria, or robust statistics, optimisation etc.

In the most celebrated and convenient cases case asymptotic bounds are about normally-distributed errors, and these are typically derived through Local Asymptotic Normality theorems. A simple and general introduction is given in Andersen et al. (1997) page 594., which applies to both i.i.d. data and dependent_data in the form of point processes. For all that it is applied, it is still stringent.

In many nice distributions, central limit theorems lead (Asymptotically) to Gaussian distributions, and we can treat uncertainty in terms of transforamtions of Gaussians.

## Fisher Information

Used in ML theory and kinda-sorta in robust estimation, and natural gradient methods. A matrix that tells is how much a new datum affects our parameter estimates. (It is related, I am told, to garden-variety Shannon information, and when that non-obvious fact is more clear to me I shall expand how precisely this is so.) 🏗

## Convolution Theorem

The unhelpfully-named convolution theorem of Hájek (1970) — is that related?

Suppose $$\hat{\theta}$$ is an efficient estimator of $$\theta$$ and $$\tilde{\theta}$$ is another, not fully efficient, estimator. The convolution theorem says that, if you rule out stupid exceptions, asymptotically $$\tilde{\theta} = \hat{\theta} + \varepsilon$$ where $$\varepsilon$$ is pure noise, independent of $$\hat{\theta}.$$

The reason that’s almost obvious is that if it weren’t true, there would be some information about $$\theta$$ in $$\tilde{\theta}-\hat{\theta}$$, and you could use this information to get a better estimator than $$\hat{\theta}$$, which (by assumption) can’t happen. The stupid exceptions are things like the Hodges superefficient estimator that do better at a few values of $$\hat{\theta}$$ but much worse at neighbouring values.

## References

Akaike, Hirotogu. 1973. In Proceeding of the Second International Symposium on Information Theory, edited by Petrovand F Caski, 199–213. Budapest: Akademiai Kiado.
Akaike, Htrotugu. 1973. Biometrika 60 (2): 255–65.
Andersen, Per Kragh, Ornulf Borgan, Richard D. Gill, and Niels Keiding. 1997. Statistical models based on counting processes. Corr. 2. print. Springer series in statistics. New York, NY: Springer.
Athreya, K. B., and Niels Keiding. 1977. Sankhyā: The Indian Journal of Statistics, Series A (1961-2002) 39 (2): 101–23.
Athreya, Krishna B, and S. N Lahiri. 2006. Measure theory and probability theory. New York: Springer.
Bacry, E., S. Delattre, M. Hoffmann, and J. F. Muzy. 2013. Stochastic Processes and Their Applications, A Special Issue on the Occasion of the 2013 International Year of Statistics, 123 (7): 2475–99.
Barbour, A. D., and Louis H. Y. Chen, eds. 2005. An Introduction to Stein’s Method. Vol. 4. Lecture Notes Series / Institute for Mathematical Sciences, National University of Singapore, v. 4. Singapore : Hackensack, N.J: Singapore University Press ; World Scientific.
Barndorff-Nielsen, O. E., and M. Sørensen. 1994. International Statistical Review / Revue Internationale de Statistique 62 (1): 133–65.
Barrio, Eustasio del, Paul Deheuvels, and Sara van de Geer. 2006. Lectures on Empirical Processes: Theory and Statistical Applications. European Mathematical Society.
Barron, Andrew R. 1986. The Annals of Probability 14 (1): -336-342.
Becker-Kern, Peter, Mark M. Meerschaert, and Hans-Peter Scheffler. 2004. The Annals of Probability 32 (1): 730–56.
Bibby, Bo Martin, and Michael Sørensen. 1995. Bernoulli 1 (1/2): 17–39.
Bréhier, Charles-Edouard, Ludovic Goudenège, and Loïc Tudela. 2016. In Monte Carlo and Quasi-Monte Carlo Methods, edited by Ronald Cools and Dirk Nuyens, 163:245–60. Springer Proceedings in Mathematics & Statistics. Cham: Springer International Publishing.
Cantoni, Eva, and Elvezio Ronchetti. 2001. Journal of the American Statistical Association 96 (455): 1022–30.
Claeskens, Gerda, Tatyana Krivobokova, and Jean D. Opsomer. 2009. Biometrika 96 (3): 529–44.
DasGupta, Anirban. 2008. Asymptotic Theory of Statistics and Probability. Springer Texts in Statistics. New York: Springer New York.
Del Moral, P., A. Kurtzmann, and J. Tugaut. 2017. SIAM Journal on Control and Optimization 55 (1): 119–55.
Duembgen, Moritz, and Mark Podolskij. 2015. Stochastic Processes and Their Applications 125 (4): 1195–1217.
Feigin, Paul David. 1976. Advances in Applied Probability 8 (4): 712–36.
Feller, William. 1951. The Annals of Mathematical Statistics 22 (3): -427-432.
Fernholz, Luisa Turrin. 1983. von Mises calculus for statistical functionals. Lecture Notes in Statistics 19. New York: Springer.
Gine, Evarist, and Joel Zinn. 1990. Annals of Probability 18 (2): 851–69.
Giraitis, L, and D Surgailis. 1999. “Central Limit Theorem for the Empirical Process of a Linear Sequence with Long Memory.” Journal of Statistical Planning and Inference 80 (1-2): 81–93.
Gribonval, Rémi, Gilles Blanchard, Nicolas Keriven, and Yann Traonmilin. 2017. arXiv:1706.07180 [Cs, Math, Stat], June.
Hájek, Jaroslav. 1970. Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete 14 (4): 323–30.
———. 1972. In. The Regents of the University of California.
Heyde, C. C., and E. Seneta. 2010. In 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.
Huber, Peter J. 1964. The Annals of Mathematical Statistics 35 (1): 73–101.
Jacob, Pierre E., John O’Leary, and Yves F. Atchadé. 2017. arXiv:1708.03625 [Stat], August.
Jacod, Jean, Mark Podolskij, and Mathias Vetter. 2010. The Annals of Statistics 38 (3): 1478–1545.
Jacod, Jean, and Albert N. Shiryaev. 1987a. Limit Theorems for Stochastic Processes. Vol. 288. Grundlehren Der Mathematischen Wissenschaften. Berlin, Heidelberg: Springer Berlin Heidelberg.
———. 1987b. In Limit Theorems for Stochastic Processes, edited by Jean Jacod and Albert N. Shiryaev, 1–63. Grundlehren Der Mathematischen Wissenschaften. Berlin, Heidelberg: Springer Berlin Heidelberg.
Janková, Jana, and Sara van de Geer. 2016. arXiv:1610.01353 [Math, Stat], October.
Karabash, Dmytro, and Lingjiong Zhu. 2012. arXiv:1211.4039 [Math], November.
Konishi, Sadanori, and Genshiro Kitagawa. 1996. Biometrika 83 (4): 875–90.
———. 2003. Journal of Statistical Planning and Inference, C.R. Rao 80th Birthday Felicitation Volume, Part IV, 114 (1–2): 45–61.
Kraus, Andrea, and Victor M. Panaretos. 2014. Biometrika 101 (1): 141–54.
Le Gland, François, Valerie Monbet, and Vu-Duc Tran. 2009. 25.
LeCam, L. 1970. The Annals of Mathematical Statistics 41 (3): 802–28.
———. 1972. In. The Regents of the University of California.
Lederer, Johannes, and Sara van de Geer. 2014. Bernoulli 20 (4): 2020–38.
Lorsung, Cooper. 2021. arXiv.
Maronna, Ricardo Antonio. 1976. The Annals of Statistics 4 (1): 51–67.
Mueller, Ulrich K. 2018. arXiv:1802.00762 [Math], February.
Ogata, Yoshiko. 1978. Annals of the Institute of Statistical Mathematics 30 (1): 243–61.
Pollard, David. 1990. Empirical Processes: Theory and Applications. IMS.
Prause, Annabel, and Ansgar Steland. 2018. Electronic Journal of Statistics 12 (1): 890–940.
Puri, Madan L., and Pham D. Tuan. 1986. Proceedings of the National Academy of Sciences of the United States of America 83 (3): 541–45.
Raginsky, Maxim, and Igal Sason. 2014. Concentration of Measure Inequalities in Information Theory, Communications, and Coding: Second Edition. Now Publishers.
Ross, Nathan. 2011. Probability Surveys 8 (0): 210–93.
Scornet, Erwan. 2014. arXiv:1409.2090 [Math, Stat], September.
Shiga, Tokuzo, and Hiroshi Tanaka. 1985. Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete 69 (3): 439–59.
Sørensen, Michael. 2000. The Econometrics Journal 3 (2): 123–47.
Stam, A. J. 1982. Journal of Applied Probability 19 (1): 221–28.
Stein, Charles. 1986. Approximate Computation of Expectations. Vol. 7. IMS.
Tropp, Joel A. 2015. An Introduction to Matrix Concentration Inequalities.
Vaart, Aad W. van der. 2007. Asymptotic statistics. 1. paperback ed., 8. printing. Cambridge series in statistical and probabilistic mathematics. Cambridge: Cambridge Univ. Press.

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