Efficient estimation of present-value distributions for long-dated contracts and functionals in the multivariate case

Tsimbalyuk, Alexandra (2020) Efficient estimation of present-value distributions for long-dated contracts and functionals in the multivariate case. PhD thesis, London School of Economics and Political Science.

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Abstract

The first chapter of this thesis focuses on the problem of estimating the joint law of a discrete-time perpetuity and underlying factors which govern the cash ow rate, in an ergodic Markovian environment. Our approach is based upon the so-called time-reversal technique which allows us to identify the joint law as a stationary distribution of an ergodic multidimensional Markov chain. Furthermore, a central limit theorem (CLT) for an estimator of the joint law is provided for a specific example of the perpetuity. Our proof of the CLT rests upon the geometric ergodicity property, which is also provided and is of independent interest. We further provide a justification for the Monte Carlo methods for approximating the joint law by sampling a single path of the reversed process. The second chapter of this thesis deals with the estimation of linear functionals in multidimensional spaces. We consider two ubiquitous statistical models: a regression model with one-sided errors and a Poisson point process (PPP) model. We consider two estimation approaches: a block-wise approach, when the estimator is an aggregate of local estimators, and a maximum likelihood approach. First, we assume the regularity of the underlying function in both models to be known. We combine the block-wise approach with martingale stopping time arguments and the PPP geometry to derive the unbiased estimators. We show that the rates of convergence of the mean squared risks match the lower bounds for the risks in both models, which are also provided and are of independent interest. In the PPP model, we show that the maximum likelihood estimator is unbiased with minimal variance among all unbiased estimators. Finally, we sketch ideas for a proof of the CLT for the estimator in the PPP model in multidimensional case and provide illustrative simulations.

Item Type:	Thesis (PhD)
Additional Information:	© 2020 Alexandra Tsimbalyuk
Library of Congress subject classification:	Q Science > QA Mathematics
Sets:	Departments > Statistics
Supervisor:	Kardaras, Kostas
URI:	http://etheses.lse.ac.uk/id/eprint/4379

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