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Current Research

Unbiased Estimation of Powers of Matrices

Advisor Dr. Sergei Kuznetsov

Beginning with an unbiased estimate for a matrix, P, the power of that estimate in turn produces a biased estimate for the power of the original matrix. Kuznetsov and Orlov proved the existence of a rich family of unbiased estimates for powers of matrices. Smith and Kuznetsov worked on extending this work to see if there was a way to demonstrate the existence of a best estimate. A new family of unbiased estimators was constructed. Smith showed that this new family induces an UMVUE in the scalar case. I am interested in consider higher dimensional cases and classifying the merits of using the new family of estimates over previous.

Past Research

The Identifiable Elicitation Complexity of the Mode is Infinite

Faculty Supervisor Dr. Rafael Frongillo
Year Summer 2017
Accepted for Publication in AISM On the indirect elicitability of the mode and modal interval

Scoring functions are commonly used to evaluate a point forecast of a particular statistical functional. This scoring function should be calibrated, meaning the correct value of the functional is the Bayes optimal forecast, in which case we say the scoring function elicits the functional. Heinrich showed that the mode is not elicitable, raising the question of the ability to do so indirectly. Our results show that it is impossible to develop a consistent loss function for evaluating point forecasts of the mode, even indirectly. The argument can also be applied to the mode interval. Moreover, they cast doubt on the existence of broadly effective empirical risk minimization schemes for estimating the mode or modal interval.

Investigating the Behavior of the Roots and Critical Values of Random Polynomials

Faculty Supervisor Dr. Sean O'Rourke
Year Summer 2016

We sought to understand a main result in Critical points of random polynomials with independent identically distributed roots by Kabluchko. Kabluchko proved that given a random polynomial P_n with n iid random roots possessing distribution μ the sequence {μ_n} converges in probability to μ when n tends to infinity. The following figure demonstrates the relationship as n increases.


We then focused our efforts on changing the hypotheses to see under what conditions a similar result still holds. We showed that when dependence amongst the roots is introduced, under some regularity conditions the result still holds.