Optimal Unbiased Estimation for the Power of an Unknown Matrix

Dearborn, Krisztina

Graduate Thesis Or Dissertation

Optimal Unbiased Estimation for the Power of an Unknown Matrix Public Deposited

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Citeable URL: https://scholar.colorado.edu/concern/graduate_thesis_or_dissertations/qv33rz143

Abstract

Consider estimating P^t ; as in a d × d matrix P raised to a known positive integer power t, given k independent observations of the matrix. In this thesis, we demonstrate the existence of an optimal estimator for P^t in the family of unbiased estimators that are symmetric in k independent and identically distributed (iid) multivariate Gaussian observations of P.

Inspired by large sample applications, Kuznetsov and Orlov defined families of unbiased estimators for any t, k ≥ 2 from iid multivariate Gaussian observations of P. Smith later developed an alternative formula for an unbiased estimator under the same assumptions. In the scalar case, Smith showed this estimator is the UMVUE by expressing it as an unbiased function of sufficient and complete statistics. As such the optimality of Smith’s matrix valued formula was conjectured. Our results establish that Smith’s estimator is indeed optimal in the matrix setting.

Minimum variance is not applicable in our setting, so we study the high-dimensional analogue. That is, for any initial vector, the expected squared prediction error for the state of the system at time t. This criterion for comparison is motivated by robust forecasting in Markov models. For any t and k with 2 ≤ k ≤ t, Smith’s estimator universally achieves the global minimum for this quantity. And, when k = t and we add a regularity condition on the distribution, it is unique in this quality.

Curiously, Smith’s estimator is not always a member of a family defined by Kuznetsov and Orlov. As such we define a quadratic program constrained to a broader family of candidate estimators. We construct candidate estimators as homogeneous degree t polynomials which are symmetric in k iid multivariate Gaussian observations of P. This family of estimators for P^t are fully characterized by a linear system that the coefficients of the polynomial must satisfy. Reduced representations of the expected squared prediction error over the family of estimators are introduced to illustrate that first and second order conditions for the quadratic program are satisfied.

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