Nonstationary Gauss-Markov Processes: Parameter Estimation and Dispersion

Abstract

This paper provides a precise error analysis for the maximum likelihood estimate â_(ML)(uⁿ₁) of the parameter a given samples uⁿ₁ = (u₁, ... , u_n)ʹ drawn from a nonstationary Gauss-Markov process U_i = aU_(i-1) + Z)_i, i ≥ 1, where U₀ = 0, a > 1, and Zi 's are independent Gaussian random variables with zero mean and variance σ². We show a tight nonasymptotic exponentially decaying bound on the tail probability of the estimation error. Unlike previous works, our bound is tight already for a sample size of the order of hundreds. We apply the new estimation bound to find the dispersion for lossy compression of nonstationary Gauss-Markov sources. We show that the dispersion is given by the same integral formula that we derived previously for the asymptotically stationary Gauss-Markov sources, i.e., |a|<1 . New ideas in the nonstationary case include separately bounding the maximum eigenvalue (which scales exponentially) and the other eigenvalues (which are bounded by constants that depend only on a) of the covariance matrix of the source sequence, and new techniques in the derivation of our estimation error bound.

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