From Parameter Estimation to Dispersion of Nonstationary Gauss-Markov Processes
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This paper provides a precise error analysis for the maximum likelihood estimate â(u) of the parameter a given samples u = (u_1, ..., u_n)^ drawn from a nonstationary Gauss-Markov process U_i = a U_i-1 + Z_i, i≥ 1, where a> 1, U_0 = 0, and Z_i's are independent Gaussian random variables with zero mean and variance σ^2. 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 derived in our previous work dispersionJournal for the (asymptotically) stationary Gauss-Markov sources, i.e., |a| < 1. New ideas in the nonstationary case include a deeper understanding of the scaling of the maximum eigenvalue of the covariance matrix of the source sequence, and new techniques in the derivation of our estimation error bound.
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