Nonstationary Gauss-Markov Processes: Parameter Estimation and Dispersion

Abstract

This paper provides a precise error analysis for the maximum likelihood estimate ${\hat{a}}_{\text{ML}}(u_1^n)$ of the parameter $a$ given samples $u_1^n=(u_1,\dots ,u_n{)}^{\prime }$ drawn from a nonstationary Gauss-Markov process $U_i=a{U}_{i-1}+Z_i,i\ge 1$, where $U_0=0$, $a>1$, and $Z_i$'s are independent Gaussian random variables with zero mean and variance ${\sigma }^2$. 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 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.

25 pages, 5 figures