On the distributions of small-sample estimates of second-order AR process parcor coefficients

Up to now, the attempts to give a complete probabilistic description of estimated reflection coefficients have been developed under simplifying assumptions, being these either long data records, allowing us to assume the estimates to be Gaussianly distributed, or white noise signals. On the contrary, the purpose of this paper is to present explicit formulae for the marginal probability density functions of the estimated reflection coefficients of an arbitrary second-order stationary Gaussian autoregressive process; such expressions are consistent with the asymptotic distributions of the estimates, and are empirically verified to be also valid for short records (some tenths of data points). The probabilistic model involves a parametric family of distributions obtained by using a classical result in Statistical Theory; its parameters are then redefined so as to achieve the desired asymptotic behaviour of the first and second order moments of the distributions.