Identification and validation of the statistics of the initial states of linear dynamic systems based on cross-sectional data

This note examines the problem of statistical inference of the initial states of a linear discrete dynamic system based on a set of cross-sectional data. Several compressed data structures are proposed to reduce the amount of the cross-sectional data obtained from multiple independent experiments. It is shown that these data structures are sufficient statistics for estimating the mean and the covariance of the initial states, given the entire raw data from multiple experiments. Thus, the identification and the validation of these parameters can be performed with reduced data structures without referring back to the entire raw data and the original dynamics. For the identification of these parameters, the procedure presented in [1] can be applied to this case. For the validation of these parameters having specified values, simple tests of "significance" type are proposed. The major advantage of these tests over the generalized likelihood ratio test is that their probability distributions are known and computable under both the null and the alternative hypotheses even for the finite sample case, i.e., the asymptotic assumption is not necessary.