Specification of initial Kalman recursions of symmetric nonlinear state-space model

A new class of nonlinear Time Series model referred to as Symmetric Nonlinear State-Space Model (SNSSM) was successfully developed using Kalman filter methodology. Some vital properties of the SNSSM such as optimal Kalman gain and optimal filter state covariance were derived. We finally initialized the filter which enabled us obtained the initial Kalman recursions under stationarity and nonstationarity assumptions. Under the former, the mean and variance were obtained unconditionally using Kronecker products and vec operator. But under the later, the mean and variance/covariance of the system were conditionally obtained using a well-known marginal and conditional property of multivariate normal distribution. It is expected that the former will be better than the later if the system is stationary, otherwise the later will be better.