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Several models are proposed which are believed to be generic for the estimation of discrete-time axial processes. By introducing axial exponential Fourier densities and axial exponential trigonometric densities, finite-dimensional recursive schemes are obtained for updating the conditional density functions. The underlying idea is the closure properties under the Bayes rule of the various combinations of these exponential densities. An estimation error criterion is introduced which is compatible with a Riemannian metric. The corresponding Optimal axial estimates be easily computed from the conditional probability distributions.