2020/10/04 by Wolfgang Scherrer, Scherrer, Wolfgang, Bernd Funovits +1
Mathematics · #FOS: Mathematics #Statistics Theory (math.ST) #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.2010.01598
arxiv created 2020/12/06 · arxiv updated 2020/12/08
We construct rational all-pass matrix functions with real-valued coefficients for mirroring pairs of complex-conjugated determinantal roots of a rational matrix. This problem appears, for example, when proving the spectral factorization theorem, or, more recently, in the literature on possibly non-invertible or possibly non-causal vector autoregressive moving average (VARMA) models. In general, it is not obvious whether the all-pass matrix function (and as a consequence the all-pass transformed rational matrix with initally real-valued coefficients) which mirrors complex-conjugated roots at the unit circle has real-valued coefficients. Naive constructions result in all-pass functions with complex-valued coefficients which implies that the real-valued parameter space (usually relevant for estimation) is left.