On the Choice of Norms in System Identification

Convergence analysis of central and minimax algorithms in scalar regressor models

“…The statistical analysis of the central algorithm becomes much simpler when f η t does not vanish at the endpoints of its support denoted by ±ε [4]. We shall assume that for all t, η t has a symmetrical pdf f η with respect to the origin that vanishes outside the open interval (−ε, ε).…”

“…In [4], formal conditions for the distribution of noise that satisfies (5) and gives faster convergence of the parameter estimate variance than O(N −1 ) were developed when the regressor signal equals one all the time. This result was extended in [2] to multi-dimensional persistently exciting regressors under the same noise assumptions.…”

“…In [4], it was also shown that the parameter estimate variance is arbitrarily bad for distributions which are extremely thin at the endpoints of their support. Our objective is to show that this result holds for the model (1) with the central and the minimax algorithms.…”

“…In these studies, disturbances are allowed to be correlated with regressors, which cannot occur in the setting of this paper. The convergence problems of the membership-set (6) and the assessment of the quality of the estimators based on this set, as well as the finite sample properties, have been studied in a stochastic setting with probabilistic assumptions on the noise in [2,[4][5][6][7][8][9][10][11][12][13]27,28].…”

“…The boundedness of these random variables facilitates integral representations for the moments of them in terms of rare event probabilities. These integrals are then used to derive upper and lower bounds on the convergence rate of the variance of the central algorithm for a specific noise pdf studied in [4] and a fairly general class of density functions. In Sect.…”