By Hirotugu Akaike (auth.), Jürgen Franke, Wolfgang Härdle, Douglas Martin (eds.)
Classical time sequence tools are according to the belief specific stochastic technique version generates the saw facts. The, most typically used assumption is that the information is a attention of a desk bound Gaussian procedure. even if, because the Gaussian assumption is a reasonably stringent one, this assumption is usually changed through the weaker assumption that the method is wide~sense desk bound and that basically the suggest and covariance series is specific. This procedure of specifying the probabilistic habit purely as much as "second order" has in fact been very hot from a theoretical standpoint be reason it has allowed one to regard a wide number of difficulties, corresponding to prediction, filtering and smoothing, utilizing the geometry of Hilbert areas. whereas the literature abounds with quite a few optimum estimation effects according to both the Gaussian assumption or the specification of second-order houses, time sequence staff haven't regularly believed within the literal fact of both the Gaussian or second-order specifica tion. they've got none-the-less under pressure the significance of such optimali ty effects, most likely for 2 major purposes: First, the consequences come from a wealthy and extremely viable idea. moment, the researchers frequently trusted a imprecise trust in a type of continuity precept based on which the result of time sequence inference might switch just a small volume if the particular version deviated just a small quantity from the assum ed model.
Read Online or Download Robust and Nonlinear Time Series Analysis: Proceedings of a Workshop Organized by the Sonderforschungsbereich 123 “Stochastische Mathematische Modelle”, Heidelberg 1983 PDF
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Additional info for Robust and Nonlinear Time Series Analysis: Proceedings of a Workshop Organized by the Sonderforschungsbereich 123 “Stochastische Mathematische Modelle”, Heidelberg 1983
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R t h . (A) for j ~ 1. ee)(A-0 ), where = r(ee ~ (A) t,~ _0 t,~ for r(h~ee)(A » t,J p+q+l _0 ~ i ~ p+q+2. 1). 1. s. s. 17). W. (1971). "The Statistical Analysis of Time Series" Wiley, New York. J. (1982). "Qualitative Robustness for General Stochastic Processes", Technical Report No. 26, Department of Statistics, University of Washington, Seattle. H. (1981). "Qualitative Robustness for General Processes". Informes de Matematica, Serie B, n!! 002/81, Instituto de Matematica Pur a e Ap1icada, Rio de Janeiro.
The maximization of L~(a) with respect to aE e may now be interpreted as the search for that function fa with a E e that approximates the calculated periodogram best (cp. 1). A brief discussion of the Whittle function and the estimate €N of a obtained by maximizing L~(a) may be found in Dzhaparidze and Yaglom (1982). Let us just remark that eN is asymptotically consistent and efficient in Fishers sense. If the integral in (2) is approximated by a sum over 21TS 0 , ... ,N- 1 t h e same resu 1 ts h 0 ld .
Follows :from the :fact that ¥ u > 0 :s;: e:, takes a unique minimum at c=O. Proo:f. (i). (ii). Ic I such that :for > 0 £ is strictly decreas- h(c,O') :for lul:s;: e: n{u) is strictly increasF{u)-F{-u) > 0 and the :fact that (:from H6). (iii). By the Lebesgue dominated convergence theorem and H3 we have = a* lim h{c,O') 0' ... 0 limh( c ,0') = -a, O'' 'CO uni:formly in a neighborhood o:f uous, there exists unique. > 0 £ and ~ 0' < i h h(c,O') = O. co. nor to We will show Icl:s;: e:, such that i:f ~ 0' ~ 0 and this ~ Then :for any • 0 and /) (iv).
