By Hannu Oja (auth.), Claudia Becker, Roland Fried, Sonja Kuhnt (eds.)

This Festschrift in honour of Ursula Gather’s sixtieth birthday offers with sleek subject matters within the box of strong statistical equipment, in particular for time sequence and regression research, and with statistical equipment for complicated facts buildings. the person contributions of top specialists offer a textbook-style evaluate of the subject, supplemented through present learn effects and questions. The statistical concept and strategies during this quantity objective on the research of information which deviate from classical stringent version assumptions, which include outlying values and/or have a fancy constitution. Written for researchers in addition to grasp and PhD scholars with a great wisdom of data.

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26) 2 Depth Statistics 31 which yields RD z | x 1 , . . , x n = inf D d r, z(t) | r, x 1 (t) , . . , r, x n (t) . 27) A slight extension of the Φ-depth is the principal components depth (Mosler and Polyakova 2012). However, certain approaches from the literature are no Φ-depths. These are mainly of two types. The first type employs random projections of the data: Cuesta-Albertos and Nieto-Reyes (2008b) define the depth of a function as the univariate depth of the function values taken at a randomly chosen argument t.

Xd are random vectors independently distributed as P , co denotes the convex hull, Vd the d-dimensional volume, and SX is defined as above. In particular, we can choose DX = ΣX . The Oja depth satisfies D1 to D5. It is continuous on z and maximum at the Oja median (Oja 1983), which is not unique; see also the contribution by Oja, Chap. 1. The Oja depth determines the distribution uniquely among those measures which have compact support of full dimension. 2 contrasts the projection depth regions with the Oja regions for our debt-unemployment data.

4. 5. The distribution of (R, Θ) is AOR. (Xn∗ )n is stable in probability. For every fixed integer 1 ≤ α ≤ n, (Xn−α+1,n )n is stable in probability. There exists θ1 such that limx→+∞ F¯θ1 (x)/F¯θ1 (x − h) = 0, for all h > 0. For all θ , limx→+∞ F¯θ1 (x)/F¯θ1 (x − h) = 0, for all h > 0. 6. 7. 8. 9. Wn,n − Wn−1,n → 0. (Wn,n )n is stable in probability. For all θ , the distribution Fθ is AOR. There exists θ1 such that the distribution Fθ1 is AOR. P Other characterizations can be found in Barme-Delcroix and Gather (2002).