By Simona Roccioletti
In this booklet Simona Roccioletti studies a number of beneficial reports approximately hazard measures and their houses; specifically she experiences the hot (and seriously mentioned) estate of "Elicitability" of a hazard degree. extra vital, she investigates the difficulty regarding the backtesting of anticipated Shortfall. the most contribution of the paintings is the applying of "Test 1" and "Test 2" constructed by means of Acerbi and Szekely (2014) on various types and for 5 worldwide marketplace indexes.
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Additional resources for Backtesting Value at Risk and Expected Shortfall
Chapter 3. 3 We will now see that they have properties similar to those of quantiles. 8) The following theorem identiﬁes the class of scoring function consistent for expectiles. The attentive reader should notice similarities with both the Bregman and the GPL families. 13. Let F be the class of the probability measures on the interval I ⊆ R with ﬁnite ﬁrst moment, and let β ∈ (0, 1). Then the following holds: (a) The β-expectile functional is elicitable relative to the class F. (b) Suppose that the scoring function S satisﬁes assumptions (1), (2) and (3) on the PO domain D = I × I.
7) where g is a nondecreasing function on I and ✶ denotes the indicator function . e. 1− the conﬁdence level. Chapter 3. Elicitability 37 (c) If g is strictly increasing, the scoring function above is strictly consistent for the α-quantile relative to the class of the probability measure F on I for which EF [g(Y )] exists and is ﬁnite. 7 generalized piecewise linear (GPL) of order α ∈ (0, 1). The motivation behind this name is a mathematical one. 7) is piecewise linear only after applying a nondecreasing transformation.
In the current context, we require that the functional T : F → P(D), F → T (F ) ⊆ D, maps each distribution F ∈ F to a subset T (F )of the domain D ⊆ R. Often T (F ) is single valued, with the set-valued quantile functionals being a leading exemption. We began this section quoting, among others, Murphy and Daan (1985, p. 391), who stressed the importance of identify what consistency for a given functional, means. It is time to give an accurate explanation of this concept. 4 (Consistency of a Scoring Function).