Marginal expected shortfall risk measure for time series

Abstract We consider the estimation of the marginal expected shortfall $${\mathbb {E}}\left( X_h | Y_0&gt;U_Y(1/p)\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mfenced> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>h</mml:mi> </mml:msub> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>Y</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>Y</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>p</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math> at extreme levels, when $$((X_t, Y_t))_{t\in {\mathbb {Z}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>Z</mml:mi> </mml:mrow> </mml:msub> </mml:math> is a strictly stationary $$\beta -$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>-</mml:mo> </mml:mrow> </mml:math> mixing time series with marginal distributions of Pareto-type, $$U_Y$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>Y</mml:mi> </mml:msub> </mml:math> is the tail quantile function associated to $$Y_t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> , h is a positive integer and $$p\in (0, 1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is such that $$p\rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . We propose an estimator for this risk measure based on a Weissman-type construction. First, in case of a non-negative time series, we establish the weak convergence of our estimator by using empirical processes arguments combined with the cluster method of Drees and Rootzén (2010). Then, we extend our result to the case of real-valued time series by using the decomposition of the original time series into the positive and negative parts, and we also propose a bootstrap procedure. The performance of our estimator is illustrated on a simulation experiment. Finally, the method is applied on river flow data.