<p>We consider the estimation of the marginal expected shortfall <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\mathbb {E}}\left( X_h | Y_0>U_Y(1/p)\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">E</mi> <mfenced close=")" open="("> <msub> <mi>X</mi> <mi>h</mi> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>Y</mi> <mn>0</mn> </msub> <mo>></mo> <msub> <mi>U</mi> <mi>Y</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> </mrow> </math></EquationSource> </InlineEquation> at extreme levels, when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$((X_t, Y_t))_{t\in {\mathbb {Z}}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mi>t</mi> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mi>t</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>t</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is a strictly stationary <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\beta -$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>mixing time series with marginal distributions of Pareto-type, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$U_Y$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>U</mi> <mi>Y</mi> </msub> </math></EquationSource> </InlineEquation> is the tail quantile function associated to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$Y_t$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Y</mi> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation>, <i>h</i> is a positive integer and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$p\in (0, 1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is such that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p\rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. 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.</p>

Marginal expected shortfall risk measure for time series

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2025年11月3日

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Yuri Goegebeur,
Armelle Guillou,
Jing Qin

摘要

We consider the estimation of the marginal expected shortfall ${\mathbb {E}}\left( X_h | Y_0>U_Y(1/p)\right) $ $E (X_{h} | Y_{0} > U_{Y} (1 / p))$ at extreme levels, when $((X_t, Y_t))_{t\in {\mathbb {Z}}}$ ${((X_{t}, Y_{t}))}_{t \in Z}$ is a strictly stationary $\beta -$ $β -$ mixing time series with marginal distributions of Pareto-type, $U_Y$ $U_{Y}$ is the tail quantile function associated to $Y_t$ $Y_{t}$ , h is a positive integer and $p\in (0, 1)$ $p \in (0, 1)$ is such that $p\rightarrow 0$ $p \to 0$ . 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.