Quantitative Statistical Robustness for Tail-Dependent Law Invariant Risk Measures

Quantitative Statistical Robustness for Tail-Dependent Law Invariant Risk Measures

\footnotetext[2]{School of Business, University of Southampton, Southampton, SO17 1BJ, UK ([email removed]).} \footnotetext[3]{Department of Systems Engineering & Engineering Management, The Chinese University of Hong Kong, Hong Kong ([email removed]), Shatin, N. T., Hong Kong.} \footnotetext[4]{School of Business, University of Southampton, Southampton, SO17 1BJ, UK ([email removed]).}

Keywords. Quantitative robustness, tail-dependent law invariant risk measures, Fortet-Mourier metric, admissible laws, index of quantitative robustness.

Introduction

One of the main purposes of quantitative modeling in finance is to quantify the loss of a financial portfolio. Over the past two decades, various risk measures have been proposed for measuring the risk of financial portfolios. A risk measure is represented as a map assigning an extended real number (a measure of risk) to each random loss under an implicit assumption that the true loss probability distribution is known. However, in practice, the true probability distribution is often unknown or it is prohibitively expensive to calculate the risk using the true distribution. Thus, in applications, evaluating the risk of a random variable representing the loss of a financial position often involves two steps: estimating the probability distribution from available observations or from the sampling data of the random financial loss via, e.g., Monte Carlo method and then plugging the estimated distribution into a risk measure to quantify the financial loss. This is because the risk measures are mostly law invariant, that is, they are determined only by the probability distributions of random variables. For the loss of a financial portfolio, a measure of risk computed based on the estimated distribution is known as a plug-in estimate for the risk measure zahle2011rates.

Let $X$ denote the random loss of a financial portfolio on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $\rho$ be a law invariant risk measure. The plug-in estimate for $\rho(X)$ is given by $\varrho(\widehat{P})$, where $\widehat{P}$ is the empirical distribution based on available observations and $\varrho$ is a risk functional defined by

see e.g. belomestny2012central,beutner2010modified. In the literature, Cont et al. cont2010robustness first study the quality of statistical estimators of the law invariant risk measures using Hampel's classical concept of qualitative robustness hampel1971general, that is, a risk functional estimator is said to be qualitatively robust if it is insensitive to the variation of the sampling data. The research is important because perceived data (particularly empirical data) may contain some noise. Without such insensitivity, financial activities based on the risk measures may cause damage. For instance, when $\rho(X)$ is applied to allocate the risk capital for an insurance company, altering the capital allocation may be costly. According to Hampel's theorem, Cont et al. cont2010robustness demonstrate that the qualitative robustness of a statistical estimator is equivalent to the weak continuity of the risk functional, and that value at risk (VaR) is qualitatively robust whereas conditional value at risk (CVaR) is not.

Kr\"atschmer et al kratschmer2012qualitative argue that the use of Hampel's classical concept of qualitative robustness may be problematic because it requires the risk measure essentially to be insensitive with respect to the tail behaviour of the random variable and the recent financial crisis shows that a faulty estimate of tail behaviour can lead to a drastic underestimation of the risk. Consequently, they propose a refined notion of qualitative robustness that applies also to tail-dependent statistical functionals and that allows us to compare statistical functionals in regards to their degree of robustness. The new concept captures the trade-off between robustness and sensitivity and can be quantified by an index of qualitative robustness. Furthermore, under the new concept, Kr\"atschmer et al kratschmer2014comparative analyze the qualitative robustness to the law-invariant convex risk measure on Orlicz spaces and show that CVaR and spectral risk measures are all qualitatively robust when the perturbation of probability distribution is restricted to a finer topological space. Alternative generalizations of Hampel's theorem can be found for strong mixing data (Z\"ahle zahle2014qualitative,zahle2015qualitative) and for stochastic processes in various ways (Boente et al boente1987qualitative and Strohriegl and Hable strohriegl2016qualitative). For comprehensive study of statistical robustness, we refer readers to huber2011robust,zahle2016definition,hampel2011robust,maronna2019robust and references therein.

In this paper, we take a step further by deriving an error bound for the plug-in estimators of law invariant risk measures in terms of the variation of data and we call the analysis quantitative because no such error bound is established in the existing qualitative robust analysis. This is achieved by adopting different metrics to measure the discrepancy of the estimators and the variation of data. Specifically, we use the Fortet-Mourier metrics as opposed to the L\'evy distance in Cont et al. cont2010robustness or the weighted Kolmogorov metric in Kr\"atschmer et al. kratschmer2012qualitative to quantify the data variation (the perturbation of the true probability distributions). Moreover, we introduce a new notion of the so-called {\em admissible laws}, which effectively restrict the scope of data variation. The new metrics enable us to establish an explicit relationship between the discrepancy of the laws of the plug-in estimators (of law invariant risk measure based on the true data and perturbed data) and the discrepancy of the associated probability distributions of the data. The research is inspired by the recent work of Guo and Xu GuX2020 where the authors derive quantitative statistical robustness for preference robust optimization models under Kantorovich metric. The main contributions of the paper can be summarized as follows.

First, we introduce the notion of admissible laws induced by a probability metric, which is a class of probability distributions whose discrepancy with the law of the Dirac measure at $0$ is finite. The admissibility effectively restricts the scope of data perturbation. Using the notion, we compare the admissibility under $\phi$-topology and the Fortet-Mourier metric.

Second, we propose to use the Fortet-Mourier metric to quantify the variation of the probability measure. The metric enables us to establish an explicit relationship between the discrepancy of the laws of the plug-in estimators of law invariant risk measure based on the true data and perturbed data by noise and the change of the true underlying probability measures when the risk functional is Lipschitz continuous on a class of admissible laws. We find that the risk functionals associated with the general moment-type convex risk measures are Lipschitz continuous.

Third, we introduce the concept of Lipschitz continuity for a general statistical functional on a class of admissible laws induced by the Fortet-Morier metric and find that for the Lipschitz continuous risk measure, the parameter of the Fortet-Mourier metric allows us to compare the tail-dependent risk measures with regard to their degree of robustness, i.e., the index of statistical robustness.

Fourth, we apply the new approach to examine the quantitative statistical robustness of a range of well known risk measures, including CVaR, optimized certainty equivalent, shortfall risk measure and conclude that under mild conditions, they are all quantitatively robust, and the indexes of quantitative robustness to them are also calculated.

The rest of the paper is organized as follows. In Section 2, we set up the background of the problem for research. In Section 3, we introduce the concept of Fortet-Mourier metric and admissible laws. In section 4, we establish the quantitative statistical robustness theory and compare with the qualitative statistical robustness theory. In section 5, we apply our theory to risk measures and give some examples. Some technical details are given in the appendix.

Problem statement

In this section, we discuss the background of statistical robustness in the context of law invariant risk measures. We begin by a brief review of law invariant risk measures and its estimation, and then move to explain the issues when the data may contain noise.

Let $(\Omega,\mathcal{F},\mathbb{P})$ be an atomless probability space, where $\Omega$ is a sample space with sigma algebra $\mathcal{F}$ and $\mathbb{P}$ is a probability measure. Let $X: (\Omega,\mathcal{F},\mathbb{P}) \rightarrow {\rm I\!R}$ be a financial loss and $F_X(x):=\mathbb{P}(X\leq x)$ be the law or the probability distribution of $X$. For $p\geq 1$, let $\mathscr{L}^p(\Omega,{\cal F},\mathbb{P})$ ($\mathscr{L}^p$ for short) denote the space of random variables mapping from $(\Omega,{\cal F},\mathbb{P})$ to ${\rm I\!R}$ with finite $p$-th order moments. We say that a map $\rho: \mathscr{L}^1 \to \overline{{\rm I\!R}}:={\rm I\!R}\cup \{+\infty\}$ is a convex risk measure\footnote{We note that the canonical model space for law invariant convex risk measure is $\mathscr{L}^1$ filipovic2012canonical.} follmer2002convex if it satisfies the following properties:

Moreover, if $\rho$ satisfies positive homogeneity, i.e., for any $\alpha \geq 0$, $\rho(\alpha X)= \alpha \rho(X)$, then $\rho$ is a coherent risk measure, see artzner1999coherent,follmer2002convex for the original definitions of these concepts. A risk measure $\rho$ is said to be law invariant if $\rho(X)=\rho(Y)$ for $X$ and $Y$ having the same law. We refer readers to F\"ollmer and Weber follmer2015axiomatic for a recent overview of risk measures.

As discussed in cont2010robustness,kratschmer2012qualitative, it is a widely-accepted procedure to estimate the risk of a financial loss by means of a Monte Carlo method or from a set of available observations. Such a procedure is particularly sensible when $\rho$ is law invariant. The following proposition states that the law invariance of a risk measure $\rho$ is equivalent to the existence of a risk functional $\varrho$ in ((ref)).

The result is well-known, see for instance Delage et al. delage2019dice for random variables defined in $\mathscr{L}^\infty$. The usefulness of the representation is that it naturally captures the law invariance and allows one to define any law invariant risk measure directly over the space of probability measures $\mathscr{P}({\rm I\!R})$ induced by random variables in $\mathscr{L}^\infty$ (also known as probability distributions), see Fritelli et al. frittelli2014risk. Dentcheva and Ruszczy\'nski dentcheva2014risk take it further to define a class of law invariant risk measures in the space of quantile functions directly. In a more recent development, Haskell et al. haskell2018preference extend the research to a broad class of multi-attribute choice functions defined over the space of survival functions. Let $P :=\mathbb{P}\circ X^{-1}$ be the push-forward probability measure on ${\rm I\!R}$ induced by $X$. Since $\mathbb{P}(X\leq x)$ coincides with $P((-\infty,x])$ ($P(x)$ for short), we also call $P$ the distribution or the law of $X$ interchangeably throughout the paper. Consequently, we can write ((ref)) as ((ref)).

In this paper, we are not concerned with the definition of risk measures over the space of probability distributions or the space of quantile functions, rather we concentrate on the stability of statistical estimators of law invariant risk measures. The risk functional $\varrho(P)$ with the law $P=\mathbb{P}\circ X^{-1}$ can be used in a natural way to construct an estimator for the risk $\rho(X)$ of $X\in \mathscr{L}^1$. All one needs to do is to take an estimate $P_N$ of $P$ based on the available observations of $X$ and then to plug this estimator into the risk functional $\varrho$ to obtain the desired estimator of $\rho(X)$, i.e.,

where in this paper, $P_N$ can be seen as the empirical distribution of an independent and identically distributed (i.i.d., for short) sequence $\xi^1,\xi^2,\ldots,\xi^N$ of historical observations or Monte Carlo simulations, i.e.,

Here and later on $\mathbf{1}_{A}$ denotes the indicator function of event $A$. Indeed, $P_N$ can be a fairly general estimates, for instance, $P_N$ can be a smoothed empirical distribution based on uncensored data or empirical distribution based on censored data, see, e.g., zahle2011rates or empirical distribution based on identically distributed dependent data, see, e.g., zahle2015qualitative.

We can see that $\widehat{\varrho}_N$ is a mapping from ${\rm I\!R}^N$ to ${\rm I\!R}$. Figure (ref) illustrates the relationship between the risk functionals, their estimators and the spaces associated.

In practice, the samples obtained from empirical data may contain noise. In that case, we might regard the samples as generated by a perturbed random variable $Y$ with law $Q$, that is, $Q=\mathbb{P}\circ Y^{-1}$. Let $\tilde{\xi}^1,\cdots,\tilde{\xi}^N$ be i.i.d samples from $Y$. Then the practical empirical distribution function for estimating the law of $X$ is

and the practical estimator is $\tilde{\varrho}_N=\widehat{\varrho}_N(\tilde{\xi}^1,\cdots,\tilde{\xi}^N):=\varrho(Q_N)$ with perceived empirical data whereas $\widehat{\varrho}_N$ is a statistical estimator with noise being detached. Since we are unable to obtain the latter, we tend to use the former as a statistical estimator of $\rho(X)$ and this works only if the two estimators are sufficiently close.

To quantify the closeness, we may look into the discrepancy between the laws of the two estimators under some metric $\mathsf {d\kern -0.07em l}$, i.e.,

where $P^{\otimes N}$ and $Q^{\otimes N}$ denote the probability measures on measurable space $\left({\rm I\!R}^N,{\cal B}({\rm I\!R})^{\otimes N}\right)$ with marginals $P$ and $Q$ on each $({\rm I\!R}, {\cal B}({\rm I\!R}))$ respectively, $\mathcal B({\rm I\!R})$ denotes the corresponding Borel sigma algebra of ${\rm I\!R}$. Since neither $P$ nor $Q$ is known, we want the discrepancy to be uniformly small for all $P$ and $Q$ over a subset of admissible laws on $\mathscr{P}({\rm I\!R})$ so long as $Q$ is sufficiently close to $P$ under some metric $\mathsf {d\kern -0.07em l}'$. The uniformity may be interpreted as robustness. {\em Qualitative robustness} refers to the case that the relationship between $\mathsf {d\kern -0.07em l}\left(P^{\otimes N}\circ \widehat{\varrho}_N^{-1}, Q^{\otimes N}\circ \tilde{\varrho}_N^{-1}\right)$ and $\mathsf {d\kern -0.07em l}'(P,Q)$ is implicit whereas {\em quantitative robustness} refers to the case that the relationship is explicit, i.e., a function of the latter can be used to bound the former, and this is what we aim to achieve in this paper because qualitative robustness have been well investigated, for instance, in cont2010robustness,kratschmer2014comparative,kratschmer2012qualitative.

$\zeta$-metrics and admissible laws

There are two essential elements in investigating both the qualitative and quantitative statistical robustness of a risk functional: One is the specific choice of probability metrics but not just the topologies generated by them, see, e.g., huber2011robust,cont2010robustness,kratschmer2012qualitative, to quantify the change of the law $P$ and to estimate the discrepancy between the laws of two estimators, i.e., ((ref)); the other is the determination of the subset $\mathscr{M}$ of admissible laws in $\mathscr{P}({\rm I\!R})$ (see, e.g., kratschmer2012qualitative,zahle2015qualitative), containing all empirical distributions: $\mathscr{M}_{1,\mathrm{emp}}\subset \mathscr{M}$, to restrict the perturbation of the law $P$. For instance, the subset $\mathscr{M}$ may be specified via some generalized moment conditions, which are interesting in econometric or financial applications.

To introduce these two essential elements thoroughly, some preliminary notions and results in probability theory and statistics such as $\phi$-weak topology are required. We first give a sketch of them to prepare our discussions in the follow-up sections. Let $\phi:{\rm I\!R}\to [0,\infty)$ be a continuous function and ${\cal M}_1^{\phi} := \left\{P'\in \mathscr{P}({\rm I\!R}): \int_{{\rm I\!R}} \phi(t)P'(dt)<\infty\right\}$. In the particular case when $\phi(\cdot):=|\cdot|^p$ and $p$ is a positive number, write ${\cal M}_1^p$ for ${\cal M}_1^{|\cdot|^p}$. Note that ${\cal M}_1^\phi $ defines a subset of probability measures in $\mathscr{P}({\rm I\!R})$ which satisfies the generalized moment condition of $\phi$. From the definition, we can see that ${\cal M}_1^{p_2} \subset {\cal M}_1^{p_1}$ for any positive numbers $p_1,p_2$ with $p_1<p_2$ due to H\"older inequality.

Clearly, $\phi$-weak topology is finer than the weak topology, and the two topologies coincide if and only if $\phi$ is bounded. It is well known (see kratschmer2012qualitative) that $\phi$-weak convergence is equivalent to weak convergence, denoted by ${P_l} \xrightarrow[]{w} P$, together with $\int_{{\rm I\!R}} \phi(t)P_l(dt) \to \int_{{\rm I\!R}} \phi(t)P(dt)$. Moreover, it follows by kratschmer2012qualitative,kratschmer2014comparative that the $\phi$-weak topology on ${\cal M}_1^\phi$ is generated by the metric $\mathsf {d\kern -0.07em l}_\phi:{\cal M}_1^\phi\times {\cal M}_1^\phi\to {\rm I\!R}$ defined by

for $P, Q \in {\cal M}_1^{\phi}$, where $\mathsf {d\kern -0.07em l}_{\mathrm{Prok}}: \mathscr{P}({\rm I\!R})\times \mathscr{P}({\rm I\!R})\to {\rm I\!R}_+$ is the Prokhorov metric defined by

where $A^\epsilon:= A + B_\epsilon(0)$ denotes the Minkowski sum of $A$ and the open ball centred at $0$ on ${\rm I\!R}$ and $\mathcal B({\rm I\!R})$ is the corresponding Borel sigma algebra on ${\rm I\!R}$. We note that the Prokhorov metric metrized the weak topology on ${\rm I\!R}$ see, e.g., gibbs2002choosing.

$\zeta$-metrics

Instead of exploiting the widely-used probability metrics such as the Prokhorov metric and the weighted Kolmogorov metric in the literature of qualitative robustness cont2010robustness,kratschmer2012qualitative, we will switch to the so-called metrics with $\zeta$-structure to establish the quantitative statistical robustness framework for a risk functional. In particular, we will use the well-known Kantorovich metric and Fortet-Mourier metrics. The new metrics enable us to establish an explicit relationship between the discrepancy of the laws of the plug-in estimators of law invariant risk measures based on the true data and perturbed data with noise and the discrepancy of the associated true probability measure. We begin with a formal definition of $\zeta$-metrics and then clarify the relationships between metrics of $\zeta$-structure and those used in mizera2010qualitative,cont2010robustness,kratschmer2012qualitative.

From the definition, we can see that $\mathsf {d\kern -0.07em l}_{\mathcal{F}}(P,Q)$ is the maximum difference of the expected values of the class of measurable functions $\mathcal{F}$ with respect to $P$ and $Q$. $\zeta$-metrics are widely used in the stability analysis of stochastic programming, see R\"omisch romisch2003stability for an excellent overview. The specific metrics with $\zeta$-structure that we consider in this paper are the Kantorovich metric and the Fortet-Mourier metric. The next definition gives a precise description of the two notions.

From the definition, we can see that for any positive numbers $p\geq p'\geq 1$,

which means that $\mathsf {d\kern -0.07em l}_{FM,p}(P,Q)$ becomes tighter as $p$ increases and they are all tighter than $\mathsf {d\kern -0.07em l}_{K}(P,Q)$. Moreover, the Fortet–Mourier metric metricizes weak convergence on sets of probability measures possessing uniformly a $p$-th moment pflug2011approximations. Notice that the function $t\rightarrow \frac{1}{p}|t|^p$ for $t\in {\rm I\!R}$ belongs to $\mathcal{F}_p({\rm I\!R})$. On ${\rm I\!R}$, the Fortet–Mourier metric may be equivalently written as

see, e.g., rachev1991probability.

In the next example, we illustrate the relationship between the existing probability metrics used in statistical robustness and the metrics with $\zeta$-structure.

Admissible laws

We now turn to discuss another important component in statistical robust analysis, that is, the subset $\mathscr{M}$ of admissible laws in $\mathscr{P}({\rm I\!R})$ which describes the scope of the perturbation of the law $P$ by a metric. This can be motivated by ensuring the finiteness of $\mathsf {d\kern -0.07em l}(P,Q)$. To this effect, we formally introduce the concept of admissible laws induced by probability metrics.

Let $\mathscr{P}_p({\rm I\!R})$ denote the admissible laws induced by the Fortet-Mourier metrics with parameter $p$ on $\mathscr{P}({\rm I\!R})$. By Definition (ref), we have

By triangle inequality, this ensures $\mathsf {d\kern -0.07em l}_{FM,p}(P,Q)<+\infty$ for any $P,Q\in \mathscr{P}_p({\rm I\!R})$.

In the following example, we compare the admissible laws induced by different probability metrics.

Relationship with $\phi$-weak topology

Since $\phi$-weak topology has been widely used for qualitative robust analysis in the literature whereas we use the topology induced by the Fortet-Mourier metrics for quantitative robust analysis, it would therefore be helpful to look into potential connections of the two apparently completely different metrics. In the next proposition, we look into such connection from admissible set perspective (which defines the space of probability measures that $P$ is perturbed in both qualitative and quantitative robust analysis), we find that $\mathscr{P}_p({\rm I\!R})$ coincides with ${\cal M}_1^\phi$ for some specific choice of $\phi$ and subsequently show that the Fortet-Mourier metric is tighter than $\mathsf {d\kern -0.07em l}_\phi$.

Part (i) of the proposition says that the admissible set $\mathscr{P}_p({\rm I\!R})$ coincides with the set of laws on ${\rm I\!R}$ satisfying the generalized moment condition of $\phi_p$. Part (ii) indicates that $\mathsf {d\kern -0.07em l}_{FM,p}$ is tighter than $\mathsf {d\kern -0.07em l}_{\phi_p}$. Part (iii) means that the $\phi_p$-weak topology on $\mathscr{P}_p({\rm I\!R})$ is generated by the metric $\mathsf {d\kern -0.07em l}_{FM,p}$.

\noindentProof. Part (i). Since for any $p\geq 1$, $\frac{1}{p}\phi_p\in \mathcal{F}_p({\rm I\!R})$, then by the definition of $\mathscr{P}_p({\rm I\!R})$, we have that $P\in \mathscr{P}_p({\rm I\!R})$ implies $P\in \mathcal{M}_1^{\phi_p}$ and subsequently, $\mathscr{P}_p({\rm I\!R})\subset \mathcal{M}_{1}^{\phi_p}$.

On the other hand, let $P\in \mathcal{M}_{1}^{\phi_p}$, then $\int_{{\rm I\!R}} \phi_p(\xi)P(d\xi)<\infty$. For any $\psi \in \mathcal{F}_{p}({\rm I\!R})$, we have

and consequently,

Therefore, we have

and consequently, $\mathcal{M}_1^{\phi_p}\subset \mathscr{P}_p({\rm I\!R})$.

Part (ii). Since $\frac{1}{p}\phi_p\in \mathcal{F}_p({\rm I\!R})$, then for any $P,Q\in \mathscr{P}_p({\rm I\!R})$,

From Example (ref)(i) and ((ref)), we have $\mathsf {d\kern -0.07em l}_{\mathrm{Prok}}(P,Q)\leq \sqrt{\mathsf {d\kern -0.07em l}_{FM,p}(P,Q)}$. Finally, by the definition of $\mathsf {d\kern -0.07em l}_{\phi_p}$, i.e., ((ref)), we obtain the conclusion.

Part (iii) follows straightforwardly from Part (ii). $\hfill \rule{2.3mm}{2.3mm}$

Proposition (ref) indicates that despite Fortet-Mourier metric $\mathsf {d\kern -0.07em l}_{FM,p}$ and $\mathsf {d\kern -0.07em l}_{\phi_p}$ are different metrics, they generate the same topology, which confirms the statement at the beginning of this section, i.e., for the qualitative robustness and the quantitative robustness, the specific choice of probability metrics matters but not the topologies generated by them. To conclude this section, we remark that the subset $\mathscr{M}$ to be used in the definition of qualitative robust analysis will be confined to the set of admissible laws when we adopt the Fortet-Mourier metric for quantitative robust analysis in the next section.

Statistical robustness

We are now ready to return our discussions to the robustness of statistical estimators of law invariant risk measures that are outlined in Section 2.

Qualitative statistical robustness

To position our research properly, we begin by a brief overview of the existing results about the qualitative statistical robustness.

The definition above captures two versions of qualitative statistical robustness proposed by Cont et al. cont2010robustness for i.i.d. observations on ${\rm I\!R}$ with $\mathsf {d\kern -0.07em l}$ and $\mathsf {d\kern -0.07em l}'$ being L\'evy distance and Kr\"atchmer et al. kratschmer2012qualitative for i.i.d. observations on ${\rm I\!R}$ with $\mathsf {d\kern -0.07em l}=\mathsf {d\kern -0.07em l}_{(\phi)}$ and $\mathsf {d\kern -0.07em l}'=\mathsf {d\kern -0.07em l}_{\mathrm{Prok}}$ respectively. Since $\mathsf {d\kern -0.07em l}_{\mathrm{Prok}}$ is tighter than $\mathsf {d\kern -0.07em l}_{\mathrm{L\acute{e}vy}}$, it means Kr\"atchmer et al. kratschmer2012qualitative examines the discrepancy of the laws with a tighter metric. On the other hand, from the definition of $\mathsf {d\kern -0.07em l}_{(\phi)}$, we can see that it is also tighter than $\mathsf {d\kern -0.07em l}_{\mathrm{L\acute{e}vy}}$ and allows one to capture the difference of distributions at the tail, it means the robust analysis in Kr\"atchmer et al. kratschmer2012qualitative is restricted to a smaller class of probability distributions when $Q$ is perturbed from $P$. This explains why CVaR is robust under the criterion of the latter but not the former.

A key result that Kr\"atchmer et al. kratschmer2012qualitative establish is the Hampel's theorem which states the equivalence between qualitative statistical robustness and stability/continuity of a risk functional (with respect to perturbation of the probability distribution) under uniform Glivenko-Cantelli (UGC) property of empirical distributions over a specified set.

The UGC property means that convergence in probability of the empirical probability measure to the true marginal distribution uniformly in $\mathscr{C}$ on $\mathscr{P}({\rm I\!R})$. Examples for metrics spaces $(\mathscr{C},\mathsf {d\kern -0.07em l})$ having the UGC property can be found in kratschmer2012qualitative. In particular, it is shown that there exists a subset of the admissible laws induced by the weigthed Kolmogorov metric enjoys the UGC property, see kratschmer2012qualitative.

Quantitative statistical robustness

We now move on to discuss our central topic, quantitative statistical robustness for the plug-in estimators of law invariant risk measures. Intuitively speaking, quantitative statistical robustness of a risk functional $\varrho$ means that for any two admissible laws $P$ and $Q$ on $\mathscr{P}({\rm I\!R})$, the distance between the laws of their plug-in estimators $\varrho(P_N)$ and $\varrho(Q_N)$ is bounded by the distance between $P$ and $Q$ when the sample size is sufficiently large.

In comparison with the qualitative statistical robustness introduced by Kr\"atchmer et al. kratschmer2014comparative or Cont et al. cont2010robustness, the definition ((ref)) here has several advantages. First, we use Kantorovich metric instead of Prokhorov metric to quantify the discrepancy between $P^{\otimes N}\circ \widehat{\varrho}_N^{-1}$ and $Q^{\otimes N}\circ \widehat{\varrho}_N^{-1}$. This enables us to capture the tail behaviour of the two laws and facilitate us to derive an explicit bound for the difference. Second, we use the Fortet-Mourier metric to quantify the perturbation of $P$, which is more sensitive than the L\'evy metric used in cont2010robustness and the weighted Kolmogorov metric in kratschmer2014comparative,zahle2014qualitative,zahle2015qualitative,zahle2016definition to the variation of the tails. Third, inequality ((ref)) gives an error bound for the discrepancy of the two laws and the bound is valid for all $Q$ in $\mathscr{M}$ instead of those in a neighborhood of $P$.

Next, we introduce a definition on the Lipschitz continuity of a general statistical mapping from $\mathscr{P}({\rm I\!R})$ to ${\rm I\!R}$, which strengthens the earlier definition of $\mathscr{C}$-continuity for a general statistical functional.

There are a few of points to note to the above definition of Lipschitz continuity:

Before presenting a proof, it might be helpful for us to explain why we consider a specific set of functions $\Psi$. For fixed $N\in \mathbb{N}$, let $\mathscr{M}_{1,\mathrm{emp}}^N$ denote the set of all empirical laws $P_{N}$ over ${\rm I\!R}$, then $\mathscr{M}_{1,\mathrm{emp}}=\bigcup_{N\in \mathbb{N}}\mathscr{M}_{1,\mathrm{emp}}^N$. Then $\Psi$ may be regarded as a set of functions derived from a class of Lipschitz continuous functional on $\mathscr{M}_{1,\mathrm{emp}}^N$ with $L=1$ and $\mathsf {d\kern -0.07em l}=\mathsf {d\kern -0.07em l}_{FM,p}$ (by writing $T(P_N)$ as a function of samples). Lemma (ref) says that for any $N\in \mathbb{N}$, the discrepancy between $P^{\otimes N}$ and $Q^{\otimes N}$ under the metric $\mathsf {d\kern -0.07em l}_{\Psi}$ can be bounded by $\mathsf {d\kern -0.07em l}_{FM,p} (P,Q)$.

\noindentProof. Let $\xi^{-j}:=\{\xi^1,\cdots,\xi^{j-1},\xi^{j+1},\cdots,\xi^N\}$, $\vec{\xi}^j:=\{\xi^1,\cdots,\xi^j\}$ and $\vec{\xi}^{-j}:=\{\xi^{j+1},\cdots,\xi^N\}$. For any $P_1,\cdots,P_N \in \mathscr{P}({\rm I\!R})$ and any $j\in \{1,\cdots,N\}$, denote \[ P_{-j}(d \xi^{-j}):=P_1(d \xi^1)\cdots P_{j-1}(d \xi^{j-1})P_{j+1}(d \xi^{j+1})\cdots P_N(d \xi^N) \] and \[ h_{{\xi}^{-j}}({\xi}^j):=\int_{{\rm I\!R}^{(N-1)}} \psi({\xi}^{-j},\xi^j) P_{-j}(d \xi^{-j}). \] Then

Let $\mathcal{H}$ denote the set of functions $h_{{\xi}^{-j}}({\xi}^j)$ generated by $\psi\in \Psi$. By the definition of $\mathsf {d\kern -0.07em l}_{\Psi}$ and the $p$-th order Forter-Mourier metric,

where the inequality is due to $N h_{\xi^{-j}}(\xi^j)\in \mathcal{F}_{p}({\rm I\!R})$ and the definition of $\mathsf {d\kern -0.07em l}_{FM,p}(P,Q)$. Finally, by the triangle inequality of the pseudo-metric, we have

The proof is complete. $\hfill \rule{2.3mm}{2.3mm}$

With the intermediate technical result, we are now ready to present our main result of quantitative statistical robustness for the plug-in estimator of a general risk functional.

\noindentProof. Since the underlying probability space is atomless, then for any $N\in \mathbb{N}$, by definition

where we write $\vec{\xi}^N$ for $(\xi^1,\cdots,\xi^N)$ and $\varrho(\vec{\xi}^N)$ for $\widehat{\varrho}_N$ to indicate its dependence on $\xi^1,\cdots,\xi^N$.

For any $\psi\in \mathcal{F}_1({\rm I\!R})$, ((ref)) ensures that

which means that $\psi(\varrho(\cdot))$ is locally Lipschitz continuous in $\vec{\xi}^N$, i.e., $\psi(\varrho(\cdot))\in \mathcal{F}_p(({\rm I\!R}^N)$ from ((ref)). Since $P, Q\in \mathscr{M}\subset \mathscr{P}_p({\rm I\!R})\subset \mathscr{P}_{K}({\rm I\!R})$ (see Example (ref)(i) and Proposition (ref)(i)), then ((ref)) is finite. The rest follows from Lemma (ref) by setting $\psi(\xi^1,\cdots,\xi^N)=\psi(\varrho(\xi^1,\cdots,\xi^N))$.

$\hfill \rule{2.3mm}{2.3mm}$

From Example (ref), we have $\mathsf {d\kern -0.07em l}_{\mathrm{Prok}}(P,Q)\leq \sqrt{\mathsf {d\kern -0.07em l}_{K}(P,Q)}$ for all $P,Q\in \mathscr{P}({\rm I\!R})$, then we have the following corollary.

Next, we take a step further to consider the index of quantitative robustness for a general statistical functional.

This index is a quantitative measurement for the degree of robustness of a statistical functional. A larger index reflects a higher degree of robustness. For a general statistical functional $\varrho$, ((ref)) may hold for uncountable many $p$, see e.g., the $2$-th moment functional $T^{(2)}$ satisfying ((ref)) for any $p\geq 2$ on $\mathscr{P}_p({\rm I\!R})=\mathcal{M}_1^p$. From Definition (ref), we conclude that the $p$-th moment functional $T^{(p)}$ has the index $\mathrm{iqr}(T^{(p)})=\frac{1}{p}$. {\color{black} Definition (ref) coincides with the index of qualitative robustness proposed by Kr\"atschmer et al. kratschmer2012qualitative when $\varrho$ is Lipschitz continuous w.r.t. $\mathsf {d\kern -0.07em l}_{FM,p}$ on $\mathscr{P}_p({\rm I\!R})$. The main advantage of Definition (ref) is that it is easy to calculate and we will illustrate this in the next section. }

Application to risk measures

As we discussed in Proposition (ref), law invariant risk measure of a random variable can be represented as a composition of a risk functional and law of the random variable. In practice, risk of a random variable is often calculated with empirical data, this is because either the true probability distribution is unknown or it might be prohibitively expensive to calculate the risk of a random variable with the true probability distribution. This raises a question as to whether the estimated risk measure based on empirical data is reliable or not. In this section, we apply the quantitative robustness results established in Theorem (ref) to some well-known risk measures. The next proposition synthesizes Proposition (ref) and Theorem (ref).

In what follows, we verify condition ((ref)) for some well-known risk measures and hence show that they satisfy the proposed quantitative statistical robustness ((ref)). To make the notation easily, we introduce the law invariant risk measure on the space of probability distributions.

In all of the above examples, the risk measures can either be represented explicitly in the form of $\int_{\rm I\!R} f(x) dP(x)$ (such as Expectation) or be obtained from solving an optimization problem where the underlying functions are represented in the expected utility form (CVaR, Certainty Equivalent and Shortfall risk measure), this is because the utility (disutility) functions are assumed to be concave (convex) and hence locally Lipschitz continuous. When growth of the Lipschitz modulus is controlled by $c_p(\xi,\xi')$, these risk measures satisfy inequality ((ref)) as we have shown. This may not work for the spectral risk measures acerbi2002spectral with unbounded risk spectrum because the latter distort the probability distribution $P(x)$. However, when the risk spectrum is bounded (such as CVaR which is a special case of spectral risk measure), we can still manage inequality ((ref)). This explains why we haven't included spectral risk measures in the examples.