Improved Confidence Intervals for Expectiles

1. Introduction

Quantile-based inference has been a long-standing interest in the financial industry and risk assessment. A few applications worth mentioning include: the widely used risk measure VaR (value at risk) is a tantamount quantile; coherent risk measures typically represent transformations of quantiles; quantile regression has been used as a tool for portfolio investment decisions.

For a continuous random variable X with a cumulative distribution function $F\left(x\right),$ density function $f\left(x\right)$ and $E\left|X\right|<\infty ,$ the $p-$th quantile is defined as

$$Q\left(p\right)=inf\{x:F(x)\ge p\}.$$

Given a sample $X_1,X_2,\cdots ,X_n,$ from $F,$ the simplest estimator of $Q\left(p\right)$ is the sample quantile. Under mild conditions, it is asymptotically normal, but its asymptotic variance ${\sigma }^2={\displaystyle \frac{p(1-p)}{f^2\left(Q\left(p\right)\right)}}$ is large, particularly in the tails. Hence, it behaves poorly for small sample sizes, and alternative estimators are needed. The kernel quantile estimator

$${\widehat{Q}}_{p,h_n}={\displaystyle \frac{1}{h_n}}{\int }_0^1{F}_n^{-1}\left(x\right)K\left({\displaystyle \frac{x-p}{h_n}}\right)dx,$$

(1)

is an obvious choice. Here $F_n^{-1}\left(x\right)$ is the inverse of the empirical distribution function, $K(.)$ is a suitably chosen kernel, and $h_n$ is a bandwidth. Conditions on bandwidth and kernel must be imposed to ensure consistency, asymptotic normality, and higher-order accuracy of ${\widehat{Q}}_{p,h_n}.$

This type of research has been of considerable interest also to the authors of the current paper. We derived in [12] higher order expansion for the standardised kernel quantile estimator thus extending long-standing flagship results of [6,7]. Our expansion is non-trivial because of the influence of the bandwidth that makes the balance between bias and variance delicate.

We then derived in [13] an Edgeworth expansion for the studentized version of the kernel quantile estimator (where the variance of the estimator is estimated using the jackknife method). Precisely this result is needed for practical applications since the variance is rarely known in practice. The inversion of the Egdeworth expansion delivered a uniform improvement in coverage accuracy compared to the inversion of the asymptotically normal approximation. Our results are applicable to improve inference for quantiles when the sample sizes are small to moderate. This situation often occurs in practice. For example, if monthly loss data is used in risk analysis, then for 10 years, the accumulated data would amount to 120 observations. There are only about 250 trading days on the stock market within a year.

After the global financial crisis, the activities of the bank regulators were directed toward proposing measures of risk that could be an alternative to the VaR. The coherency requirement for a risk measure in finance was first formulated in the seminal paper [1] and has been widely used since then. We note that VaR is not a coherent risk measure mainly because it does not satisfy the subadditivity property. The AVaR (average value at risk) does satisfy the subadditivity and turns out to be a coherent risk measure. It was considered around 2009 as an alternative risk measure by the Basel Committee of Banking Supervision. Meanwhile, academic research on the properties of risk measures continued. The elicitability property was pointed out as another essential requirement in [8]. The latter property is an important requirement associated with effective backtesting. It then turned out that expectiles “have it all" as they are simultaneously coherent and elicitable. Moreover, it was shown (see, for example, [17]) that expectiles are the only law-invariant risk measures that are simultaneously coherent and elicitable.

It was natural for us then to turn to expectiles and to propose methods for improved inference about expectiles for small to moderate sample sizes. The current paper suggests such a methodology and illustrates its effectiveness. The paper is organized as follows. In the next Section 2 we introduce some notations, definitions, and auxiliary statements we need in the sequel. Section 3 presents our main results about the Edgeworth expansion of the asymptotic distribution of the estimator. This section is subdivided into three subsections. The first subsection deals with the standardized kernel-based expectile estimator. The second subsection discusses the related results for the studentized kernel-based expectile estimator. While the first subsection is of theoretical interest mainly, the results of the second subsection can be directly applied to derive more accurate confidence intervals for the expectile of the population for small to moderate samples. The third subsection discusses a Cornish-Fisher-type approximation for the quantile of the kernel-based expectile estimator. The application for accurate confidence interval construction presents the main purpose of our methodology. Its efficiency is illustrated numerically in the next Section 4. Section 5 summarizes our findings. Technical results and proofs of the main statements of the paper are postponed to the Appendix.

2. Materials and Methods

From a methodological standpoint, ${\widehat{Q}}_{p,h_n}$ is an L-estimator: it can be written as a weighted sum of the order statistics $X_{\left(i\right)},i=1,2,\cdots ,n:$

$${\widehat{Q}}_{p,h_n}=\sum _{i=1}^n{v}_{i,n}{X}_{\left(i\right)},v_{i,n}={\displaystyle \frac{1}{h_n}}{\int }_{{\displaystyle \frac{i-1}{n}}}^{{\displaystyle \frac{i}{n}}}K\left({\displaystyle \frac{x-p}{h_n}}\right)dx.$$

(2)

Consider a random variable $X\in {\mathcal{L}}^2(\Omega ,\mathcal{F},P).$ Using the notations $x_{+}=max(x,0),x_{-}=max(-x,0),$ Newey and Powel introduced in [15] the expectile $e_{\tau }$ as the minimizer of the asymmetric quadratic loss

$$e_{\tau }\left(X\right)={\mathrm{argmin}}_{y\in R}{\left[\tau \right||(X-y)}_{+}{\left|\right|}_2^2+{(1-\tau )\left|\right|(X-y)}_{-}{\left|\right|}_2^2],$$

(3)

and we realize that its empirical variant can also be represented as an L-statistic. When $\tau =1/2$ we get $e_{1/2}\left(X\right)=E\left(X\right)$ which means that the expectiles can be interpreted as an asymmetric generalization of the mean. In addition, it has been shown in several papers (see for example [2]) that the so-called expectile-VaR ($EVa{R}_{\tau }=-e_{\tau }\left(X\right)$ is a coherent risk measure when $\tau \in (0,1/2]$ as it satisfies the four coherency axioms from [1]. Any value of $\tau \in (0,1)$ can be used in the definition of the expectile but for the reason mentioned in the previous sentence, we will assume that $\tau \in (0,1/2]$ in the theoretical developments of this paper.

The asymptotic properties of L-statistics are usually discussed by first decomposing them into an U-statistic plus a small-order remainder term and then applying asymptotic theory of the U-statistic. For starters, the L statistic is written as ${\int }_0^1{F}_n^{-1}\left(u\right)J\left(u\right)du$ where $F_n$ is the empirical distribution function and the score function$J\left(u\right)$ does not involve $n.$ For the presentation (2), however, such a decomposition is impossible as the “score function” becomes a delta function in the limit. Therefore, a dedicated approach is needed. For the case of quantiles, details about such an approach are given in [13]. Our current paper shows how the issue can be resolved in the case of expectiles. The main tools in our derivations are some large deviation results on U-statistics from [14].

Remark.

We mention in passing that in the paper [10], it is shown that expectiles of a distribution F are in a one-to-one correspondence to quantiles of another distribution G that is related to F by an explicit formula. It might be tempting to use this relation to utilize our results from [13] in the construction of confidence intervals for the expectiles of $F.$ We have examined this option and realize that the correspondence is quite complicated involving functionals of F that need to be estimated by the data. It turns out that proceeding in this way is not an option to construct precise confidence intervals for the expectiles. Hence, our way to proceed is to deal directly from the very beginning with the definition of the expectile of $F.$

We start with some initial notations, definitions, and auxiliary statements.

We consider a sample $X_1,\dots ,X_n$ of n independently and identically distributed random variables with density and cumulative distribution functions $f,F,$ respectively.

Let us define

$$I_{\tau }(x,y)=\tau (y-x)I(y\ge x)-(1-\tau )(x-y)I(y<x)$$

where $I(·)$ is an indicator function. Looking at the original definition (3) we can define the (true theoretical) expectile $y^{*}$ as a solution to the Equation

$$E\left[I_{\tau }(y^{*},X_1)\right]=0.$$

(4)

Using the relation ${(y-x)}_{+}={(x-y)}_{-}$ we realize that the defining Equation (4) leads to the same solution as the defining Equation (2) in [2] or the defining Equation (2) in [9].

As discussed in [9], the $\tau$-expectile $y^{*}$ satisfies the Equation

$$\tau {\int }_{y^{*}}^{\infty }\{1-F\left(s\right)\}ds-(1-\tau ){\int }_{-\infty }^{y^{*}}F\left(s\right)ds=0$$

(5)

Using integration by parts, we have the following proposition.

Proposition 1.

Assume that $E|X_1|<\infty$. Then we have

$$\tau (\mu -y^{*})-(1-2\tau )F^{\left[2\right]}\left(y^{*}\right)=0$$

(6)

where $\mu =E\left(X_1\right)$ and $F^{\left[2\right]}\left(x\right)={\int }_{-\infty }^{x}F\left(s\right)ds.$Thus an estimator of the expectile is given by a solution of the equation

$$\tau (\overline{X}-\tilde{y})-(1-2\tau ){\int }_{-\infty }^{\tilde{y}}{F}_n\left(s\right)ds=0$$

where $\overline{X}$ is the sample mean and $F_n(·)$ is the empirical distribution function.

Holzmann and Klar in [9] have proved a uniform consistency and asymptotic normality of $\tilde{y}$. In this paper, we study the higher-order asymptotic properties of the expectile estimator. To study the higher-order asymptotics, we use a kernel-type estimator $\widehat{F}(·)$ of the distribution function $F(·)$, instead of $F_n(·)$.

Let us define kernel estimators of the density and distribution function

$$\begin{array}{ccc}\hfill \widehat{f}\left(x\right)& =& {\displaystyle \frac{1}{nh}}\sum _{i=1}^{n}K\left({\displaystyle \frac{x-X_i}{h}}\right),\hfill \\ \hfill \widehat{F}\left(x\right)& =& {\displaystyle \frac{1}{n}}\sum _{i=1}^{n}W\left({\displaystyle \frac{x-X_i}{h}}\right)\hfill \end{array}$$

where $K(·)$ is a kernel function and $W(·)$ is an integral of $K(·)$. We assume that

$$\left(a1\right)K\left(s\right)\ge 0,{\int }_{-\infty }^{\infty }K\left(s\right)=1,K(-s)=K\left(s\right),$$

and $W\left(x\right)={\int }_{-\infty }^{x}K\left(s\right)ds.$ Here h is a bandwidth where $h\to 0,nh\to \infty$. Hereafter we assume that the bandwidth $h=O\left(n^{-{\displaystyle \frac{1}{4}}}{(logn)}^{-1}\right)$.

As in the case of quantile estimation, we are using a kernel-smoothed estimator of the cumulative distribution function in the construction of the expectile estimator. The reason for switching to the kernel-smoothed version of the empirical distribution function in the definition of our expectile estimator is that only for this version is it possible to show the validity of the Edgeworth expansion. As discussed in detail in [13], if we use a kernel estimator with an informed choice of bandwidth and a suitable kernel, then the resulting expectile estimator can be easily studentized; the Edgeworth expansion up to order $o\left(n^{-1/2}\right)$ for the studentized version will be derived and the theoretical quantities involved in the expansion can be shown to be easily estimated. In addition, a Cornish-Fisher inversion can be used to construct confidence intervals for the expectile (which is the main goal of the inference in this paper). The resulting confidence intervals are more precise than the intervals obtained via inversion of the normal approximation and can be used to improve the coverage accuracy for moderate sample sizes.

Hence from now on we will discuss the higher order asymptotic properties of the estimator $\widehat{y}$ of $y^{*}$ that satisfies

$$\tau {\int }_{\widehat{y}}^{\infty }\{1-\widehat{F}\left(s\right)\}ds-(1-\tau ){\int }_{-\infty }^{\widehat{y}}\widehat{F}\left(s\right)ds=0.$$

Let us define

$$A\left(t\right)={\int }_{-\infty }^{t}W\left(s\right)ds,{\widehat{F}}^{\left[2\right]}\left(s\right)={\displaystyle \frac{1}{n}}\sum _{i=1}^{n}hA\left({\displaystyle \frac{s-X_i}{h}}\right).$$

Then similarly to Proposition 1, we have the following Proposition.

Proposition 2.

If the kernel function satisfies the condition (a1), the kernel expectile estimator $\widehat{y}$ is given by the solution of the equation

$$\tau (\overline{X}-\widehat{y})-(1-2\tau ){\widehat{F}}^{\left[2\right]}\left(\widehat{y}\right)=0.$$

(7)

For our further discussion we define the following quantities.

Definition 1.

$$\begin{array}{c}\\ \hfill u_1\left(X_i\right)& =& -(1-2\tau )\left\{hA\left({\displaystyle \frac{y^{*}-X_i}{h}}\right)-F^{\left[2\right]}\left(y^{*}\right)\right\}+\tau (X_i-\mu ),\hfill \\ \hfill b_{1n}& =& -(1-2\tau )\left\{E\left[hA\left({\displaystyle \frac{y^{*}-X_i}{h}}\right)\right]-F^{\left[2\right]}\left(y^{*}\right)\right\},\hfill \\ \hfill {\tilde{u}}_1\left(X_i\right)& =& u_1\left(X_i\right)-b_{1n},\hfill \\ \hfill {\xi }_n^2& =& E\left({\tilde{u}}_1^2\left(X_1\right)\right),\hfill \\ \hfill b_{2n}& =& E\left[W\left({\displaystyle \frac{y^{*}-X_i}{h}}\right)\right]-F\left(y^{*}\right),\hfill \\ \hfill \tilde{W}\left(X_i\right)& =& W\left({\displaystyle \frac{y^{*}-X_i}{h}}\right)-F\left(y^{*}\right)-b_{2n},\hfill \\ \hfill b_{3n}& =& E\left[{\displaystyle \frac{1}{h}}K\left({\displaystyle \frac{y^{*}-X_i}{h}}\right)\right]-f\left(y^{*}\right),\hfill \\ \hfill \tilde{K}\left(X_i\right)& =& {\displaystyle \frac{1}{h}}K\left({\displaystyle \frac{y^{*}-X_i}{h}}\right)-f\left(y^{*}\right)-b_{3n},\hfill \\ \hfill C& =& \tau +(1-2\tau )F\left(y^{*}\right)\hfill \\ \hfill \tilde{C}& =& \tau +(1-2\tau )\widehat{F}\left(y^{*}\right),\hfill \\ \hfill \widehat{C}& =& \tau +(1-2\tau )\widehat{F}\left(\widehat{y}\right).\hfill \end{array}$$

Note that $E\left[{\tilde{u}}_1\left(X_i\right)\right]=E\left[\tilde{W}\left(X_i\right)\right]=E\left[\tilde{K}\left(X_i\right)\right]=0$ holds and that $b_{1n},b_{2n}$ and $b_{3n}$ denote the biases of the kernel estimators of $F^{\left[2\right]}\left(y^{*}\right),F\left(y^{*}\right)$ and $f\left(y^{*}\right).$

Using the equations (6) and (7), we have

$$\tau (\overline{X}-\mu )-\tau (\widehat{y}-y^{*})-(1-2\tau )\left\{{\widehat{F}}^{\left[2\right]}\left(\widehat{y}\right)-F^{\left[2\right]}\left(y^{*}\right)\right\}=0.$$

(8)

Since we intend to discuss the Edgeworth expansion with a residual term $o\left(n^{-1/2}\right)$, we will obtain the asymptotic representation with residual term $o_L\left(n^{-1/2}\right)$ where

$$P\left\{|o_L\left(n^{-1/2}\right)|\ge n^{-1/2}{\epsilon }_n\right\}=o\left(n^{-1/2}\right)$$

as ${\epsilon }_n\to 0$. When we obtain the Edgeworth expansion until the order $n^{-1/2}$, it follows from Esseen’s smoothing lemma that we can ignore the terms of order $o_L\left(n^{-1/2}\right)$.

Similarly to the $o_L\left(n^{-1/2}\right)$ notation, we will also be using the $o_{\ell }\left(n^{-1/2}\right)$ notation that follows the definition

$$P\left\{|o_{\ell }\left(n^{-1/2}\right)|\ge n^{-1/2}{(logn)}^{-1}\right\}=o\left(n^{-1/2}\right).$$

Note that we can also ignore the $o_{\ell }\left(n^{-1/2}\right)$ terms when we discuss the Edgeworth expansion with residual term $o\left(n^{-1/2}\right)$.

3. Results

3.1. Edgeworth Expansion for the Standardized Expectile

In this subsection, we will get an asymptotic representation of the standardized expectile ${\displaystyle \frac{\sqrt{n}C}{{\xi }_n}}(\widehat{y}-y^{*})$ and its Edgeworth expansion. This expansion is of theoretical interest mainly as the constant C and the normalizing quantity ${\xi }_n$ involved depend on parameters of the unknown population distribution. Later, in SubSection 3.2 we will formulate the related but more practicable expansions of the studentized expectile. They do not depend on unknown population parameters and can also be inverted to deliver accurate asymptotic confidence intervals for the expectile.

We assume that the following conditions hold:

$$\begin{array}{ccc}\hfill \left(a2\right)& & C\ne 0,\hfill \\ \hfill \left(a3\right)& & E|X_1{-\mu |}^{5+\delta }<\infty .\hfill \end{array}$$

Theorem 1.

We assume the conditions $\left(a1\right),\left(a2\right),\left(a3\right)$ hold. Further, assume that $K(·)$, the derivatives $f^{\prime }(·)$ and $K^{\prime }(·)$ are bounded, and $\int s^{4}K\left(s\right)ds<\infty$. Then:

(1) For the standardized expectile estimator, we have

$$\begin{array}{ccc}& & {\displaystyle \frac{\sqrt{n}C}{{\xi }_n}}(\widehat{y}-y^{*})\hfill \\ & =& n^{1/2}{\displaystyle \frac{b_{1n}}{{\xi }_n}}+n^{-1/2}{\displaystyle \frac{{\nu }_B}{{\xi }_n}}+n^{-1/2}\sum _{i=1}^n{\displaystyle \frac{{\tilde{u}}_1\left(X_i\right)}{{\xi }_n}}\hfill \\ & & +n^{-3/2}\sum _{1\le i<j\le n}{\displaystyle \frac{{\nu }_2(X_i,X_j)}{{\xi }_n}}+o_L\left(n^{-1/2}\right)\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill {\nu }_B& =& -(1-2\tau )\left\{{\displaystyle \frac{f\left(y^{*}\right)}{2C^2}}E\left[{\tilde{u}}_1^2\left(X_1\right)\right]+{\displaystyle \frac{1}{C}}E\left[\tilde{W}\left(X_1\right){\tilde{u}}_1\left(X_1\right)\right]\right\}\hfill \\ \hfill {\nu }_2(X_i,X_j)& =& -(1-2\tau )\left\{{\displaystyle \frac{f\left(y^{*}\right)}{C^2}}{\tilde{u}}_1\left(X_i\right){\tilde{u}}_1\left(X_j\right)\right.\hfill \\ & & \left.+{\displaystyle \frac{1}{C}}\left[\tilde{W}\left(X_i\right){\tilde{u}}_1\left(X_j\right)+\tilde{W}\left(X_j\right){\tilde{u}}_1\left(X_i\right)\right]\right\}.\hfill \end{array}$$

(2)The asymptotic variance of $\widehat{y}$ is equal to ${\displaystyle \frac{1}{C^2}}Var\left[\tilde{u}\left(X_1\right)\right]$ where

$$Var\left[\tilde{u}\left(X_1\right)\right]=(2-4\tau )F^{\left[3\right]}\left(y^{*}\right)+{\tau }^2{(y^{*}-\mu )}^2+{\tau }^2{\sigma }^2+O\left(h\right),$$

${\sigma }^2=V\left(X_1\right)$, and

$$F^{\left[3\right]}\left(x\right)={\int }_{-\infty }^x{F}^{\left[2\right]}\left(s\right)ds.$$

(3) The Edgeworth expansion is given by

$$P\left\{{\displaystyle \frac{\sqrt{n}C}{{\xi }_n}}(\widehat{y}-y^{*})\le x\right\}=Q\left(x-{\displaystyle \frac{n^{1/2}{b}_{1n}}{{\xi }_n}}\right)+o\left(n^{-1/2}\right)$$

where

$$\kappa ={\displaystyle \frac{1}{{\xi }_n^3}}E\left[{\tilde{u}}_1^3\left(X_1\right)\right]+{\displaystyle \frac{3}{{\xi }_n^3}}E\left[{\tilde{u}}_1\left(X_1\right){\tilde{u}}_1\left(X_2\right){\nu }_2(X_1,X_2)\right]$$

and

$$Q\left(x\right)=\Phi \left(x\right)-\varphi \left(x\right)n^{-1/2}\left\{{\displaystyle \frac{{\nu }_B}{{\xi }_n}}+{\displaystyle \frac{\kappa (x^2-1)}{6}}\right\}.$$

(9)

Remark 1

It follows from the results of Holzmann and Klar in [9] that

$$\begin{array}{ccc}& & E\left[I_{\tau }^2(y^{*},X_1)\right]\hfill \\ & =& E[{\tau }^2{(X_1-y^{*})}^2{I}_{(X_1\ge y^{*})}+{(1-\tau )}^2{(y^{*}-X_1)}^2{I}_{(X_1<y^{*})}]\hfill \\ & =& {\tau }^2{\int }_{y^{*}}^{\infty }{(s-y^{*})}^{2}f\left(s\right)ds+{(1-\tau )}^2{\int }_{-\infty }^{y^{*}}{(y^{*}-s)}^{2}f\left(s\right)ds\hfill \\ & =& {\tau }^2{\int }_{-\infty }^{\infty }{(s-y^{*})}^{2}f\left(s\right)ds+(1-2\tau ){\int }_{-\infty }^{y^{*}}{(y^{*}-s)}^{2}f\left(s\right)ds\hfill \\ & =& {\tau }^2{\int }_{-\infty }^{\infty }{(s-\mu +\mu -y^{*})}^{2}f\left(s\right)ds\hfill \\ & & +(1-2\tau )\left\{{\left[{(y^{*}-s)}^{2}F\left(s\right)\right]}_{-\infty }^{y^{*}}+2{\int }_{-\infty }^{y^{*}}(y^{*}-s)F\left(s\right)ds\right\}\hfill \\ & =& {\tau }^2{\sigma }^2+\tau {(\mu -y^{*})}^2+(1-2\tau )\left\{{\left[2(y^{*}-s)F^{\left[2\right]}\left(s\right)\right]}_{-\infty }^{y^{*}}+2{\int }_{-\infty }^{y^{*}}{F}^{\left[2\right]}\left(s\right)ds\right\}\hfill \\ & =& {\tau }^2{\sigma }^2+{\tau }^2{(\mu -y^{*})}^2+(2-4\tau )F^{\left[3\right]}\left(y^{*}\right).\hfill \end{array}$$

Comparing with the result of Corollary 4 on p. 2359 in [9] we realize that, as expected, the first-order approximations of the asymptotic variances of the kernel-smoothed expectile estimator in our paper and of the empirical distribution-based estimator discussed in [9] coincide.

It is easy to see that

$$\begin{array}{ccc}& & {\displaystyle \frac{3}{{\xi }_n^3}}E\left[{\tilde{u}}_1\left(X_1\right){\tilde{u}}_1\left(X_2\right){\nu }_2(X_1,X_2)\right]\hfill \\ & =& -{\displaystyle \frac{3(1-2\tau )}{{\xi }_n^3}}\left\{{\displaystyle \frac{{\xi }_n^{4}f\left(y^{*}\right)}{C^2}}+{\displaystyle \frac{{\xi }_n^2}{C}}E\left[\tilde{W}\left(X_1\right){\tilde{u}}_1\left(X_1\right)\right]\right\}\hfill \\ & =& -3(1-2\tau )\left\{{\displaystyle \frac{{\xi }_{n}f\left(y^{*}\right)}{C^2}}+{\displaystyle \frac{1}{C{\xi }_n}}E\left[\tilde{W}\left(X_1\right){\tilde{u}}_1\left(X_1\right)\right]\right\}\hfill \end{array}$$

and this relation can be used to write down the formula for $\kappa$ in (9) in an alternative way.

The proof of the above Theorem will be presented in the Appendix. It relies heavily on some large deviations results for U-statistics that we summarized in Lemma A1 (whose formulations and proof are also postponed to the Appendix). Using the results of the latter Lemma, we can obtain the evaluation of the order of the asymptotic approximations and expansions of the differences $(\widehat{y}-y^{*}),{(\widehat{y}-y^{*})}^2,{(\widehat{y}-y^{*})}^3$ (also to be presented in Lemma A2 in the Appendix). Combining the evaluations from Lemma A2 essentially guarantees the asymptotic representation in the standardized case given in Theorem 1.

3.2. Edgeworth Expansion for the Studentized Expectile

As pointed out by many papers, the Edgeworth expansion of the studentized estimator is more important than the expansion of the standardized estimator. In addition, it is the studentized version that could be inverted in practical settings to deliver confidence intervals for the unknown expectile. In order to get the asymptotic representation of the studentized estimator, we need to construct suitable estimators $\widehat{C}$ and ${\widehat{\xi }}_n$. $\widehat{C}$ is given in (Definition 1) already. Next we proceed to get an estimator of $E\left[u_1^2\left(X_1\right)\right]$. Similarly as the ordinal estimation of the population variance, putting

$${\widehat{u}}_1\left(X_i\right)=-(1-2\tau )hA\left({\displaystyle \frac{\widehat{y}-X_i}{h}}\right)+\tau X_i,$$

we can obtain a consistent estimator

$${\widehat{\xi }}_n^2={\displaystyle \frac{1}{n-1}}\sum _{i=1}^n{\widehat{u}}_1^2\left(X_i\right)-{\displaystyle \frac{n}{n-1}}{\left\{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\widehat{u}}_1\left(X_i\right)\right\}}^2.$$

Let us now analyze the following studentized expectile estimator

$${\displaystyle \frac{\widehat{C}}{{\widehat{\xi }}_n}}(\widehat{y}-y^{*}).$$

We introduce our next set of notations.

Definition 2.

Let us define

$$\begin{array}{ccc}& & \zeta \left(X_i\right)={\displaystyle \frac{1}{2{\xi }_n^3}}\{{\tilde{u}}_1^2\left(X_i\right)-{\xi }_n^2\},\hfill \\ & & B_n={\displaystyle \frac{1-2\tau }{2C^2}}f\left(y^{*}\right){\xi }_n-E\left[{\tilde{u}}_1\left(X_1\right)\zeta \left(X_1\right)\right],\hfill \\ & & u_1^{*}\left(X_i\right)={\displaystyle \frac{{\tilde{u}}_1\left(X_i\right)}{{\xi }_n}},\hfill \\ & & u_2^{*}(X_i,X_j)={\displaystyle \frac{1-2\tau }{C^2{\xi }_n}}f\left(y^{*}\right){\tilde{u}}_1\left(X_i\right){\tilde{u}}_1\left(X_j\right)-{\tilde{u}}_1\left(X_i\right)\zeta \left(X_j\right)-{\tilde{u}}_1\left(X_j\right)\zeta \left(X_i\right).\hfill \end{array}$$

Similarly to the standardized expectile estimator, we can derive the asymptotic representation of the studentized estimator in the next Theorem.

Theorem 2.

Under the same assumptions as in Theorem 1:

(1) For the studentized expectile estimator, we get

$$\begin{array}{ccc}\hfill {\displaystyle \frac{n^{1/2}\widehat{C}(\widehat{y}-y^{*})}{{\widehat{\xi }}_n}}& =& {\displaystyle \frac{n^{1/2}{b}_{1n}}{{\xi }_n}}+n^{-1/2}{B}_n+n^{-1/2}\sum _{i=1}^n{u}_1^{*}\left(X_i\right)\hfill \\ & & +n^{-3/2}\sum _{1\le i<j\le n}{u}_2^{*}(X_i,X_j)+o_{\ell }\left(n^{-1/2}\right).\hfill \end{array}$$

(10)

(2) We have the Edgeworth expansion with residual term $o\left(n^{-1/2}\right)$

$$P\left\{{\displaystyle \frac{\sqrt{n}\widehat{C}(\widehat{y}-y^{*})}{{\widehat{\xi }}_n}}\le x\right\}=Q_S\left(x-{\displaystyle \frac{n^{1/2}{b}_{1n}}{{\xi }_n}}\right)+o\left(n^{-1/2}\right)$$

(11)

where

$${\kappa }_S=E\left[{\left\{u_1^{*}\left(X_1\right)\right\}}^3\right]+3E\left[u_1^{*}\left(X_1\right)u_1^{*}\left(X_2\right)u_2^{*}(X_1,X_2)\right]$$

and

$$Q_S\left(x\right)=\Phi \left(x\right)-n^{-1/2}\varphi \left(x\right)\left\{B_n+{\displaystyle \frac{{\kappa }_S(x^2-1)}{6}}\right\}.$$

The proof of Theorem 2 is presented in the appendix.

3.3. Cornish-Fisher-Type Approximation of the $\alpha$-Quantile of the Studentized Expectile

Here we will obtain an approximation of $\alpha$-quantile $e_{\alpha }$ of the studentized expectile where

$$\alpha =P\left\{{\displaystyle \frac{\sqrt{n}\widehat{C}(\widehat{y}-y^{*})}{{\widehat{\xi }}_n}}\le e_{\alpha }\right\}=Q_S\left(e_{\alpha }-{\displaystyle \frac{n^{1/2}{b}_{1n}}{{\xi }_n}}\right)+o\left(n^{-1/2}\right).$$

Let us define

$$e_{\alpha }^{*}=e_{\alpha }-{\displaystyle \frac{n^{1/2}{b}_{1n}}{{\xi }_n}}.$$

Then expanding around the $\alpha$-quantile $z_{\alpha }$ of $N(0,1)$, we have

$$\begin{array}{ccc}\hfill e_{\alpha }^{*}& =& z_{\alpha }+n^{-1/2}{\displaystyle \frac{\varphi \left(z_{\alpha }\right)\left\{B_n+P_S\left(z_{\alpha }\right)\right\}}{Q_S^{\prime }\left(z_{\alpha }\right)}}+O\left(n^{-1}\right)\hfill \\ & =& z_{\alpha }+n^{-1/2}\left\{B_n+P_S\left(z_{\alpha }\right)\right\}+O\left(n^{-1}\right)\hfill \end{array}$$

where $P_S\left(x\right)={\displaystyle \frac{{\kappa }_S(x^2-1)}{6}},{\kappa }_S=3{\displaystyle \frac{(1-2\tau )f\left(y^{*}\right)}{C^2}}{\xi }_n-{\displaystyle \frac{2E\left[{\tilde{u}}_1^3\left(X_1\right)\right]}{{\xi }_n^3}}.$ For the $\alpha$-quantile $e_{\alpha }$, we have

$$e_{\alpha }=z_{\alpha }+n^{1/2}{\displaystyle \frac{b_{1n}}{{\xi }_n}}+n^{-1/2}\left\{B_n+P_S\left(z_{\alpha }\right)\right\}+O\left(n^{-1}\right).$$

Since

$$b_{1n}=-h^2{\displaystyle \frac{(1-2\tau )f\left(y^{*}\right)}{2}}{\int }_{-\infty }^{\infty }{u}^{2}K\left(u\right)du+O\left(h^4\right),$$

we have an estimator of $b_{1n}$

$${\widehat{b}}_{1n}=-h^2{\displaystyle \frac{(1-2\tau )\widehat{f}\left(\widehat{y}\right)}{2}}{\int }_{-\infty }^{\infty }{u}^{2}K\left(u\right)du.$$

It is easy to see that

$$B_n={\displaystyle \frac{1-2\tau }{2C^2}}f\left(y^{*}\right){\xi }_n-{\displaystyle \frac{E\left[{\left\{{\tilde{u}}_1\left(X_1\right)\right\}}^3\right]}{2{\xi }_n^3}}.$$

Thus we have estimators of $B_n$ and ${\nu }_3$ as follows:

$$\begin{array}{ccc}\hfill {\widehat{B}}_n& =& {\displaystyle \frac{1-2\tau }{2{\widehat{C}}^2}}\widehat{f}\left(\widehat{y}\right){\widehat{\xi }}_n-{\displaystyle \frac{{\widehat{\mu }}_3}{2{\widehat{\xi }}_n^3}},\hfill \\ \hfill {\widehat{\kappa }}_S& =& 3{\displaystyle \frac{(1-2\tau )}{{\widehat{C}}^2}}\widehat{f}\left(\widehat{y}\right){\widehat{\xi }}_n-{\displaystyle \frac{2{\widehat{\mu }}_3}{{\widehat{\xi }}_n^3}}\hfill \end{array}$$

where

$${\widehat{\mu }}_3={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left\{\widehat{u}\left(X_i\right)-{\displaystyle \frac{1}{n}}\sum _{j=1}^n\widehat{u}\left(X_j\right)\right\}}^3.$$

Therefore, we have an estimator of the $\alpha$-quantile $e_{\alpha }$

$${\widehat{e}}_{\alpha }=z_{\alpha }+n^{1/2}{\displaystyle \frac{{\widehat{b}}_{1n}}{{\widehat{\xi }}_n}}+n^{-1/2}\left\{{\widehat{B}}_n+{\displaystyle \frac{{\widehat{\kappa }}_S(z_{\alpha }^2-1)}{6}}\right\}.$$

(12)

4. Discussion

Given that the main application domain of expectiles has been in risk management, we also want to illustrate the application of our methodology in this area. As discussed in Section 2, $EVaR=-e_{\tau }\left(X\right)$ is a coherent risk measure when $\tau \in (0,1/2].$ It is easy to check (or compare p. 46 of ([3])) that $-e_{1-\tilde{\tau }}\left(X\right)=e_{\tilde{\tau }}(-X)$ holds. In addition, most interest in risk management is in the tails. If the random variable of interest X represents an outcome, then $-X$ represents a loss, and one would be interested in losses in the tail. To illustrate the effectiveness of our approach for constructing improved confidence intervals, we need to compare simulation outcomes with the population distribution for which the true expectile is known precisely. Such examples are relatively scarce in the literature. A small number of suitable exceptions are discussed in [2]. One of these exceptions is the exponential distribution, which we chose for our illustrations below.

Setting $Z=(-X)$ to be standard exponentially distributed, we have for the values of $\tilde{\tau }$ the relation

$$e_{\tilde{\tau }}\left(Z\right)=1+\mathbb{W}\left\{{\displaystyle \frac{2\tilde{\tau }-1}{(1-\tilde{\tau })e}}\right\},$$

where $\mathbb{W}(·)$ is the Lambert $\mathbb{W}$ function defined implicitly by means of the equation

$$\mathbb{W}\left(z\right)exp\left(\mathbb{W}\right(z\left)\right)=z$$

(and we note that $\mathbb{W}\left(x\right)\sim log\left(x\right)$ for $x\to \infty$ holds). We have used a symmetric compactly supported on $(-1,1)$ kernel $K\left(x\right).$ It is said to be of order m is the mth derivative $K^{\left(m\right)}\in Lip\left(\beta \right)$ for some $\beta >0$ and

$${\int }_{-1}^{1}K\left(x\right)dx=1,{\int }_{-1}^1{x}^{i}K\left(x\right)dx=0,i=1,2,\cdots ,m-1,{\int }_{-1}^1{x}^{m}K\left(x\right)dx\ne 0.$$

In our numerical experiments, we used the classical second-order Epanechnikov kernel

$$K\left(x\right)={\displaystyle \frac{3}{4}}(1-x^2)I\left(\right|x|\le 1).$$

With it, the factor in the definition of the estimator ${\widehat{b}}_{1n}$ becomes ${\int }_{-\infty }^{\infty }{u}^{2}K\left(u\right)du=0.2$

There are at least two ways to produce accurate confidence intervals at level $(1-\alpha )$ for the expectile when exploiting the Edgeworth expansion of its studentized version. One approach (we call it the CF method) is based on using the estimated values $({\widehat{e}}_{\alpha /2}$ and ${\widehat{e}}_{1-(\alpha /2)})$ obtained by using the formula (12). Then the left-and right-hand sides of the $(1-\alpha )\times 100\%$ confidence interval for $EVaR=-e_{\tau }\left(X\right)$ at given $\tau$ are obtained as

$$(\widehat{y}-{\widehat{e}}_{1-{\displaystyle \frac{\alpha }{2}}}{\widehat{\xi }}_n/\left({\widehat{C}}_n\sqrt{n}\right),\widehat{y}-{\widehat{e}}_{{\displaystyle \frac{\alpha }{2}}}{\widehat{\xi }}_n/\left({\widehat{C}}_n\sqrt{n}\right)).$$

Another approach (we call it numerical inversion) is to use numerical root-finder procedures to solve the two equations

$${\widehat{Q}}_S({\eta }_1-{\displaystyle \frac{n^{1/2}{\widehat{b}}_{1n}}{{\widehat{\xi }}_n}})={\displaystyle \frac{\alpha }{2}},{\widehat{Q}}_S({\eta }_2-{\displaystyle \frac{n^{1/2}{\widehat{b}}_{1n}}{{\widehat{\xi }}_n}})=1-{\displaystyle \frac{\alpha }{2}},$$

and construct the confidence interval as $({\eta }_1,{\eta }_2).$

These two methods should be asymptotically equivalent but would deliver different intervals for small sample sizes, with the numerical inversion delivering significantly better results in terms of closeness to the nominal coverage level.

With the standardized version, we achieved a better approximation and more precise coverage probabilities across very low sample sizes such as $n=10,12,15,\mathrm{or}20,$ across a range of $\tau$ values such as $\tau =0.1,0.2,0.3,0.4$ and a range of values of $\alpha$ such as $\alpha =0.1,0.05,0.01$ for the $(1-\alpha )*100\%$ confidence intervals. The approximations were extremely accurate for such small sample sizes. We do not reproduce all these here as our main goal is to investigate the practically more relevant studentized case. We only include one graph (Figure 1) where the case $n=12,\tau =0.2$ is demonstrated graphically. The “true" cdf of the estimator for the purpose of comparison was obtained via the empirical cdf based on 50000 simulations from standard exponentially distributed data of size $n=12.$ The resulting confidence intervals at a nominal $90\%$ level had actual coverage of 0.90 for the Edgeworth and 0.9014 for the normal approximation. At nominal $95\%$ they were 0.9490 and 0.9453, respectively. At nominal $99\%$ they were 0.9858 for Edgeworth versus 0.9804 for the normal approximation.

In the practically relevant studentized case, we are unable to obtain such good results for sample sizes as low as the ones from the standardized case. This is of course to be expected as in this case there is a need to estimate the ${\xi }_n$ and the C quantities using the data. The moderate sample sizes at which the CF and numerical inversion methods deliver significantly more precise results depend, of course, on the distribution of C itself. For the exponential distribution these turned out to be in the range of 20, 50, 100, 150 to about 200. For larger sample sizes, all three methods- the normal theory-based confidence intervals, the ones obtained by the numerical inversion and the CF-based intervals become very accurate but the discrepancy between their accuracy becomes negligibly small and for that reason we do not report it here.

Before presenting thorough numerical simulations, we include one illustrative graph (Figure 2) where the case $n=50,\tau =0.3$ where the studentized case is demonstrated. The graph demonstrates the virtually uniform improvement when the Edgeworth approximation is used instead of the simple normal approximation. The comparison is made with the “true" cdf (obtained via the empirical cdf based on 50000 simulations from standard exponentially distributed data of size $n=50$). We found that at the 50000 replications a stabilization occurs and further increase of the replications seems unnecessary. The resulting confidence intervals at a nominal $90\%$ level had actual coverage of 0.877 for the numerical inversion of the Edgeworth, with 0.869 for the normal approximation. At $95\%$ nominal level, the actual coverage was 0.921 for the numerical inversion of the Edgeworth versus 0.917 for the normal approximation.

Next, we include Tables 1 and 2 showing the effects of applying our methodology for constructing confidence intervals for the expectiles. The moderate samples included in the comparison are chosen as 20, 50, 100, 150, and 200. The two tables illustrate the results for two common confidence levels used in practice ($90\%$ for Table 1 and $95\%$ for Table 2). The better performer in each row is in bold font. Examination of Tables 1 and 2 shows that the “true" coverage probabilities approach the nominal probabilities when the sample size increases. As expected, the discrepancies in accuracy between the different approximations also decreases when the sample size n increases. As the confidence intervals are based on asymptotic arguments, this demonstrates the consistency of our procedure. For the chosen levels of confidence, our new confidence intervals virtually always outperform the ones based on the normal approximation. There appears to be a downward bias in the coverage probabilities across the tables for both normal and Edgeworth-based methods. This bias is getting smaller as n increases above 200. We observe that the value of $\tau$ also influences the bias with smaller values of $\tau$ impacting the bias more significantly. This is expected as these values of $\tau$ lead to expectiles that are further in the right tail of the distribution of the loss variable X and hence are more difficult to estimate. As is known, the Edgeworth approximation’s strength is in the central part of the distribution. However, at small values of $\tau$ we are focusing on the tail where it does not necessarily improve over the normal approximation.

5. Conclusions

Edgeworth expansions are known to deliver better approximations to the standardized and studentized versions of estimators of parameters of interest. The inversion of these expansions can be applied for constructing more accurate confidence intervals for these parameters when sample sizes are moderate. To justify the validity of these expansions, one needs to switch to using a kernel-smoothed version of the empirical distribution function in the definition of the estimator. We apply the technique for the estimation of the expectile which has found fruitful applications in risk management recently. We illustrated the advantages of our procedure on simulations that utilize the exponential distribution as an example. We chose this distribution because the expectile is known precisely (not only approximately) for it. However, we stress, that our procedure is fully nonparametric and can be applied for any distribution as long as the conditions of Theorem 2 are satisfied.