On a family of weighted Cramér–von Mises goodness-of-fit tests in operational risk modeling

Kirill Mayorov; James Hristoskov; Narayanaswamy Balakrishnan

1 Introduction

1.1 Background

Financial institutions bear a number of operational risks while conducting business activities. These risks can often result in potentially significant loss events that may degrade the firm’s assets to levels that are lower than liabilities. Under such circumstances, financial institutions will not have adequate backing to make whole depositors and lenders, and will usually face imminent default. In order to mitigate the potential for default from adverse operational risk events, firms are required to keep sufficient equity capital that can foreseeably absorb all significant losses without jeopardizing liabilities and causing bankruptcy.

Definition 1.1.

Operational risk (OpRisk) is defined as the risk of loss resulting from inadequate or failed internal processes, people and systems, or from external events. This definition includes legal risk but excludes strategic and reputational risk.

Some examples of OpRisk events (McNeil et al 2015) are the following:

• fraud (internal and external); losses due to IT failure; errors in settlements of transactions; losses due to external events such as flooding, fire, earthquakes or terrorism;

• system errors, such as the estimated US$440 million loss from a computer-trading glitch at Knight Capital Group in 2012; and

• legal losses and penalties, such as Bank of America’s August 2014 record fine of US$16.65 billion for the misselling of financial products on the basis of inaccurate or misleading information about their risks.

Regulators typically require financial institutions to hold equity capital for OpRisk at a level that will ensure the total yearly loss from extreme or on aggregate large operational events does not exceed capital with a chance of 0.1% over a one-year time horizon. As such, to ensure capital adequacy, financial institutions attempt to determine the one-year, 99.9th-percentile aggregate loss that could possibly occur and ensure that equity capital is at a level that is sufficient to cover such loss.

The 99.9th-percentile aggregate loss is specified by the Basel Committee on Banking Supervision (2004) as representing the target for minimum capitalization under Pillar 1 of its three-pillar system. At present, Pillar 1 minimum capitalization for OpRisk can be determined via three different approaches: a basic indicator approach (BIA), the standardized approach (TSA) and the advanced measurement approach (AMA). The AMA is driven by regulatory principles prescribing a general framework, which requires the application of frequency and severity distributions to complete an aggregate loss distribution under the LDA.

Under Basel II, Pillar 2 directives, financial institutions can apply internal models to complement and test the minimum capital requirements set under Pillar 1 (either through economic capital or stress testing frameworks). Being typically more advanced, Pillar 2 models for OpRisk generally follow LDA prescriptions and require well-calibrated severity models, as the loss targets for capitalization can exceed the 99.95 percentile.

Given the high-percentile loss targets for capitalization, under both pillars, a major challenge for LDA modeling has been to accurately model loss severity with limited publicly available or internal operational loss data for more extreme events. Often the underlying problem is that less than a minimum set of historic losses have been collected for adequate statistical modeling. In addition, wrong candidate severity distributions may be selected. The combination of these problems can lead to significant errors in modeling for capital adequacy.

In addressing the problem with more accurate severity distribution selection modeling under the LDA approach, we discuss below the distributional properties of a family of weighted Cramér–von Mises (WCvM) goodness-of-fit (GoF) test statistics, with weight function $\psi(t)=1/(1-t)^{\beta}$ , that are suitable for the more accurate selection of best-fit severity distributions. We demonstrate that, for weights where $\beta\geq 2$ , the WCvM test statistic does not have limiting distributions, which may limit its practical utility. In order to rectify the problem, we provide a normalization for $\beta=2$ , which leads to a nondegenerate asymptotic distribution. Notwithstanding the normalization, our analysis will show that the tests provide greater utility when $\beta<1.5$ , and that utility is questionable for $\beta\geq 1.5$ , as only Monte Carlo schemes, which are shown to be very slow in approaching the asymptotic regime, are practical even for very large samples.

1.2 Suitable GoF tests for OpRisk modeling

Under the LDA, financial institutions model frequencies and magnitudes of losses that can result from potential future OpRisk events. The various magnitudes of potential losses and their relative probabilities of occurrence are assumed to be best captured and modeled through parametric severity distributions. In general, the profile of historic losses is highly leptokurtic, which makes weighted GoF test statistics that are more sensitive to tails, such as the WCvM test statistic, popular for selecting candidate severity distributions. In order to convey the importance of severity distribution selection on capital modeling, below we show how, under a generalized capital approximation regime, the upper quantiles of the severity distribution directly affect capital requirements.

Definition 1.2.

The aggregate loss distribution (ALD) is the distribution of the aggregate (compound) loss $L=X_{1}+\cdots+X_{N}$ , where the frequency $N$ is a discrete random variable and $X_{1},\dots,X_{N}$ are positive continuous random severities.

OpRisk capital is then a quantile of the ALD, given by

\mathrm{VaR}_{\delta}(L)=\inf\{x\in\mathbb{R}\mid P(L>x)\leq 1-\delta\},

where confidence level $\delta\in(0,1)$ . Typically, $\delta\in\{0.999,0.9995,0.9997\}$ .

If $F_{L}(x)=P(L\leq x)$ is continuous and strictly increasing, then $\mathrm{VaR}_{\delta}(L)=F_{L}^{-1}(\delta)$ .

Frequencies of losses are usually assumed to follow a Poisson distribution; they are also generally well behaved and do not exhibit leptokurtic features. Therefore, with the exception of cases in which frequencies of losses are extremely low and jumps may occur (eg, where the Poisson lambda is less than 0.25 per annum), new loss event occurrences usually do not change the profile of frequency distributions, and the impact of new occurrences on capital requirements is typically modest.

This is due to the so-called single loss approximation (Böcker and Klüppelberg 2005) capital $\mathrm{VaR}_{\delta}(L)\approx F^{-1}(1-((1-\delta)/\lambda))$ , where $\lambda$ is the mean of the Poisson-distributed frequency of loss event occurrence, and $F(\cdot)$ is the severity cumulative distribution function (CDF).

For regulatory capital, $\delta=0.999$ . Even a $\lambda$ of moderate magnitude, say, 25, 50 or 100, leads to the calculation of 99.996, 99.998 or 99.999% quantiles of the severity distribution, respectively.

Consequently, the final choice of a severity distribution is crucial, and capital requirements are highly sensitive to upper tail behavior of the selected severity distribution. This supports the notion that the use of appropriately tuned GoF test statistics for severity distribution selection is of significant importance. Our analysis, therefore, will focus on presenting findings that allow for the development of practical utility properties for WCvM test statistics. Based on the developed properties, we are able to indicate weight boundaries for the test statistics which ensure that practical utility exists.

To this end, below we provide a brief description of some concepts and techniques that are key to the developments presented in subsequent sections.

Definition 1.3.

Given a random sample $x_{1},\dots,x_{n}$ of independent observations with CDF $F$ , the empirical (sample) distribution function (EDF) is defined as

F_{n}(t)=\frac{1}{n}\sum\mathbb{I}_{\{x_{i}\leq t\}},

where $\mathbb{I}_{A}(\cdot)$ is the indicator function on event A.

Some basic properties of the EDF are as follows:

• $\mathbb{E}(F_{n}(t))=F(t)$ and $\operatorname{Cov}(F_{n}(t_{1}),F_{n}(t_{2}))=(1/n)F(t_{1})(1-F(t_{2}))$ for $0\leq t_{1}\leq t_{2}\leq 1$ .
• $\sqrt{n}(F_{n}(t)-F(t))\xrightarrow{\mathrm{d}\,}N(0,\sqrt{F(t)(1-F(t))})$ as $n\rightarrow\infty$ .

We wish to test the null hypothesis

H_{0}\colon F(x)=F_{0}(x)\text{ versus }H_{1}\colon F(x)\neq F_{0}(x)

(1.1)

for a continuous CDF $F_{0}$ .

The parameters of the CDF $F_{0}$ may be completely known, or they may have to be estimated. The former case is known as a simple hypothesis, and the latter is known as a composite hypothesis. In what follows, we will consider the case of a simple hypothesis; the discussion of a composite hypothesis is intended for our future work.

Definition 1.4.

EDF-based GoF tests are tests for assessing $H_{0}$ that are based on a comparison of $F_{0}(x)$ with $F_{n}(x)$ .

The Kolmogorov–Smirnov and Cramér–von Mises (CvM) test statistics proposed with regard to this are

	$\displaystyle D_{n}$	$\displaystyle=\sqrt{n}\sup_{x\in\mathbb{R}}\|F_{n}(x)-F_{0}(x)\|$
and
	$\displaystyle\omega_{n}^{2}$	$\displaystyle=n\int_{-\infty}^{\infty}(F_{n}(x)-F_{0}(x))^{2}\,\mathrm{d}F_{0}% (x).$

These statistics are classical representatives of two classes of EDF-based GoF tests: the supremum and integral-based tests.

If a test statistic, denoted by $K_{n}$ , is adopted, the hypothesis is rejected for the samples for which $K_{n}$ is greater than some $C_{n,\alpha}$ .

The value $C_{n,\alpha}$ is to be chosen so that when the hypothesis $H_{0}$ is true, the probability of rejection is some specified number, referred to as a significance level and denoted by $\alpha$ . Usually, $\alpha$ takes values from $\{0.01,0.05,0.10\}$ . $C_{n,\alpha}$ is then called the critical value of the test statistic at significance level $\alpha$ . The critical value $C_{n,\alpha}$ is usually determined by Monte Carlo simulations.

We now define the limiting distribution of a test statistic.

Definition 1.5.

Let $\smash{K_{n}\xrightarrow{\mathrm{d}\,}K_{\infty}}$ as $n\rightarrow\infty$ , ie, $F_{K_{n}}(t)\rightarrow F_{K_{\infty}}(t)$ at continuity points of $F_{K_{\infty}}(\cdot)$ . If $K_{\infty}$ is nondegenerate (ie, almost surely not a constant or $\pm\infty$ ), then $F_{K_{\infty}}(\cdot)$ is referred to as the (nondegenerate) limiting distribution.

Definition 1.6.

In accordance with the notation of Definition 1.5,

• where $K_{\infty}$ does not exist, or exists but is degenerate, $K_{n}$ will be said not to possess a limiting distribution;
• if there is no limiting distribution but there exist deterministic sequences $a_{n}\in\mathbb{R}$ and $b_{n}>0$ , such that

$\frac{K_{n}-a_{n}}{b_{n}}\xrightarrow{\mathrm{d}\,}K_{\infty}^{*}\quad\text{as% ~{}}n\rightarrow\infty,$

and $K_{\infty}^{*}$ is nondegenerate, then $K_{n}$ will be said to possess an asymptotic distribution under positive affine normalization.

In sufficiently large samples, it is common practice to approximate the critical values of $K_{n}$ by their analogs from limiting (or asymptotic) distributions:

C_{n,\alpha}\approx C_{\infty,\alpha}

(respectively, $C_{n,\alpha}\approx b_{n}C_{\infty,\alpha}^{*}+a_{n}$ ), where

	$\displaystyle C_{n,\alpha}$	$\displaystyle=F_{K_{n}}^{-1}(1-\alpha)$
and
	$\displaystyle C_{\infty,\alpha}$	$\displaystyle=F_{K_{\infty}}^{-1}(1-\alpha)$

(respectively, $C_{\infty,\alpha}^{*}=F_{K_{\infty}^{*}}^{-1}(1-\alpha)$ ).

Therefore, the limiting properties of distributions of GoF test statistics are important, as they are instrumental in computing critical values.

The classic CvM test places equal weight on the main part and the tails of the distribution under $H_{0}$ . To weight the deviations between the EDF and $F_{0}$ according to the importance attached to various portions of the CDF, let us consider WCvM statistics:

W_{n}^{2}(\psi)=n\int_{-\infty}^{\infty}\psi(F_{0}(x))(F_{n}(x)-F_{0}(x))^{2}% \,\mathrm{d}F_{0}(x),

(1.2)

where weight $\psi\colon[0,1]\rightarrow\mathbb{R}_{+}$ .

Notable examples include the following choices of the weighting function: $\psi(t)\equiv 1$ (Cramér 1928; von Mises 1928), $\psi(t)=1/(t(1-t))$ (Anderson and Darling 1952), $\psi(t)=1/t$ , $\psi(t)=1/(1-t)$ (Sinclair et al 1990; Scott 1999), $\psi(t)=1/t^{\beta}$ for $\beta<2$ (Deheuvels and Martynov 2003) and $\psi(t)=1/(1-t)^{\beta}$ for $\beta<2$ (Feuerverger 2015).

The following test statistics have been proposed in Chernobai et al (2005), Chernobai (2006), Chernobai et al (2015) and Luceño (2006):

$\displaystyle\mathrm{AD}^{\ast}$	$\displaystyle=\sqrt{n}\sup_{x\in\mathbb{R}}\frac{\|F_{n}(x)-F_{0}(x)\|}{\sqrt{F_% {0}(x)(1-F_{0}(x))}},$	(1.3)
$\displaystyle\mathrm{AD}_{\mathrm{up}}$	$\displaystyle=\sqrt{n}\sup_{x\in\mathbb{R}}\frac{\|F_{n}(x)-F_{0}(x)\|}{(1-F_{0}% (x))},$	(1.4)
$\displaystyle\mathrm{AD}_{\mathrm{up}}^{2}$	$\displaystyle=n\int_{-\infty}^{\infty}\frac{(F_{n}(x)-F_{0}(x))^{2}}{(1-F_{0}(% x))^{2}}\,\mathrm{d}F_{0}(x).$	(1.5)

A useful, related statistic will be denoted by $\mathrm{AD}_{\mathrm{down}}^{2}$ :

\mathrm{AD}_{\mathrm{down}}^{2}=n\int_{-\infty}^{\infty}\frac{(F_{n}(x)-F_{0}(% x))^{2}}{(F_{0}(x))^{2}}\,\mathrm{d}F_{0}(x).

(1.6)

Statistic $\mathrm{AD}_{\mathrm{up}}^{2}$ is $W_{n}^{2}(\psi)$ with $\psi(t)=1/(1-t)^{2}$ . The $\mathrm{AD}_{\mathrm{up}}^{2}$ test statistic is popular in OpRisk modeling for severity distribution selection (Mignola and Ugoccioni 2006; Turk 2009; Wahlström 2013; Lavaud and Lehérissé 2014; Feuerverger 2015; Chernobai et al 2015). It is also known in engineering and meteorological applications under a different name: the right-tail Anderson–Darling test of second degree (Zhou 2013; Drobinski et al 2015).

Until recently, to the best of our knowledge, the OpRisk literature has not questioned the limiting properties of (1.3)–(1.6). Chernobai et al (2015), when briefly speaking about the mean of the asymptotic distribution of $\mathrm{AD}_{\mathrm{up}}^{2}$ , take for granted the very existence of such an asymptotic distribution (see Chernobai et al 2015, footnote 2, p. 585). In fact, in the terminology of Definition 1.5, Chernobai et al (2015) mean the limiting distribution. In this paper, however, we will show that no limiting distribution exists for the $\smash{\mathrm{AD}_{\mathrm{up}}^{2}}$ statistic.

Moreover, we discuss limiting properties for (1.3) and (1.4) and contribute to the analysis of the limiting properties of a subfamily of WCvM GoF test statistics (1.2), with weight function $\psi(t)=1/(1-t)^{\beta}$ , which will be denoted throughout by

W_{n}^{2,\beta}=n\int_{-\infty}^{\infty}\frac{(F_{n}(x)-F_{0}(x))^{2}}{(1-F_{0% }(x))^{\beta}}\,\mathrm{d}F_{0}(x),

for $\beta\in\mathbb{R}$ . Statistics $W_{n}^{2,\beta}$ are suitable for a more accurate selection of severity distributions because of the emphasis they put on the right tail of the distributions.

The rest of this paper is organized as follows. In Section 2, we collect results from the literature related to the supremum and quadratic class of upper-tail test statistics. This section reveals that (1.3)–(1.6) do not have well-defined limiting distributions. In Section 3, we provide a positive affine normalization, which makes the statistic $\mathrm{AD}_{\mathrm{up}}^{2}$ have a standard normal asymptotic distribution. We also obtain some results on the spectrum of a certain integral operator associated with the statistic. In Section 4, we provide evidence to demonstrate that, for $W_{n}^{2,\beta}$ , $1.5\leq\beta\leq 2$ , the tests’ practical utility may be limited due to a very slow rate of convergence of the finite-sample distribution to the asymptotic regime. Finally, Section 5 provides some concluding remarks.

2 Facts from theory of empirical processes

2.1 Preliminaries

Let $U_{1},\dots,U_{n}$ be independent $U(0,1)$ random variables, with corresponding order statistics $U_{(1)}\leq\cdots\leq U_{(n)}$ .

Definition 2.1.

For

E_{n}(s)=\begin{cases}0,&0\leq s<U_{(1)},\\ k/n,&U_{(k)}\leq s<U_{(k+1)},\quad k=1,\dots,n-1,\\ 1,&U_{(n)}\leq s\leq 1,\end{cases}

the uniform empirical process is defined as $e_{n}(s)=\sqrt{n}(E_{n}(s)-s)$ , $0\leq s\leq 1$ .

As shown in Shorack and Wellner (2009), pp. 3–5, and Csörgő and Horváth (1993), pp. 365–367, for a continuous $F_{0}(\cdot)$ , the probabilistic study of

\sqrt{n}(F_{n}(x)-F_{0}(x)),\quad x\in\mathbb{R},

is equivalent to that of $e_{n}(F_{0}(x))$ , $x\in\mathbb{R}$ , under the null hypothesis (1.1), as in this case $U=F_{0}(X)$ is $U(0,1)$ -distributed.

It is known (see Csörgő and Horváth 1993, Formula (5.1.65)) that

\{e_{n}(t)\psi(1-t),\,t\in(0,1)\}\overset{\mathrm{d}}{=}\{e_{n}(1-t)\psi(1-t),% \,t\in(0,1)\}

for each $n\geq 1$ , where $\overset{\mathrm{d}}{=}$ denotes equality in distribution.

Hence, in particular,

\int_{0}^{1}(e_{n}^{2}(t)/t^{2})\,\mathrm{d}t\overset{\mathrm{d}}{=}\int_{0}^{% 1}(e_{n}^{2}(t)/(1-t)^{2})\,\mathrm{d}t.

Thus, $\mathrm{AD}_{\mathrm{down}}^{2}\overset{\mathrm{d}}{=}\mathrm{AD}_{\mathrm{up}% }^{2}$ .

A standard Brownian motion will be denoted by $W=\{W(t)\mid W(0)=0,\,t\geq 0\}$ , with state space $\mathbb{R}$ and a standard Brownian bridge

B\overset{\mathrm{d}}{=}W(t)-tW(1)

for $t\in[0,1]$ .

Of key interest is an adaptation of Csörgő et al (1993), Theorem 1.1.

Proposition 2.2.

$W_{n}^{2}(\psi)\xrightarrow{\mathrm{d}\,}\int_{0}^{1}\psi(t)B^{2}(t)\,\mathrm{% d}t$ as $n\rightarrow\infty$ if and only if

\int_{0}^{1}\psi(t)t(1-t)\,\mathrm{d}t<\infty.

It follows that not all weights are suitable to produce a well-defined limiting distribution for the corresponding test statistic.

If $\int_{0}^{1}\psi(t)t(1-t)\,\mathrm{d}t<\infty$ , an application of Mercer’s theorem and the Karhunen–Loève theorem leads to the following classic representation (Shorack and Wellner (2009), pp. 201–213):

\int_{0}^{1}\psi(t)B^{2}(t)\,\mathrm{d}t\overset{\mathrm{d}}{=}\sum^{\infty}_{% k=1}\lambda_{k}X_{k},

where $X_{k}$ are independent random variables distributed as $\chi_{1}^{2}$ , ie, a chi-squared distribution with one degree of freedom.

Here, $\lambda_{k}$ are eigenvalues of the $L^{2}(0,1)$ integral operator

(K\varphi)(x)=\int_{0}^{1}k(x,y)\varphi(y)\,\mathrm{d}y,

(2.1)

with kernel $k(x,y)=(\min(x,y)-xy)\sqrt{\psi(x)\psi(y)}$ . That is, $\lambda_{k}$ solve $(K\varphi)(x)=\lambda\varphi(x)$ .

For $\psi(t)=1/(1-t)^{\beta}$ , $\beta<2$ , Deheuvels and Martynov (2003) demonstrates that $\lambda_{k}=(2\nu/z_{\nu,k})^{2}$ , where $z_{\nu,k}$ is the $k$ th positive zero of the Bessel function $J_{\nu}(\cdot)$ of the first kind, and $\nu=1/(2-\beta)$ .

In other words, in the case of $\psi(t)=1/(1-t)^{\beta}$ with $\beta<2$ , the spectrum of integral operator (2.1) is purely point. In contrast, in Section 3, we show that for $\beta=2$ the spectrum is purely absolutely continuous.

2.2 Classical results

Throughout, the limits are taken as $n\rightarrow\infty$ unless otherwise stated. We refer to known and proven results as propositions, and to important new results emerging from this work as theorems.

Proposition 2.3.

(i)

$\displaystyle\sup_{0\leq t\leq 1}\frac{|e_{n}(t)|}{(1-t)^{\beta}}\xrightarrow{% \mathrm{d}\,}\infty$ for any $\beta\geq 1/2$ .
(ii)

$\displaystyle\sup_{0\leq t\leq 1}\frac{|e_{n}(t)|}{\sqrt{t(1-t)}}\xrightarrow{% \mathrm{d}\,}\infty$ .

Proof.

Part (i) follows from Mason (1985), while part (ii) follows from Chibisov (1964), Theorem 2. ∎

Corollary 2.4.

Test statistics $\mathrm{AD}^{*}$ (1.3) and $\mathrm{AD}_{\mathrm{up}}$ (1.4) do not have well-defined limiting distributions.

Proposition 2.5.

(i)

$\displaystyle\int_{0}^{1}\frac{e_{n}^{2}(t)}{(1-t)^{\beta}}\,\mathrm{d}t% \xrightarrow{\mathrm{d}\,}\int_{0}^{1}\frac{B^{2}(t)}{(1-t)^{\beta}}\,\mathrm{% d}t$ if and only if $\beta<2$ .
(ii)

$\displaystyle\int_{0}^{1}\frac{e_{n}^{2}(t)}{(1-t)^{\beta}}\,\mathrm{d}t% \xrightarrow{\mathrm{d}\,}\infty$ for any $\beta\geq 2$ .

Proof.

Part (i) follows from Proposition 2.2, while part (ii) is a consequence of Csörgő et al (1993), Theorem 1.3, or, for $\beta=2$ , from the results of Shepp (1966), p. 353. ∎

Corollary 2.6.

Test statistics $\mathrm{AD}_{\mathrm{up}}^{2}$ (1.5) and $\mathrm{AD}_{\mathrm{down}}^{2}$ (1.6) do not have well-defined limiting distributions.

As discussed in Csörgő and Horváth (1988, 1993),

\psi(t)=\frac{1}{\sqrt{(t(1-t))}}\quad\text{and}\quad\psi(t)=\frac{1}{(t(1-t))% ^{2}}

are special weight functions, in the sense that they separate light weights from heavy ones in the supremum and integral class. In Csörgő and Horváth (1993), Chapter 5, it is emphasized that the asymptotic behavior of a weighted EDF-based test statistic will be determined by that of the uniform empirical process $e_{n}(t)$ on a subinterval $I_{n}\subseteq[0,1]$ for $t$ in the tails if $I_{n}$ is either $(0,a_{n}]$ or $[1-a_{n},1)$ , or in the middle, ie, if $I_{n}$ is $[a_{n},1-a_{n}]$ , where $a_{n}\downarrow 0$ .

Propositions 2.7 and 2.9 make this specific for the weight function

\psi(t)=\frac{1}{(1-t)^{\beta}}.

Proposition 2.7.

For any sequence of real numbers $\{a_{n}\}_{n=1}^{\infty}$ such that $a_{n}\rightarrow 0$ and $na_{n}\rightarrow\infty$ , we have the following.

(i)

If $-\infty<\beta<1/2$ , then

$a_{n}^{\beta-(1/2)}\sup_{1-a_{n}\leq t<1}\frac{|e_{n}(t)|}{(1-t)^{\beta}}% \xrightarrow{\mathrm{d}\,}\sup_{t\in[0,1]}\frac{|W(t)|}{t^{\beta}}.$
(ii)

If $\beta=1/2$ , then

$A(\tfrac{1}{2}\ln(na_{n}))\sup_{1-a_{n}\leq t<1}\frac{|e_{n}(t)|}{\sqrt{t(1-t)% }}-D(\tfrac{1}{2}\ln(na_{n}))\xrightarrow{\mathrm{d}\,}Y,$

where

$\displaystyle A(x)$ $\displaystyle=\sqrt{2\ln(x)},$

$\displaystyle D(x)$ $\displaystyle=2\ln(x)+\frac{\ln(\ln(x))}{2}-\frac{\ln(\pi)}{2}$

and $Y$ is the Gumbel random variable, which has a probability density function (PDF) of $\exp(-2\exp(-x))$ , $x\in\mathbb{R}$ .
(iii)

If $\beta>1/2$ , then

$n^{1/2-\beta}\sup_{1-a_{n}\leq t<1}\frac{|e_{n}(t)|}{(1-t)^{\beta}}% \xrightarrow{\mathrm{d}\,}\sup_{t\in(0,\infty)}\frac{|N(t)-t|}{t^{\beta}},$

where $\{N(t),t\geq 0\}$ is a Poisson process with $\mathbb{E}[N(t)]=t$ .
(iv)

If $\beta>1/2,$ then

$n^{1/2-\beta}\sup_{1-a_{n}\leq t<U_{(n)}}\frac{|e_{n}(t)|}{(1-t)^{\beta}}% \xrightarrow{\mathrm{d}\,}\sup_{t\in[S(1),\infty)}\frac{|N(t)-t|}{t^{\beta}},$

where $S(1)$ is the time of the first jump of $N(t)$ .

Proof.

See Csörgő and Horváth (1993), Theorem 1.2, p. 265. ∎

Remark 2.8.

It is known that (Mason 1985)

\sup_{t\in(0,\infty)}\frac{N(t)-t}{t}\overset{\mathrm{d}}{=}X,

where $X$ has a CDF of $P(X\leq x)=x/(1+x)$ for $x\geq 0$ , and $0$ otherwise. This is a Lomax distribution (Johnson et al 1994).

Proposition 2.9.

For any sequence of real numbers $\{a_{n}\}_{n=1}^{\infty}$ such that $a_{n}\rightarrow 0$ and $na_{n}\rightarrow\infty$ , we have the following.

(i)

If $-\infty<\beta<2$ , then

$a_{n}^{\beta-2}\int_{1-a_{n}}^{1}\frac{e_{n}^{2}(t)}{(1-t)^{\beta}}\,\mathrm{d% }t\xrightarrow{\mathrm{d}\,}\int_{0}^{1}\frac{W^{2}(t)}{t^{\beta}}\,\mathrm{d}t.$

(ii)

If $\beta=2$ , then

	$\displaystyle\frac{1}{2\sqrt{\ln(na_{n})}}$	$\displaystyle\bigg(\int_{1-a_{n}}^{1}\frac{e_{n}^{2}(t)}{t^{2}(1-t)^{2}}\,% \mathrm{d}t-\ln(na_{n})\bigg)\xrightarrow{\mathrm{d}\,}N(0,1)$
and
	$\displaystyle\frac{1}{2\sqrt{2\ln(n)}}$	$\displaystyle\bigg(\int_{0}^{1}\frac{e_{n}^{2}(t)}{t^{2}(1-t)^{2}}\,\mathrm{d}% t-2\ln(n)\bigg)\xrightarrow{\mathrm{d}\,}N(0,1).$

(iii)

If $\beta>2$ , then

$n^{2-\beta}\int_{1-a_{n}}^{1}\frac{e_{n}^{2}(t)}{(1-t)^{\beta}}\,\mathrm{d}t% \xrightarrow{\mathrm{d}\,}\int_{0}^{\infty}\frac{(N(t)-t)^{2}}{t^{\beta}}\,% \mathrm{d}t.$

Proof.

See Csörgő and Horváth (1993), Theorem 3.2, p. 325, and Csörgő and Horváth (1993), Equation (5.3.113), p. 335. ∎

3 Main results

In this section, we establish the following main theorems. Our proofs and auxiliary results that are of independent interest are given in the online appendix.

Theorem 3.1.

(i)

Statistics $\mathrm{AD}_{\mathrm{up}}^{2}$ and $\mathrm{AD}_{\mathrm{down}}^{2}$ possess asymptotic distributions under positive affine normalization:

$\frac{\mathrm{AD}_{\mathrm{up}}^{2}-\ln(n)}{2\sqrt{\ln(n)}}\overset{d}{% \rightarrow}N(0,1)\quad\text{and}\quad\frac{\mathrm{AD}_{\mathrm{down}}^{2}-% \ln(n)}{2\sqrt{\ln(n)}}\xrightarrow{\mathrm{d}\,}N(0,1).$
(ii)

For all real $x$ and $y$ ,

$P\bigg(\frac{\mathrm{AD}_{\mathrm{up}}^{2}-\ln(n)}{2\sqrt{\ln(n)}}\leq x,\frac% {\mathrm{AD}_{\mathrm{down}}^{2}-\ln(n)}{2\sqrt{\ln(n)}}\leq y\bigg)\to\varPhi% (x)\varPhi(y).$

That is, the random variables

$\frac{\mathrm{AD}_{\mathrm{up}}^{2}-\ln(n)}{2\sqrt{\ln(n)}}\quad\text{and}% \quad\frac{\mathrm{AD}_{\mathrm{down}}^{2}-\ln(n)}{2\sqrt{\ln(n)}}$

are asymptotically independent.

Corollary 3.2.

We have

\frac{\mathrm{AD}_{\mathrm{up}}^{2}-\ln(n)}{2\sqrt{\ln(n)}}+\frac{\mathrm{AD}_% {\mathrm{down}}^{2}-\ln(n)}{2\sqrt{\ln(n)}}\xrightarrow{\mathrm{d}\,}\sqrt{2}N% (0,1).

Theorem 3.3.

Statistic $W_{n}^{2,2}$ admits the decomposition $W_{n}^{2,2}=A_{n}+B_{n}$ , such that

\frac{A_{n}+2n}{2\sqrt{n}}\xrightarrow{\mathrm{d}\,}-X\quad\text{and}\quad% \frac{B_{n}-2n}{2\sqrt{n}}\xrightarrow{\mathrm{d}\,}X,

with $X$ being distributed as the standard normal distribution.

Theorem 3.4.

Let $k(x,y)=(\min(x,y)-xy)/((1-x)(1-y))$ for $x$ and $y\in(0,1)$ . Define the integral operator $K$ with kernel $k(x,y)$ as $(K\varphi)(x)=\smash{\int_{0}^{1}k(x,y)\varphi(y)\,\mathrm{d}y}$ . Then, the spectrum of this operator is simple, purely absolutely continuous, filling in the interval $[0,4]$ .

4 Practical utility of $W_{n}^{2,\beta}$ test statistics

Commonly used GoF test statistics have been observed to possess the following properties or stylized facts.

Property 1. The statistic has a finite mean and variance in finite samples.

Property 2. It possesses a nondegenerate limiting distribution.

Property 3. The limiting distribution has a finite mean and variance.

Property 4. The finite-sample distributions rapidly converge to the limiting distribution.

For example, the Kolmogorov–Smirnov, CvM $W_{n}^{2,0}$ , Anderson–Darling and modified Anderson–Darling $W_{n}^{2,1}$ test statistics (see Marsaglia et al 2003; Csörgő and Faraway 1996; Stephens 1974; Sinclair et al 1990) satisfy the properties for practically important upper-tail probabilities (0.8 and higher).

One may argue that the existence of the first two moments is of little to no importance in practical applications, where one is primarily concerned with computing a critical value that is a quantile. However, for a nonnegative random variable, the finiteness of its mean ensures that the random variable is well defined, ie, finite with probability $1$ (Gut 2013, Theorem 4.4, p. 52).

A conventional way to determine a finite-sample critical value is via Monte Carlo experiments. Although the processing power and available RAM of modern computers have significantly increased the computational tractability of Monte Carlo methods, the evaluation of critical values for sufficiently large samples still faces the problem of long execution time. In these circumstances, it is common to approximate the critical value by an asymptotic one.

In this section, we investigate if the class of $W_{n}^{2,\beta}$ test statistics for $\beta>1$ possesses Properties 1–4.

Consider testing $H_{0}\colon F(x)=F_{0}(x)$ versus $H_{1}\colon F(x)\neq F_{0}(x)$ for a continuous CDF $F_{0}(\cdot)$ with specified parameters. We take a sample of independent random variables $X_{1},\dots,X_{n}$ , distributed according to $F_{0}(\cdot)$ . If $X_{(1)}\leq\cdots\leq X_{(n)}$ are the corresponding order statistics, let $z_{j}=F_{0}(X_{(j)})$ .

In Feuerverger (2015), the following computational formula for $W_{n}^{2,\beta}$ is established:¹¹Note that the expression given in Feuerverger (2015) contains typos; it must be multiplied by $n$ to give the correct result (4.1).

W_{n}^{2,\beta}=\mathrm{Const}+\frac{2}{2-\beta}\sum_{j=1}^{n}(1-z_{j})^{2-% \beta}+\frac{1}{(\beta-1)n}\sum_{j=1}^{n}\frac{1+2(n-j)}{(1-z_{j})^{\beta-1}},

(4.1)

where $\mathrm{Const}=n((1/(3-\beta))-(2/(2-\beta))+(1/(1-\beta)))$ .

In Chernobai et al (2005), the computational formula for $W_{n}^{2,2}$ is given:

W_{n}^{2,2}=2\sum_{j=1}^{n}\ln(1-z_{j})+\frac{1}{n}\sum_{j=1}^{n}\frac{1+2(n-j% )}{(1-z_{j})}.

(4.2)

Observe that $W_{n}^{2,2}$ can be viewed as a limit of $W_{n}^{2,\beta}$ as $\beta\uparrow 2$ (cf. Feuerverger 2015).

Under the assumption that $F_{0}(\cdot)$ is fully specified, $z_{j}$ , $j=1\dots n$ , will follow a uniform distribution in $[0,1]$ . As such, $W_{n}^{2,\beta}$ is independent of the null distribution. After some straightforward algebra, it is not hard to determine the first two moments of $W_{n}^{2,\beta}$ . Indeed,

\mathbb{E}[W_{n}^{2,\beta}]=\frac{1}{(2-\beta)(3-\beta)}

for $\beta<2$ , and $+\infty$ otherwise. Similarly,

\mathbb{V}[W_{n}^{2,\beta}]=\frac{2}{(2-\beta)(5-2\beta)(3-\beta)^{2}}

for $\beta<2$ , and $+\infty$ otherwise. In Deheuvels and Martynov (2003), it is shown that the limiting distributions of $W_{n}^{2,\beta}$ ( $\beta<2$ ) have the same moments as their finite-sample analogs. As mentioned above, this means that $W_{n}^{2,\beta}$ (for $\beta<2$ ), and their limiting random variables, are well defined.

Although $W_{n}^{2,2}$ has no moments, it is still well defined, as is demonstrated in Lemma 4.1 below.

Lemma 4.1.

Under the null hypothesis,

P(W_{n}^{2,2}<\infty)=1.

Proof.

Observe that

\bigcup_{m=0}^{\infty}\{W_{n}^{2,2}\leq m\}=\{W_{n}^{2,2}<\infty\}

for all $n\in\mathbb{N}$ . Then (see Gut 2013, Theorem 3.1, p. 11), $P(W_{n}^{2,2}\leq m)\to P(W_{n}^{2,2}<\infty)$ as $m\to\infty$ . Clearly, $P(W_{n}^{2,2}\leq 0)=0$ . From (4.2), it is readily seen that $W_{n}^{2,2}\leq n/(1-z_{n})$ for all $n\in\mathbb{N}$ . Hence, $P(W_{n}^{2,2}\leq m)\geq P(n/(1-z_{n})\leq m)$ . But $P(n/(1-z_{n})\leq m)=(1-n/m)^{n}$ implies $P(n/(1-z_{n})\leq m)\to 1$ as $m\to\infty$ . This completes the proof. ∎

The results of Section 3 imply that, since no limiting distribution exists for $W_{n}^{2,2}$ , one may speak of there being no moments of the limiting distribution whatsoever. Therefore, while $\smash{W_{n}^{2,\beta}}$ ( $\beta<2$ ) enjoys the first three GoF properties identified above, $W_{n}^{2,2}$ , albeit well defined, fails each of them.

To study the convergence of the finite-sample distributions of $\smash{W_{n}^{2,\beta}}$ ( $\beta<2$ ) and $\smash{(W_{n}^{2,2}-\ln{(n)})/(2\sqrt{\ln{(n)}})}$ to their limiting distributions, we considered testing

H_{0}\colon F(x)=\varPhi(x)\quad\text{versus}\quad H_{1}\colon F(x)\neq\varPhi% (x)

for the standard normal CDF $\varPhi(\cdot)$ at a significance level of $\alpha=0.05$ .

For a range of sample sizes $n$ from $10$ to $50\,000$ , and various $\beta$ from 1 to 2, with an increment of $0.05$ , we tabulated $W_{n}^{2,\beta}$ using (4.1) and (4.2) by extensive Monte Carlo simulations ( $10^{6}$ trials for each sample size).²²Note that (4.1) is not applicable for $\beta=1$ . For this case, the computational formula is given in Sinclair et al (1990). We then computed critical values $\smash{C_{n,\alpha}^{2,\beta}}$ for $\smash{W_{n}^{2,\beta}}$ (see Figure 1).

Figure 1: Critical values $C_{n,\alpha}^{\text{2},\beta}$ for $W_{n}^{\text{2},\beta}$ by Monte Carlo simulations for selected values of $\beta\in[\text{1,2})$ , and for selected sample sizes ( $\text{10}^{\text{6}}$ trials for each sample size). Here, $\alpha=\text{0.05}$ , and the $\text{2.0}^{\ast}$ exponent refers to the normalized version of $W_{n}^{\text{2},\beta}$ .

Figure 2: Asymptotic critical values $C_{\infty,\alpha}^{\text{2},\beta}$ as a function of $\beta$ . Here, $\beta\in[\text{1,2})$ and $\alpha=\text{0.05}$ .

Figure 3: Monte Carlo finite-sample critical values and asymptotic critical value $\varPhi^{-1}(1-\alpha)$ (straight line) for the normalized version of $W_{n}^{\text{2,2}}$ for $\alpha=\text{0.05}$ .

The asymptotic critical values $C_{\infty,\alpha}^{2,\beta}$ , as a function of $\beta$ , are shown in Figure 2.³³This computation is based on the limiting distribution established in Deheuvels and Martynov (2003), Formula (1.4), p. 67. See also Table 1.1 therein. It is readily seen that $C_{\infty,\alpha}^{2,\beta}$ explode as they approach $\beta=2$ .

It is also informative to plot the finite-sample and asymptotic critical values together for the normalized version of the $W_{n}^{2,2}$ test statistic. It is seen that the finite-sample critical values $(C_{n,\alpha}^{2,2}-\ln{(n)})/(2\sqrt{\ln{(n)}})$ converge extremely slowly to the asymptotic critical values $\varPhi^{-1}(1-\alpha)$ . See Figure 3, where the case of $\alpha=0.05$ is depicted.

We considered a partition of the range of sample sizes $n$ from 10 to 50 000 into disjoint intervals $[n_{1},n_{2})$ . We also fitted a simple linear regression to the finite-sample critical values $\smash{C_{n,\alpha}^{2,\beta}}$ : $\smash{C_{n,\alpha}^{2,\beta}=an+b+\epsilon}$ in each interval. Here, $\epsilon$ is the error term.

We then assessed

H_{0}\colon a=0\quad\text{versus}\quad H_{1}\colon a\neq 0.

The $p$ -values are presented in Table 1 for $\alpha=0.05$ and selected $\beta$ values.

Table 1:

p

-values for testing

H_{\text{0}}\colon a=\text{0}

versus

H_{\text{1}}\colon a\neq\text{0}

at

\alpha=\text{0.05}

for

C_{n,\alpha}^{\text{2},\beta}=an+b+\epsilon

. [Here, samples whose size

n

falls within

[n_{1},n_{2})

are considered. The

\text{2.0}^{\ast}

exponent refers to the normalized version of

W_{n}^{\text{2},\beta}

.]

$n_{\textbf{1}}$	$n_{\textbf{2}}$	1.0	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	1.9	2.0	$\textbf{2.0}^{\ast}$
		$\beta$

0010	01 200	0.39	0.01	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
1200	08 000	0.19	0.20	0.12	0.09	0.02	0.00	0.00	0.00	0.00	0.00	0.00	0.00
8000	50 000	0.66	0.56	0.31	0.21	0.15	0.01	0.00	0.00	0.00	0.00	0.00	0.00

It is noteworthy that in all cases but one the $p$ -values behave as expected: they increase row-wise and decrease column-wise. In practice, for a given event type, the size of the available internal loss data of financial institutions does not typically exceed 1000 events, while for external data the size does not exceed 50 000 points. Thus, Table 1 suggests that there is insufficient evidence that $\smash{C_{n,\alpha}^{2,\beta}}$ , $1.5\leq\beta\leq 2$ , and the normalized version $\smash{(C_{n,\alpha}^{2,2}-\ln{(n)})/(2\sqrt{\ln{(n)}})}$ flatten over the entire range of sample sizes important in OpRisk applications. As such, the knowledge of the theoretical limiting distribution is of little utility. Instead, a Monte Carlo method should be applied to determine critical values even for large sample sizes.

5 Conclusions

In this paper, motivated by the problem of selecting parametric distributions for modeling OpRisk loss severities more adequately in the upper tails, we have examined a class of weighted EDF-based GoF test statistics $W_{n}^{2,\beta}$ when $\beta\leq 2$ . As $\beta$ gets closer to $2$ , $W_{n}^{2,\beta}$ puts more weight on the upper tail of the distribution under the null hypothesis. The statistic $W_{n}^{2,2}$ is used for severity distribution selection by some OpRisk modelers in the financial industry.

Given the importance of this class of test statistics in the severity selection process, we have more carefully studied the behavior of $W_{n}^{2,\beta}$ for $1.5\leq\beta\leq 2$ and discovered that the practical utility of the statistics is likely limited. In fact, it has been established that $W_{n}^{2,\beta}$ for $\beta\geq 2$ has no well-defined limiting distribution.

Therefore, OpRisk modelers applying LDA principles should be highly cautious when using $W_{n}^{2,\beta}$ for $1.5\leq\beta<2$ , and exercise extreme caution when attempting to use the statistic for $\beta\geq 2$ .

Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of this paper. The views expressed in this paper are the authors’ own and do not represent the views of their respective employers or any other institution.

Acknowledgements

The results in this paper are part of the first author’s work on his PhD thesis at McMaster University (Mayorov 2017). The authors wish to thank Andrei Biryuk for stimulating technical discussions. The authors also thank the anonymous referee for comments and suggestions for the improvement of the presentation of this paper.

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