MODELS WITH HIDDEN REGULAR VARIATION: GENERATION AND DETECTION

We review the notions of multivariate regular variation (MRV) and hidden regular variation (HRV) for distributions of random vectors and then discuss methods for generating models exhibiting both properties concentrating on the non-negative orthant in dimension two. Furthermore we suggest diagnostic techniques that detect these properties in multivariate data and indicate when models exhibiting both MRV and HRV are plausible ﬁts for the data. We illustrate our techniques on simulated data, as well as two real Internet data sets.

management [9,19]. Often the observed data are multidimensional and generated by complex systems. Empirical evidence often indicates heavy-tailed marginal distributions and the dependence structure between the various components must be discerned.
Analysis of multivariate heavy-tailed models is facilitated by knowledge of model generation methods. Generation methods often lead to efficient simulation algorithms, suitable statistical models, appropriate estimation techniques, and control policies. In a risk-assessment setting, generation techniques help in stress-testing worst-case scenarios.
We pursue two broad themes in this paper: First, we adapt a standard general model generation technique based on the polar coordinate transform [26, p. 198] that produces tractable models in the particular cases of regular variation on E := R 2 + \ {0} and E 0 := (0, ∞) 2 . We discuss proposals that produce models with both MRV and HRV and outline their strengths and weaknesses. Second, we discuss diagnostics for exploratory detection and identification of multivariate heavy-tailed models prior to estimating model parameters. These ideas can be adapted for dimensions higher than two but we do not pursue this adaptation here. 1.1. Motivation. Suppose we have a vector Z = (Z 1 , Z 2 ) giving the risk of two assets and we wish to estimate the probability P[Z ∈ R] of a remote risk region beyond the range of observed data. A solution based on the asymptotic assumption of data being heavy tailed is to assume convergence of the measures to a limit measure ν(·) for some limit function b(n) → ∞ as n → ∞. Temporarily ignore technical issues such as the meaning of the arrow "→", how we estimate ν(·), how to get b(n), and which sets are allowed to be inserted in place of (·) in (1.1). A quasi-solution in this simple view is to estimate with where hats indicate quantities needing estimates. We wonder if risk contagion is present and if both components can be simultaneously large; that is, whether P[Z 1 > x 1 , Z 2 > x 2 ] > 0? For models with asymptotic independence (which means ν((0, ∞ 2 )) = 0; see Section 1.4), we estimate this probability as 0 because ν(·) concentrates all its mass on the axes. Does this mean that the risk is actually zero or merely that we used the wrong asymptotic regime for the estimation? Perhaps the scaling function b(t) was too big. Should the state space for the risk estimation problem be R 2 + or (0, ∞) 2 ? In general if R is disjoint from the support of ν(·), our risk estimate is zero and we wonder if we chose an asymptotic method ill-suited to the purpose. The idea behind hidden regular variation is if the support of ν(·) is small (eg. the two co-ordinate axes from 0) and does not contain R, we further concentrate on the complement of the support (eg. (0, ∞) 2 ) for a second, more appropriate regular variation property where the limit measure has support intersecting R.

1.2.
Outline. The mathematical framework for the study of multivariate heavy tails is regular variation of measures. The theory is flexible when given for closed subcones of metric spaces [21], but we specialize to subcones of R 2 + where statistical results are most readily exhibited. Statistical extensions to higher dimensions will be discussed elsewhere. We list needed notation in Section 1.3 for reference. The definitions of multivariate regular variation (MRV) and hidden regular variation (HRV) are reviewed in Section 1.4 where general concepts are adapted for subcones in two dimensions. In Section 2, assuming asymptotic limit measures are specified, we adapt the standard multiplicative method for generating regularly varying models based on the generalized polar coordinate transform [10,21,26] to E = R 2 + \ {0} and E 0 = (0, ∞) 2 producing relatively tractable models.
In Section 3 we discuss generation of models that exhibit both MRV and HRV. When both MRV and HRV are present, one must be careful to properly take into account their interaction since otherwise estimation procedures will be misinterpreted. We review two model generation methods that yield both MRV on E and HRV on E 0 and discuss properties of each method. These methods are (i) the mixture method, and (ii) the additive method. We give particular attention to the recently proposed additive generation method (ii) of [32] and study why interaction of MRV and HRV means asymptotic parameters may not be coming from the anticipated summand of the representation. Accompanying simulation examples illustrate our discussion. Section 4 gives techniques for detecting when data is consistent with a model exhibiting MRV and HRV. These techniques rely on the fact that under broad conditions, if a vector X has a multivariate regularly varying distribution on a cone C, then under a generalized polar coordinate transformation (see (1.4)), the transformed vector satisfies a conditional extreme value (CEV) model for which detection techniques exist from [8]. Our methodology is more reliable than one dimensional techniques such as checking if one dimensional marginal distributions are heavy tailed or checking whether one dimensional functions of the data vector such as the maximum and the minimum component are heavy tailed.
In Section 5, we give two examples of our detection and model estimation techniques applied to Internet downloads and HTTP response data. Concluding comments are in Section 6 and Section 7 contains proofs of the propositions in Section 3.
1.3. Basic notation. We summarize some notation and concepts here. For this paper, we have dimension d = 2 unless otherwise specified. We use bold letters to denote vectors, with capital letters for random vectors and small letters for non-random vectors, e.g., y = (y 1 , y 2 ) ∈ R 2 . We also define 0 = (0, 0) and ∞ = (∞, ∞). Vector operations are always understood component-wise, e.g., for vectors x and y, x y means x i y i for i = 1, 2. Some additional notation follows with explanations that are amplified in subsequent sections. Detailed discussions are in the references provided.

RV β
Regularly varying functions with index β > 0; that is, functions We can and do assume such functions are continuous and strictly increasing. See [2,11,27].
The set of all non-zero measures on C \ C 0 which are finite on subsets bounded away from the forbidden zone C 0 . C(C \ C 0 ) Continuous, bounded, positive functions on C \ C 0 whose supports are bounded away from the forbidden zone C 0 . Without loss of generality [21], we may assume the functions are uniformly continuous. [10,18,21] and Definition 1.1.
The random elements X and Y are independent.
multivariate regular variation; for this paper, it means regular variation on E. HRV hidden regular variation; for this paper, it means regular variation on E 0 . GPOLAR Polar co-ordinate transformation relative to the deleted forbidden zone C 0 , ). See [10,21].
1.4. Regularly varying distributions on cones. We review material from [10,18,21] describing the framework for the definition of MRV and HRV specialized to two dimensions. Note that the convergence concept used for defining regular variation is M-convergence which is slightly different from vague convergence traditionally used in such cases. Reasons for preferring M-convergence are discussed in Remark 1.1 below and in [10,21].
Consider R 2 + as a metric space with Euclidean metric d(x, y). A subset C ⊂ R 2 + is a cone if it is closed under positive scalar multiplication: if x ∈ C then c x ∈ C for c > 0. A proper framework for discussing regular variation is measure convergence defined by M-convergence [10,21] on a closed cone C ⊂ R 2 + with a closed cone C 0 ⊂ C deleted. Call the deleted cone C 0 the forbidden zone. The two cases of interest in this paper are is the space for defining M-convergence appropriate for regular variation of distributions of positive random vectors. The forbidden zone is the origin . Then E 0 := C \ C 0 = (0, ∞) 2 , the first quadrant without its axes, is the space for defining M-convergence appropriate for HRV. The forbidden zone is the set of axes emanating from the origin in the positive direction.
Let M(C \ C 0 ) be the set of Borel measures on C \ C 0 which are finite on sets bounded away from the forbidden zone C 0 [10,18,21]. We think of sets bounded away from the forbidden zone C 0 as tail regions. We now formulate M-convergence which becomes the basis for our definition of multivariate regular variation.
dμ for all bounded, continuous, non-negative f on C \ C 0 whose support is bounded away from C 0 . Definition 1.2. A random vector Z 0 is regularly varying on C \ C 0 with index α > 0 if there exists b(t) ∈ RV 1/α , called the scaling function, and a measure ν(·) ∈ M(C \ C 0 ), called the limit or tail measure, such that as t → ∞, We write Z ∈ MRV (α, b(t), ν, C \ C 0 ) to emphasize that regular variation depends on an index α, scaling function b ∈ RV 1/α , limit measure ν, and state space C \ C 0 . Since b(t) ∈ RV 1/α , the limit measure ν(·) has a scaling property, c>0.

Regular variation under polar coordinate transformation.
It is convenient to transform (1.2) and (1.3) using generalized polar coordinates [10,21]. We define this for general cones of the form C \ C 0 , but will restrict our attention eventually to E and E 0 . Set ℵ C 0 = {x ∈ C \ C 0 : d(x, C 0 ) = 1}, the locus of points at distance 1 from the deleted forbidden zone C 0 . Define GPOLAR : Consequently, the inverse GPOLAR ← : (0, ∞) × Then (1.2) and (1.3) are equivalent to is a probability measure on ℵ C 0 [10,21]. The transformation GPOLAR depends on the forbidden zone C 0 and this dependence should be understood from the context.
Unpacking the notation we obtain the two regular variation limits Using GPOLAR, separately for the two cones (1.7) and (1.8) we get for any norm · , (1.10) where S and S 0 are probability measures on ℵ 0 and ℵ [axes] respectively. Note So we may rewrite (1.10) as two statements: for x 1, In terms of G 1 , G 2 , p and q, (1.10) is equivalent to and we will often use (1.13) in place of (1.10).
Remark 1.1. Traditionally, regular variation on E relied on vague convergence, the polar coordinate transform x → ( x , x/ x ) and Radon measures being finite on relatively compact sets; see [26]. In order to make the natural tail regions appearing in practice to be relatively compact, the theory required one point uncompactification of a compactified version of E; see [26] for further details. On E = [0, ∞) 2 this works fine because {x ∈ E : x = 1} is compact and lines through ∞ cannot carry mass. However, on E 0 the traditional unit sphere {x ∈ E 0 : x = 1} is no longer compact. Hence, Radon measures on {x ∈ E 0 : x = 1} may not be finite (eg. ν 0 (x, ∞) = (x 1 x 2 ) −1 ) and for estimation problems the approach relying on vague convergence is a dead end if estimation of a possibly infinite measure is required. Since ℵ [axes] is bounded away from the forbidden zone, limit measures on this set are finite. By using ℵ [axes] as the new unit sphere and the equivalent convergence condition (1.13) we are able to estimate the hidden measure (at least a transformed version) by estimating probability measures irrespective of whether the hidden angular measure is finite or not. More details on why M-convergence, an approach without compactification, is desirable are in [10,18,21].

Generating regularly varying models.
In this section we discuss methods for generating regularly varying models on cones in R 2 + . We adapt a standard scheme for generating regularly varying distributions. This approach generates the full totality of asymptotic limits but not the full totality of pre-asymptotic models; so there can be many other ways to get the same asymptotic models. This approach is a multiplicative method relying on the polar co-ordinate transformation of the random vectors.
2.1. Generating regular variation on E. The easiest way to obtain a regularly varying model on E with scaling function b(t) and limit measure ν(·) = ν α × S • GPOLAR is as follows: Suppose R is a random element of (0, ∞) with a regularly varying tail and scaling function b(t): and consequently (1.7) hold. Reminder: GPOLAR is defined relative to the deleted forbidden zone {0} in this case.

2.2.
Generating regular variation on E 0 (and sometimes also on E). As suggested in [22], we may follow the same scheme as in Section 2.1. Let R 0 be a random element of (0, ∞) that is regularly varying with index α 0 and scaling function b 0 (t). Let Θ 0 be a random element of ℵ [axes] with distribution S 0 and independent of R 0 . Then is regularly varying with scaling function b 0 (t) and limit measure ν 0 := (ν α 0 × S 0 ) • GPOLAR on E 0 because (1.10) and therefore (1.8) hold. Reminder: GPOLAR in this case is defined relative to the deleted forbidden zone consisting of [axes].
In practice we specify the measure S 0 on ℵ [axes] as follows: Let G 1 , G 2 be two probability measures on (1, ∞) and define smears probability mass on the horizontal line emanating from (1, 1) and G 2 does the same thing for the vertical line. For estimation purposes, note for s > 1 that and is thus a somewhat restricted case. Regular variation on E 0 by itself does not in general imply one dimensional regular variation of the marginals. Moreover, if the tails of G i are heavier than the tail of R, we can have regular variation on E 0 with index α 0 but the tails of Z 1 and Z 2 may be regularly varying with a smaller index α. Full discussion is in [22].
3. Generating models that have both MRV on E and HRV on E 0 . Section 2 discussed multiplicative methods to generate regularly varying models separately on E and E 0 . Here we pursue the first theme of the paper, namely methods that generate models with both regular variation on E and hidden regular variation on E 0 . We give two methods for generating such models that we call the mixture method and the additive method. The mixture method is somewhat easier to analyze and the majority of the section is devoted to the nuances of using the additive method under different convergence regimes in E and E 0 . We pay particular attention to interaction between limit measures which are generated by separately generating regularly varying random vectors in E and E 0 .
3.1. Mixture method. This method [22,26] expresses the random vector Z as a mixture where Y gives the regular variation on E and V gives the regular variation on E 0 and B is a Bernoulli mixing variable with P[B = 1] = 1 − P[B = 0]. Since HRV implies that MRV on E must induce asymptotic independence [25,26], we need Y to model MRV with index α on E and have asymptotic independence. So we take Y to concentrate on the set [axes] and by the scheme of Section 2.2 to be regularly varying on E 0 with limit measure ν 0 and scaling function b 0 (t). Provided that the resulting Z has both MRV on E and HRV on E 0 :

3.2.
Additive method. In their paper, [32] advocate an additive model of the form The idea is the limit measure of Z on E should come from Y and the limit measure ν 0 on E 0 should come from V . They argue that this representation has advantages for parameter estimation and the additive model overcomes the undesirable and usually unrealistic feature of the mixture method where generated points are installed directly on the axes. However, while this additive model is a nice pre-asymptotic model, it is does not always successfully separate the limit measure on E and the hidden measure on E 0 in an identifiable manner.
We consider the additive model under three separate scenarios. Proofs for the claims under these scenarios are deferred to Section 7.
(a) Y has form (3.1) and therefore has MRV but not HRV and V has MRV on E without asymptotic independence and therefore has HRV on E 0 . In this case Z is regularly varying on E with asymptotic independence and has HRV with scaling function, index and limit measure same as that of V . (b) Y is not necessarily of form (3.1) but has MRV while V has MRV on E (and hence HRV on E 0 ) but not asymptotic independence. There is a tail condition (3.5) as well. Then Z is regularly varying on E with asymptotic independence and has HRV on E 0 with scaling function, index and limit measure same as that of V . (c) Y has MRV but not HRV and V has both MRV on E and HRV on E 0 .
Identifiability issues are rampant for this case.
Case (a): Y has no HRV and V has MRV on E. We start with this simplest result. The following proposition summarizes the findings. (1) Y has the structure given in (3.1) (so that Y has no HRV) and (3.2) holds.
, limit measure ν 0 ∈ M(E) and no asymptotic independence. Regular variation of V on E has the consequence that for i = 1, 2, Remark 3.1. Condition (2) in Proposition 3.1 is equivalent to the hidden limit measure ν 0 having finite spectral measure with respect to the conventional unit sphere since V has MRV on E. So the construction in Proposition 3.1 yields only a special case of HRV since there are many cases where (3.4) fails.
Consider an example to make the ideas clearer.
This construction makes the marginals of V = (V 1 , V 2 ) regularly varying with index α 0 which is consistent with V being MRV on E rather than just E 0 : where the ∼ can be verified either directly or by applying Breiman's theorem [4] on products.
To check whether we can identify the distributions of Y and V from a data sample of Z = Y + V , we simulate data following this model for three different choices of α while keeping α 0 fixed. We then check whether we can estimate back the values of α and α 0 . In all three cases α 0 = 2 with Θ 1 d = Θ 2 and (Θ 1 − 1), (Θ 2 − 1) are iid standard exponential distributions, and p = 0.5. In each case we simulate 10000 iid samples from Z. Then we create Hill plots for the marginals of Z 1 and Z 2 to identify the value of α. To detect the hidden part we create a Hill plot for min(Z 1 , Z 2 ) to find the value of α 0 . Referencing (1.13), we also make a QQ plot of max(Z 1 /Z 2 , Z 2 /Z 1 ) for the 100 highest values of min(Z 1 , Z 2 ) against the quantiles of standard exponential which is the distribution of Θ 1 and Θ 2 . We discuss the cases below.
• Case 1: α = 1. The top panel of Figure 3 indicates that we can identify the tails of Z to be heavy tailed. The correct index α = 1 is slightly overestimated. The Hill plot of min(Z 1 , Z 2 ) also indicates HRV on E 0 with index close to α 0 = 2. The QQ plot of max(Z 1 /Z 2 , Z 2 /Z 1 )  thresholded by the 100 largest values of min(Z 1 , Z 2 ) against standard exponential shows a decent fit. • Case 2: α = 1.5. The top panel of Figure 4 again indicates that we can identify the tails of Z to be heavy tailed. The index α is again overestimated, this time more than in the previous case, perhaps because of the closeness of α to α 0 . The Hill plot of min{Z 1 , Z 2 } also indicates HRV on E 0 with index close to α 0 = 2. The QQ plot of max{Z 1 /Z 2 , Z 2 /Z 1 } thresholded by the 100 largest values of min{Z 1 , Z 2 } against standard exponential shows a decent fit again. • Case 3: α = 0.5. In this case too, the top panel of Figure 5 indicates heavy-tailed Z. The Hill plot of min(Z 1 , Z 2 ) also indicates hidden regular variation. The indices α = 0.5 and α 0 = 2 are reasonably estimated here, presumably because the original values of α and α 0 are far apart. However, the exponential QQ plot of max{Z 1 /Z 2 , Z 2 /Z 1 } for the 100 largest values of min{Z 1 , Z 2 } struggles to indicate an exponential fit.

Case (b): Y has MRV and V has MRV.
For Case (b) we remove the restriction in Proposition 3.1 that Y = (Y 1 , Y 2 ) concentrates on the axes. However, to guarantee that the tails of V and Y do not interact in such a way so as to obscure the fact that the hidden measure of Z is that of V we need a tail condition comparing the tails of Y with V . Continue to suppose Y ⊥ ⊥ V .  MRV(α, b(t), ν, E) and exhibits asymptotic independence.

The interaction of the tails of Y and V is controlled by the condition
, ν, E) and asymptotic independence. 2. HRV on E 0 with index α 0 , scaling function b 0 (t), limit measure ν 0 restricted to E 0 . [25,26] with index 2α and condition (3.5) is needed to guarantee that the HRV in Z comes from V and not Y . Condition (3.5) is equivalent in this case to and it is sufficient that This is seen by noting that for Y 1 , Y 2 iid regularly varying with index α, and since b ← (b 0 (t)) → ∞ and b(·) is the scaling function of Y 1 , this is asymptotic to We need lim t→∞ t/(b ← (b 0 (t))) 2 = 0 and unwinding this condition yields (3.6).
Case (c): Y does not have HRV and V has both MRV on E and HRV on E 0 . A problem with the additive model is that the tail weights contributing to MRV on E and HRV on E 0 can be confounded between Y and V and it is possible for V to have MRV on E, HRV on E 0 but the hidden measure of Z = Y + V is not the hidden measure of V .
To focus on the influence of V , we again assume Y satisfies (3.1) as in Proposition 3.1.

For specificity, specify the scaling functions
and define another scaling function h(t) through its inverse h ← (t) by .
with asymptotic independence and has HRV on E 0 with index α + α * and limit measure (different than the hidden measure of V ): and Z has asymptotic independence. Z has HRV on E 0 with hidden limit measure ν 0 and scaling function b 0 (t) equal to those of V . A sufficient condition then Z ∈ MRV(α, b(t), ν, E) with asymptotic independence and Z has HRV with index α + α * , scaling function b 0 (t) and hidden measure ν Z which is a linear combination of the measure given in (3.9) and ν 0 , the hidden measure of V , where C 0 = c 1/α 0 . A sufficient condition for (3.10) is α * = α 0 − α. See Section 7 for the proof. We discuss an example to clarify ideas here.
Example 3.2. We illustrate instances of the three cases given in Proposition 3.3. We simulate data samples from three different regimes as discussed in the Proposition 3.3 and estimate back the parameters of the additive model from which the data was generated. We can check that the limit measure ν Z,hidden in (3.9) has density from which one can readily compute G 1 from (1.11) for s > 1 as A similar calculation will lead to G 2 (s) = (const)s −1/2 , s > 1 meaning both G 1 and G 2 have regularly varying tail distributions with index 1/2. In fact they are both Pareto (1/2) distributions. We generate 10000 iid samples following the construction of Z = Y + V described above and check whether we can estimate the regular variation index α = 0.5, the hidden regular variation index α + α * = 1.5 and the tail index of G 1 and G 2 from the sample. Figure 6 shows Hill plots for Z 1 and Z 2 in the top panel, both of which indicate that the marginals are heavy tailed with parameter α = 0.5. The Hill plot of min{Z 1 , Z 2 } correctly identifies the HRV parameter α + α * = 1.5. The final Hill plot of max{Z 1 /Z 2 , Z 2 /Z 1 } for the 200 highest order statistics of min{Z 1 , Z 2 } clearly indicates a heavy tail with a tail index of 1/2 for both G 1 and G 2 . Note since G 1 = G 2 , (1.13) allows doing the estimation using the thresholded maxima of the component ratios. We generate 10000 iid samples using the construction of Z = Y + V and from this sample we estimate the regular variation index α = 0.5, the hidden regular variation index α 0 = 1.25 and the tail index of G 1 and G 2 which is 1. The top panels in Figure 7 display Hill plots for Z 1 and Z 2 that indicate the same tail index of α = 0.5. The Hill plot for min{Z 1 , Z 2 } correctly indicates a tail index of α 0 = 1.25. Finally, the Hill plot of max{Z 1 /Z 2 , Z 2 /Z 1 } for the 200 highest order statistics of min{Z 1 , Z 2 } indicates a tail index of α * = 1 for both G 1 ≡ G 2 . • Case 3: α + α * = α 0 . Let α = 0.5, α * = 1, α 0 = 1.5 which satisfies α + α * = 1.5 = α 0 . We generate Y as in Case 1 or 2 and generate V using the method of Case 2, except that now R is generated from a Pareto α 0 = 1.5 distribution. We verify that Z = Y + V ∈ MRV(α = 0.5, t 2 , {0} × ν 1/2 + ν 1/2 × {0} , E) and Z = Y + V ∈ MRV(1.5, t 1/1.5 , ν Z , E 0 ). Getting the distribution of G 1 and G 2 is more difficult in this case since the hidden limit measure for Z is more complicated as can be seen in (3.12). A careful calculation shows that G 1 and G 2 have regularly varying tails with index 0.5. We generate 10000 iid samples of Z = Y + V using this model. In Figure 8 the Hill plots for Z 1 and Z 2 are in the neighborhood of α = 0.5 and the Hill plot for min{Z 1 , Z 2 } correctly indicates a tail index of α 0 = 1.5 The Hill plot of max{Z 1 /Z 2 , Z 2 /Z 1 } for the 200 highest order statistics of min{Z 1 , Z 2 } indicates a tail index of α * = 0.5 for both G 1 ≡ G 2 which was what we were expecting.

Detection and estimation: Regular variation and hidden regular variation.
What diagnostic tools exist to help us verify that multivariate data come from a distribution possessing regular variation on some domain? Since regular variation is only an asymptotic tail property, the task of deciding to use a multivariate regularly varying model is challenging.
Suppose we have Z = (Z 1 , Z 2 ) multivariate regularly varying on E = [0, ∞) 2 \ {0}. Under the transformation GPOLAR as defined in (1.4), Z is regularly varying with some tail index α and (1.6) holds. Diagnostics that investigate if Z is regularly varying often reduce the data to one dimension for instance by taking norms or max-linear combinations of Z [26,Chapter 8] and then apply one dimensional heavy-tail diagnostics such as Hill or QQ plotting. We propose further diagnostics for the viability of a multivariate regularly varying model using the GPOLAR transformation since GPOLAR converts a regularly varying model to a conditional extreme value (CEV) model for which detection techniques exist [8].
4.1. Detecting multivariate regular variation using the CEV model. The conditional extreme value model [7,8,16] requires at least one of the marginal distributions be in the domain of attraction of an extreme value distribution. In this section we discuss a modified version of the CEV model for bivariate random vectors whose first components are non-negative and where convergences are described by M-convergence [10,21]. Define Additionally assume that Then we say (ξ, η) satisfies a conditional extreme value model and write (ξ, η) ∈ CEV(a, b, μ). Note that assumption (b) in the definition is a normalization, and the theory would work assuming H to be a finite measure.

If a(t) = b(t), t > 0 then (ξ, η)
is multivariate regularly varying on E with limit measure μ. (In such a case μ cannot be a product measure).
Remark 4.2. Statistical plots that check whether bivariate data can be modelled by a CEV model were derived in [7] and are based on the Hillish, Pickandsish and Kendall's Tau statistics. If data is generated from a CEV model, these statistics tend to a constant as the sample size increases. We concentrate on the Hillish and Pickandsish statistics for this paper. We will further specialize to the case where μ is a product measure μ = ν α × H for reasons that will be clear in the next subsection.
The η-variable corresponding to ξ (i) , also called the concomitant of ξ (i) .
Hillish statistic. For 1 k n, the Hillish statistic is
The proof follows from Propositions 2.2 and 2.3 in [8]. The only difference here is the use of measure μ instead of μ * and the roles of the first and the second components are switched.
Pickandsish statistic. This statistic gives another way to check the suitability of the CEV assumption and to detect a product measure in the limit. The Pickandsish statistic is based on ratios of differences of ordered concomitants and is patterned on the Pickands estimate for the scale parameter of an extreme value distribution (Pickands [23], de Haan and Ferreira [11, page 83], Resnick [26, page 93]). For notational convenience for s t write η * s:t := η * s : t . We define the Pickandsish statistic for 0 < q < 1 as

is a product measure if and only if
Pickandsish k,n (q) P → 0 The proof follows from Proposition 2.4 in [8]. The second part is immediate from (4.5).

4.2.
Relating MRV and CEV. We have methods to detect a CEV model and indicate when the limit is a product measure. What is the connection with multivariate regular variation? This connection is given in (1.7)-(1.10). Regular variation of a vector Z on E and E 0 with scaling functions b(t) ∈ RV 1/α and b 0 (t) ∈ RV 1/α 0 respectively with 0 < α α 0 is equivalent to: in M((0, ∞) × ℵ 0 ) and (4.6) (4.7) If ℵ 0 and ℵ [axes] were subsets of R we could conclude that (4.6) and (4.7) describe CEV models and modest changes, described in the next two results, allow use of the CEV model diagnostics.
Proposition 4.3 is easily deducible from the relationship between S and S and Proposition 4.4 follows from the connection between S 0 and G 1 , G 2 .

Testing for MRV and HRV: data examples.
Here we analyze data sets to see whether a multivariate regularly varying model is a valid assumption. We also look for asymptotic independence and if it exists we test for the existence of hidden regular variation. The relevant plots for this section appear at the end of the paper.
Example 5.1 (Boston University: HTTP downloads.). The first data set is obtained from a now classical Boston University study [5] which suggested self-similarity and heavy-tails in web-traffic data. Our dataset was created from HTTP downloads in sessions initiated by logins at a Boston University computer laboratory. It consists of 8 hours 20 minutes worth of downloads in February 1995 after applying an aggregation rule to downloads to associate machine-triggered actions with human requests and is discussed in [15, page 176]. The result of the aggregation is 4161 downloads which are characterized by the following variables: • S = the size of the download in kilobytes, • D = the duration of the download in seconds, • R = throughput of the download; that is, = S/D.
Previous studies [26, page 299, 316] indicate heavy-tailed behavior of all three variables and asymptotic independence between D and R. We concentrate on the variables (D, R) so our data is {(D i , R i ); 1 i 4161}. Moreover the rank-transformed variables are denoted: for 1 i 4161 with the generic rank-transformed variables denoted D * and R * respectively.
In Figure 9 we plot Hill estimates of the tail parameters of D and R for increasing number of order statistics of their respective univariate data values. Both plots are consistent with D and R being heavy tailed with tail parameters α D and α R greater than 1. (This is confirmed [12,26,29]  The angular density plot of (D * , R * ) shows data concentration near 0 and π/2 consistent with asymptotic independence of the quantities. Asymptotic independence does not automatically imply HRV so we check for HRV on E 0 . We proceed by testing the following: 1. Is the variable A = min{D * , R * } regularly varying with parameter greater than 1? The bottom right plot in Figure 9 plots Hill estimates for increasing number of order statistics of A and stabilizes between 2 and 3 indicating the desired heavy-tail behavior. 2. For D * > R * , we check whether (A, θ 1 ) := (min{D * , R * }, D * R * ) follows a CEV model. In the top panel of Figure 10, the Hillish plots of (A, θ 1 ) and (A, −θ 1 ) are close to 1 near the left side of their plots. Moreover we observe that the Pickandsish estimate at q = 0.8 also remains near 0. From Propositions 4.1 and 4.2, this is consistent with (A, θ 1 ) ∈ CEV(b 0 , 1, μ 0 ) with a limit measure of the CEV being a product measure. 3. For D * < R * , we similarly check whether (A, θ 2 ) := (min{R * , D * }, R * D * ) follows a CEV model. In the bottom panel of Figure 10 we observe that the Hillish plots of (A, θ 2 ) and (A, −θ 2 ) are close to 1 near the left side of their plots. We also observe that the Pickandsish estimate at q = 0.8 remains near 0. Hence we again conclude that the evidence is consistent with (A, θ 2 ) ∈ CEV(b 0 , 1, μ 0 ) with a limit measure of the CEV being a product measure.
The rank transformation causes (D * , R * ) to be standard regularly varying with α = 1 and Proposition 4.4 implies (D * , R * ) has hidden regular variation on E 0 if (and only if) for some function b 0 . Thus modeling the joint distribution of (D, R) using MRV and HRV is consistent with the data. The next step is to estimate the distributions of θ 1 ∼ G 1 and θ 2 ∼ G 2 as well as q defined in Proposition 4.4. Figure 11 plotŝ (2) . . . form order statistics from A i ; 1 i 4161. Observing Figure 11 for k near 0, an estimate of q isq = 0.6.
To find the distribution of θ 1 we make a standard exponential QQ plot of log(D , which serves as an exploratory diagnostic for heavy tails. We also create Hill plots for D * i /R * where A i > A (k) for two choices of k. The top panels of Figure 12 give the QQ plot for k = 100 (left) and the Hill plots for k = 100 and 400 (middle and right). The bottom panels in Figure 12 have a histogram of D * i /R * i for A i > A (100) (left) and kernel density plots of D * i /R * i for A i > A (100) (middle) and A i > A (100) (right). The plots indicate G 1 is heavy tailed with an index between 1.5 and 2 and we can provide a density estimate for the distribution of θ 1 .
We create the same set of plots for finding G 2 in Figure 13 which also indicates towards a similar conclusion of heavy-tailed behavior for G 2 with an index close to but less than 2.
Example 5.2 (UNC Chapel Hill HTTP response data). A response is the data transfer resulting from an HTTP request. The data set [17] consists of 21,828 thresholded responses bigger than 100 kilobytes measured between 1:00pm and 5:00pm on 25th April, 2001. We use similar notation as in Example 5.1.
• S = HTTP response size; total size of packets transferred in kilobytes, • D = the elapsed duration between first and last packets in seconds of the response, • R = throughput of the response = S/D.
Thus, the data set consists of {(S i , D i , R i ); 1 i 21828}. Our interest is in the variables (S, R) which exhibit heavy tails and asymptotic independence [17]. Denote the rank-transformed variables: with the generic rank-transformed variables denoted S * and R * respectively. The top left plots in Figure 14 give Hill plots of the tail indices of the distributions of S and R and suggest these indices are between 1 and 2. Asymptotic independence of S, R is exhibited in the angular density plot (top middle plot) for (S * , R * ). We next inquire if HRV exists on E 0 . The Hill plot for min(S * , R * ) on the upper right panel of Figure 14 gives a tail estimateα 0 clearly greater than 1 and is consistent with HRV. We transform the data {(S * , R * ); 1 i 21828} with the transformation GPOLAR) to obtain: model with product limit measure. The Hillish plots are reasurringly hovering at height 1 and the Pickandsish plots center at 0. So we have accumulated evidence that the data is consistent with an HRV model on E 0 . Now we proceed to provide some estimates on the structure of the hidden angular measure, which boils down to estimating three things 1. The proportion q appearing in μ 0 in Proposition 4.4: this can be estimated byq 21, 828. where (2) . . . form order statistics from A i ; 1 i 21, 828 as in Figure 15. Looking at the plot for k near zero, we can estimatep = 0.55. 2. The distribution of θ 1 ∼ G 1 : see Figure 16. First we make a standard exponential QQ plot of log(S * i /R * i ) when A i > A (100) . This acts as a diagnostic for heavy tails. This plot clearly indicates against heavy tails as does a Hill plot of S * i /R * i when A i > A (100) . A histogram and kernel density estimate plot of (S * i /R * i ) for A i > A (100) points towards a light-tailed distribution. 3. The distribution of θ 2 ∼ G 2 : see Figure 17. As before, first we make a standard exponential QQ plot of log(R * i /S * i ) when A i > A (100) , and the points nicely hug a straight line which indicates presence of heavy tails. The Hill plots of R * i /S * i when A i > A (100) and A i > A (400) provide an estimate of the tail index to be between 1 and 1.5. The histograms and kernel density estimates seem to support that the distribution of G 2 is heavy tailed.

Conclusion.
In this paper we have discussed different techniques to generate models which exhibit both regular variation and hidden regular variation. We have seen some simulated examples where we can estimate the parameters of both MRV and HRV but there are also examples where it is difficult to correctly estimate parameters. We restricted ourselves to the two dimensional non-negative orthant here, but clearly some of the generation techniques can be extended to higher dimensions. Moreover, the detection techniques for HRV on E 0 using the CEV model can also be extended to detect HRV on other types of cones especially in two dimensions but perhaps even more. Overall this paper serves as a starting point for methods of generating and detecting multivariate heavy-tailed models having tail dependence explained by HRV.
Proof of Proposition 3.2. As in Proposition 3.1, we focus on the HRV claim. Again assume that f ∈ C((0, ∞) 2 ) where f is uniformly continuous, bounded by a constant f and for some η > 0. We need to show (7.1). For any small δ > 0 with δ < η, the absolute value of the difference in (7.1) is bounded by since for the second term, the only way the difference can be non-zero is if both V 1 and V 2 are sufficiently large. Observe that → 0, as δ → 0.
For I we have I Ia + Ib + Ic. where The term Ia can be quickly killed, from (3.5). The term Ib also tends to zero since The term Ic is handled similarly.
The convergence in (7.3) can be similarly proved.