On Parameter-Mixing of Dependence Parameters

Transcription

1 On Parameter-Mixing of Dependence Parameters by Murray D Smith 1 and Xiangyuan Tommy Chen 2 Econometrics and Business Statistics The University of Sydney This draft prepared for the ESRC Econometric Study Group Conference Bristol, July 2006 Abstract: The method of parameter mixing has served to introduce new distributions into statistical practice. These distributions are usually more flexible in the sense that they may contain an increased number of parameters as compared to that of the unmixed, parent distribution. The classic example of parameter mixing is the Beta-Binomial distribution, a univariate distribution formed by assigning a Beta distribution to the success probability of the Binomial distribution. The object of interest in this article is the dependence structure of a collection of two or more random variables, where this is measured by a copula function, the parameters of which are termed dependence parameters. The key issue is to establish to what extent does parameter-mixing of dependence parameters contribute, or enhance, the dependence structure, and promote improved outcomes to statistical modelling. The results derived here may have further implications because of recent proposals (e.g. Patton [5]) to model dependence parameters in terms of covariates of the random variables of interest. Keywords: Dependence parameter; Copula; Parameter-mix; AMH family. 1 Address for correspondence July January 2007: Murray D Smith, Visiting Fellow, Department of Economics and Related Studies, The University of York, Heslington, York YO10 5DD, UK. (Murray.Smith@econ.usyd.edu.au) 2 Econometrics and Business Statistics, The University of Sydney, Sydney NSW 2006, Australia. (tommy_chen@student.usyd.edu.au) 1

2 1 Introduction In multivariate contexts, the dependence parameters (i.e. the parameters of the copula function) typically enter in classic pairwise measures of association between random variablessuchasspearman srhos ρ and Kendall s tau τ. For example, the dependence parameter θ [ 1, 1] of the Farlie-Gumbel-Morgenstern family of 2-copulas (FGM hereafter) C θ (u, v) =uv(1 + θ(1 u)(1 v)) (1) where (u, v) II 2 =[0, 1] [0, 1], findsitprominentintherelevantformulas: S ρ = θ/3 and τ =2θ/9. With data, estimation of dependence parameters (along with any other margin parameters) may often proceed using standard techniques, the point being that these unknown parameters are assumed fixed in the population of interest. One way in which parametric statistical models have been imbued with greater flexibility is to relax the notion of parameter fixity. One such technique that is well known throughout the statistical literature is (termed here) parameter-mixing. Roughly, the basic idea is to assign the parameter of interest a distribution, such that the latter itself depends on a parameter set larger in dimension than the base, or parent distribution. For extensive discussion and numerous examples see Johnson et al [3, Chps 8-9]. Perhaps the most well-known example of parameter-mixing is in the generalisation of the 2-parameter univariate Binomial(n, p) distribution to the 3-parameter univariate Beta-Binomial(n, α, β) distribution, where the latter is formed by assigning a Beta(α, β) distribution to the success probability of the former. Put formally, this parameter-mix is defined as the following expectation with respect to the parameter distribution: Beta-Binomial(n, α, β)=binomial(n, P ) Beta(α, β) P = E[Binomial(n, P )] where the operator is a popularly used notation to denote parameter-mixing. Here, the Binomial(n, p) parent is interpreted as a conditional distribution, where conditioning is on a value p assigned to the variable P. In this instance, parameter-mixing achieves a remarkable success, because what began as a 2-parameter statistical model has now, through mixing, been grown to a new statistical model whose three parameters are formally identified. 2

3 Parameter-mixing ideas applied to dependence parameters would at first sight appear to give the same opportunity to enhance the flexibility of the resulting copula function. To establish notation, for a (possibly vector-valued) dependence parameter Θ = θ ascribed the distribution F(λ), whereλ are themselves parameters assumed to have dimensionality at least that of Θ (preferably more because the desired aim is to enhance parametric flexibility), the F parameter-mix of the 2-copula C θ is given by 3,4 C 0 λ(u, v)=c Θ (u, v) Θ F(λ) = E [C Θ (u, v)] Z = C θ (u, v)df (θ; λ) ZΘ = C θ (u, v)f(θ; λ)dθ (2) Θ where F (θ; λ) denotes the cdf (cumulative distribution function) of Θ and f(θ; λ) the pdf (probability density function) of Θ; the last line assumes continuity of the mixing distribution. The outcome Cλ 0 (u, v) is a 2-copula. Finally, having averaged over the members of the C θ family, it is clear that the dependence structure of Cλ 0 can cover no more than that of C θ, and can be less if F(λ) is used to represent prior information. The firstreadilyobviousresulttoflow from (2) occurs if the parent copula C θ is linear in θ, for then, provided μ F = E[Θ] exists, the parameter-mixed copula is equivalent to the parent copula, apart from its parameterisation. Thus, there can be no gain made from parameter-mixing. Indeed, if the parameter space induced through the new dependence parameter μ F exceeds that of θ, then the additional parameters cannot be identified. To illustrate this, compare the functional form of the following parameter-mixed copula with that of the FGM parent (1) that is linear in θ. Let X F = Beta(α, β), for which the transform 2X 1 projects onto the same support as that of θ. Then, consider the parameter-mix: Z 1 FGM (2X 1) = uv(1 + (2x 1)(1 u)(1 v)) xα 1 (1 x) β 1 dx Θ 0 B(α, β) = uv (1 + μ(1 u)(1 v)) 3 Extending parameter-mixing to an arbitrary dimension copula is straightforward, but as the main ideas can be conveniently expressed in two dimensions then only 2-copulas will be considered in this article. 4 An alternative nomenclature is the convex sum of C θ with respect to F; see Nelsen [4, Sec ]. 3

4 where μ = α β. Clearly, the FGM dependence structure has been preserved under α+β parameter-mixing, but under this specification of F both induced dependence parameters α and β cannot separately be identified. The re-parameterisation in X that sets α = 1+μ β 1 μ would serve to isolate the object μ as the dependence parameter and eliminate the superfluous parameter β, but in doing so nothing whatsoever is gained by parameter-mixing. A second result, somewhat similar to the first, is that if the parameter-mix Cλ 0 is of the same family as C θ, thenthereisnogaintobemadefromparameter-mixing. Consider, for example, Mardia s family of comprehensive copulas, indexed by θ [ 1, 1] : C θ (u, v) = 1 2 θ2 (1 + θ)m +(1 θ 2 )Π θ2 (1 θ)w where M =min(u, v) and W =max(u + v 1, 0) are the extremal 2-copulas, and Π = uv is the Product copula. Despite being non-linear in θ, parameter-mixing applied to the Mardia family serves merely to recover the parent, differing only in its parameterisation. If parameter-mixing is to enhance the flexibility of the resulting copula function, from this and the previous example it is clear that mixing must generate a family of copulas that are functionally different to the parent. 2 Example: The Gumbel-Barnett Family of Copulas Consider the Gumbel-Barnett family of 2-copulas (GB hereafter) C θ (u, v) =Π exp( θ(log u)(log v)) (3) where the family is indexed by values assigned to θ (0, 1]. Obviously, lim θ 0 C θ (u, v) =Π and C θ (u, v) < Π, so the GB family covers a region of negative dependence. In terms of Spearman s rho, coverage is S ρ < 0. The lower bound is found by substitution of θ =1into Z S ρ =12 C θ (u, v)dudv 3 I 2 =12θ 1 e 4/θ G (4/θ) 3 where G(z) = R e t t 1 dt is a special case of the incomplete gamma function such that, z for θ (0, 1], 0 <G(4/θ) G(4) = In this example, parameter-mixing yields a family of copulas different to that of the parent family, with interest here centreing on whether the number of dependence parameters can be increased from the original singleton, and still be formally identified. For 4

5 a distribution F with pdf f(θ; λ), the F parameter-mix of the GB family of copulas is, applying (2), C 0 λ(u, v)=c Θ (u, v) Θ F(λ) = Π Z 1 0 exp( θ(log u)(log v))f(θ; λ)dθ = Π mgf F ( (log u)(log v)) where mgf F denotes the moment generating function of F, i.e. E[exp(tΘ)]. In particular, for F = Beta(α, β), where α>0 and β > 0 are parameters, with mgf given by the confluent hypergeometric function 1 F 1 (α; α + β; t),t IR, then the copula of the Beta parameter-mix of the GB family is given by, C 0 α,β(u, v)=π 1 F 1 (α; α + β; (log u)(log v)) where the second line uses Kummer s relation = Π exp( (log u)(log v)) 1 F 1 (β; α + β;(logu)(log v)) 1F 1 (p; q; x) =e x 1F 1 (q p; q; x). For further details on the confluent hypergeometric function see, for example, Slater [8]. It is evident that limiting cases applied to the parameters (α, β) correspond to the extremes of dependence that Cα,β 0 covers. For instance, allowing α to be free and letting β 0 finds Cα,β 0 (u, v) C 1(u, v). Equally, α and β free finds Cα,β 0 (u, v) C 1(u, v). Likewise, allowing α to be free but letting β finds Cα,β 0 (u, v) Π, as too α 0 and β free finds Cα,β 0 (u, v) Π. That we cannot distinguish between, for example, α becoming large and β becoming small, is an indication that the introduction of additional dependence parameters does not lead to added flexibility, for those parameters are not identified. This can be formalised by examining the Fisher Information matrix for (α, β), I α,β = i α i αβ i αβ i β where its elements can be obtained using Theorem 1 of Smith [9], valid under any pair of continuous margins bound together by an assigned copula. To illustrate, the element corresponding to α is given by Z µ 2 1 i α = I c 0 2 α,β (u, v) α c0 α,β(u, v) dudv 5

6 where the copula density c 0 α,β(u, v) = 2 u v C0 α,β(u, v). Explicit expressions in terms of (α, β) for (i α,i β,i αβ ) cannot be obtained; however, numerical integration can be used to approximate the Fisher Information matrix at given values of (α,β). The following table lists a small selection of results in the parameter space: Table: Fisher Information matrix and associated eigenvalues (to 3dp) (α, β) I α,β Eigenvalues (1, 1) (2, 4) i α i αβ TBA i αβ i β TBA (2, 10) i α i αβ TBA i αβ i β TBA To numerical precision, the (scaled) second eigenvalue is so small in each case that this is strong evidence that both parameters (α, β) are not identified in the Beta parameter mix of the GB family of copulas. 3 Example: The Ali-Mikhail-Haq family of copulas In this example, parameter-mixing yields a family of copulas different to that of the parent family, with interest here centreing on measuring the improvement in modelling, if any, with artificially generated data. Consider the Ali-Mikhail-Haq family of 2-copulas (AMH hereafter) C θ (u, v) = uv 1 θ(1 u)(1 v) (4) where the family is indexed by values assigned to the dependence parameter θ [ 1, 1]. Let X F = Beta(α, b), where parameter α>0and scalar b>0is known; clearly, the support of 2X 1 projects onto the same range of values as that of the dependence parameter. The Beta(α, b) parameter-mix of the AMH family of copulas is, applying (2), 6

7 defined as Cα(u, 0 v; b)=c Θ (u, v) (2X 1) Θ Z 1 uv = 0 1 (2x 1)(1 u)(1 v) 1 uv = Beta(α, b) 1+(1 u)(1 v) Z 1 µ 1 x α 1 (1 x) b 1 dx Beta(α, b) 1 2(1 u)(1 v) 0 1+(1 u)(1 v) x x α 1 (1 x) (α+b) α 1 dx uv = 1+(1 u)(1 v) 2 F 1 (1,α; α + b; s) uv = 1 (1 u)(1 v) 2 F 1 (1,b; α + b; s) (5) where 2(1 u)(1 v) s = 1+(1 u)(1 v) is such that 0 s 1 when (u, v) II 2, and the solution to the integral can be deduced as a special case of the single-valued, analytic definition of the Gaussian hypergeometric function given by Euler (e.g. see Rainville [6, p.47]): 2F 1 (p, q; r; s) = 1 Beta(q, r q) Z 1 0 (1 sx) p x q 1 (1 x) r q 1 dx provided arg(1 s) <π.note that the last line of (5) is obtained by using Euler s transformation: 2F 1 (p, q; r; s) =(1 s) p 2F 1 µp, r q; r; s. s 1 To establish the dependence coverage of the Beta mixed family of AMH copulas, observe that C 0 α(u, v; b) =C 1 (u, v) 2 F 1 (1,α; α + b; s) with, for 0 <α<, the Gaussian hypergeometric function satisfying 1 < 2 F 1 (1,α; α + b; s) < 1 F 0 (1; s) (6) where 1F 0 (1; s) =(1 s) 1 1+(1 u)(1 v) = 1 (1 u)(1 v) = C +1(u, v) C 1 (u, v). (7) For fixed u, v and b, 2 F 1 (1,α; α + b; s) takes smaller (larger) values corresponding to smaller (larger) values of α. If the limit cases corresponding to α 0 and α are 7

8 included, the range of dependence coverage of the mixed family of copulas is, as expected, equivalent to that of the parent family 5 ; i.e. C 1 (u, v) C 0 α(u, v; b) C +1 (u, v) where these inequalities can be found by substituting (7) into (6) and multiplying through by C 1 (u, v); note that C 1 (u, v) = Π 2 Σ + Π and C +1 (u, v) = Π Σ Π where Σ = u+v. In terms of Spearman s measure S ρ, the mixed family covers the interval ( , ), to all intents the same coverage as that of the parent AMH family of copulas. The models considered in the experiment that follows are the AMH and the Powermixed AMH, where Power refers to the Power distribution that is equivalent to the Beta(α,1) distribution. In this experiment, both margins are Uniform(0, 1) distributed, in which case the copula of the joint distribution is equivalent to the joint cdf. Thus, the joint cdf of the AMH model is given by (4), and the joint cdf of the Power-mixed AMH model is given by (5), with b =1. Each model has one parameter, a dependence parameter, being θ in the AMH model, and α in the Power-mixed AMH model. For a random sample of size n on (U, V ), the log-likelihood for θ (i.e. for the AMH model) is denoted by log L = nx log 2 u v C θ(u i,v i ) and the log-likelihood for α (i.e. for the Power-mixed AMH) is denoted by i=1 log L 0 = nx log 2 u v C0 α(u i,v i ;1) i=1 where, for the sake of brevity, the detailed expressions for these are omitted. In this experiment, sample size was fixed at n =1000, with data pairs pseudo-randomly generatedfromeachmodelbasedontheamhalgorithmgivenbynelsen[4,p.101]. Parameter values are assigned such that a large portion of the dependence structure is 5 Although the coverage range of Cα(u, 0 v; b) matches that of the parent family C θ (u, v), one important member of the latter is excluded from the former, namely Π. 8

9 covered. The simulation results are reported in Table 1, where the reported parameter estimates bα and β b along with associated standard errors (in braces) and maximised loglikelihood values are averages taken over 200 independent replications. The first set of results (those labelled α =0.05,..., 50) correspond to when the Power-mixed AMH is the true data-generating process, while the second set (those labelled θ = 0.9,..., 0.9) are when the AMH is the true data-generating process. On a comparison of the fits alone, as measured by the maximised log-likelihood values, the performance of the true data generating process is superior to that of the misspecified model whichever way around the results are viewed; the Power-AMH provides a better fit whenitisthetruemodel,whiletheamhhastheedgewhenitisthetruemodel. This tends to dispel the view that parameter-mix models must generalise the parent because the dependence parameter can be viewed as varying under the former, but fixed under the latter. The parent family of copulas is not a baseline from which improvement in fit will be made possible through parameter-mixing applied to the dependence parameters, rather the models are competing on an equal footing in the parametrically non-nested sense because the parameter space that indexes both families is of the same dimension (to ensure identification). Any differences in fit as measured by the maximised log-likelihoods vanish as α tends to either extreme of the parameter space when the Power-mixed AMH is true, as too thesameoccursasθ ±1 when the AMH is true. This is to be expected, for both competing families of copulas have the same extremals, C 1 (u, v) and C +1 (u, v), and therefore becoming increasingly difficult to distinguish despite the fact that one of them is always misspecified. The fit of the true model exceeds the fit of the misspecified model by the greatest amount when α =2under the Power-mix AMH, and when θ =0.2 under the AMH. Expressed in terms of Spearman s rank correlation S ρ, both of these values coincide reasonably close to S ρ =0.1, with the latter happening to be close to the mid-point of thedependencecoverage ofbothmodels[ , ]. This suggests that should any advantage be derived by applying parameter-mixing to dependence parameters, then that will require the data to cooperate by exhibiting sample dependence in the middle of the range of dependence coverage of the parent family of copulas. 9

10 4 Empirical Application It is interesting to observe that while the AMH family (4) nests the Product copula as a special case, i.e. Π = uv = C 0 (u, v), the Beta mixed family (5) does not. For Cα(u, 0 v; b) to nest Π, the Gaussian hypergeometric function 2 F 1 (1,α; α + b; s) would have to be simplifiable to (1 1 2 s) 1 =1+(1 u)(1 v) for some α at every given b, but no such set of pairings can be found. This implies that even though the point at which the mixed copula generates zero dependence, that point does not represent independence; it is not Π. For the mixed AMH copula to retain independence as a special case, a mixture that restricts attention to either positive or negative ranges of dependence is required. Such informative mixtures will retain the Product copula as a special case provided it is generated at an extreme of the parameter space; i.e. provided Π is at the edge of the support of the mixing distribution then it will not be mixed away. Furthermore, informative mixtures are of interest in their own right for modelling purposes as will be illustrated. To construct a positive informative mixture, contrast the derivation of (5) with the following Beta parameter-mix of the AMH copula: C 0+ α (u, v; b)=c Θ (u, v) Θ X Z 1 uv x α 1 (1 x) b 1 = dx 0 1 x(1 u)(1 v) Beta(α, b) = uv 2 F 1 (1,α; α + b;(1 u)(1 v)) (8) that covers positive dependence Π <C 0+ α (u, v; b) <C +1 (u, v) where parameter α>0; the limit cases are lim α 0 Cα 0+ (u, v; b) =Π and lim α Cα 0+ (u, v; b) = C +1 (u, v). A negative-coverage parameter mix can be constructed by reflecting the mixing Beta distribution about the origin: C 0 α (u, v; b)=c Θ (u, v) Θ ( X) Z 1 uv x α 1 (1 x) b 1 = dx 0 1+x(1 u)(1 v) Beta(α, b) = uv 2 F 1 (1,α; α + b; (1 u)(1 v)) (9) where parameter α>0, with limit cases lim α 0 C 0 α (u, v; b) =Π and lim α C 0 α (u, v; b) = 10

11 C 1 (u, v). Another negative-coverage parameter-mix shifts the mixing Beta: Z 1 C Θ (u, v) (X 1) = Θ 0 = uv x α 1 (1 x) b 1 dx 1 (x 1)(1 u)(1 v) Beta(α, b) µ uv 1+(1 u)(1 v) 2 F 1 1,α; α + b; (1 u)(1 v) 1+(1 u)(1 v) where the Product copula Π corresponds to the limit case α, and C 1 (u, v) as the limit case α 0. Clearly, informative mixing is one means by which prior information can be imposed. The following bivariate data are analysed to investigate the properties of mixing, especially informative mixing, in empirical modelling. The data set is constructed from the career average statistics for players in the National Basketball Association (NBA) in the United States, where the period from which the data are drawn dates from the season through to the season. The two variables of interest are Assists Per Minute (AP M) and Points Per Minute (PPM); both measures are commonly used as indications of player quality in terms of, respectively, passing and scoring, see Chatterjee [1] 6. Figure 1 provides a scatter plot of these n =1988pairs, as well as a histogram of each margin. Two observations in particular are highlighted, these being (to 1995) the career high PPM record that is held by Michael Jordan, and the career high AP M record that is held by John Stockton. Simonoff [7, Chp.4] examined a subset of these data; namely, AP M and PPM for 96 NBA guards in the season. Using kernel smoothing methods, he detected bimodality in the joint distribution of AP M and PPM; a finding that motivated partitioning players into two groups according to their AP M score. On doing so, Simonoff detected positive correlation amongst the guards for which their AP M < 0.2, and negative correlation amongst those for which AP M > 0.2. Our larger data set, on the other hand, contains records on all NBA players, not just guards, as too it records career statistics, not single season averages. While for all n =1988pairs the sample Spearman rank correlation coefficient S r =0.0482, a search instead across AP M-ordered pairwise data partitions that maximises the difference in S r uncovers the opposite findingtosimonoff s. 6 The raw data assisted points, points scored and minutes played are obtained from Microsoft Complete NBA (1994) CD-ROM. Because minutes played were not recorded prior to the season, any assists and points recorded on a player in seasons prior to this have been excluded from their AP M and PPM career averages. 11

12 Amongst those n + =1299players for whom AP M < (hereafter termed Partition 1), finds S r = While for the remaining n =689players (hereafter termed Partition 2), their S r = Ten models based on the AMH copula and the Power-mixed AMH copulas are fitted to these data, two for the full set of n = 1988 data and eight for the four partitioned data sets. In each case, the two-step IFM estimation method is used whereby the margins are estimated first followed by estimation of the dependence parameter of the joint model, but with the margins fixed at their first round estimates. Joe [2, Chp.10] discusses the IFM estimator at length, demonstrating that it is consistent and asymptotically normal under standard regularity conditions. Following Joe s suggestion, the variance-covariance matrix of the asymptotic distribution is estimated by using a randomised block Jackknife approach. The estimation results appear in Table 2. The first outcome is that there is a decisive improvement in fit as a result of partitioning the data according to the rule AP M < , than in not doing so. This can be seen by comparing the maximised likelihood of either the AMH or the better-fitted Power-mixed AMH ( ) with the sum of the maximised log-likelihoods of, say, the informative Power-mixed AMH sforeachpartition( = ). The difference is already so heavily in favour of partitioned models that imposing a standard AIC or BIC penalty adjustment for the extra five parameters they contain will not change the outcome. Preferred marginal models are sought amongst a range of univariate models 7.Forthe AP M margin of each data partition, the preferred models in both cases is the truncated standard Beta distribution. For Partition 1 data, the Beta is truncated on the right at , and this is denoted by Beta r, while for Partition 2 data, the Beta is truncated on the left at , andthisisdenotedbybeta. For PPM, the preferred models for Partition1 and Partition 2 data are respectively the Beta and the Normal. Point estimates and associated standard errors are given in Table 2, while Figure 2 plots frequency polygons of the data for each margin in each partition, overlaying these with the preferred fit (the dashed lines). The joint models reported in Table 2 include the AMH and the informative Power- 7 The univariate models considered for fitting the margins included the Normal, Beta, Inverse Gamma, Inverse Gaussian, Lognormal, Weibull and a Pearson system fit. 12

13 mix AMH distributions. With the margins fixed,themodelscompeteonthebasisof copula specification. For Partition 1, the AMH family of copulas provides a worse fit ( ; fitted S b ρ =0.1694) than the positive-coverage Power-mixed AMH ( ; fitted S b ρ =0.1634), but not greatly so. This outcome is consistent with the findings of the Monte Carlo experiment in the previous section, because the sample S r = lies reasonably well away from the extremes of the dependence coverage of the competing families of copulas. On the other hand, because S r = for Partition 2 is reasonably close to the negative extreme of S ρ = for C 1 (u, v), there can be little difference expected between both sets of results. Indeed, the results for Partition 2 in Table 2 bear that out. References [1] Chatterjee, S., and Yilmaz, M.R. (1999). The NBA as an evolving multivariate system, The American Statistician, 53, [2] Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall: London. [3] Johnson, N.L., Kotz, S., and Kemp, A.W. (1993). Univariate Discrete Distributions. Wiley: New York. [4] Nelsen, R. B. (2006). An Introduction to Copulas. 2nd edition. Springer-Verlag: New York. [5] Patton, A. J. (2006). Modelling asymmetric exchange rate dependence. International Economic Review, 47, [6] Rainville, E. D. (1960). Special Functions. MacMillan: New York. [7] Simonoff, J. S.(1996). Smoothing Methods in Statistics. Springer-Verlag: New York. [8] Slater, L. J. (1960). Confluent Hypergeometric Functions. Cambridge University Press: Cambridge. [9] Smith, M. D. (2005). Invariance theorems for Fisher Information. Unpublished manuscript. 13

14 Table 1: Estimates of the AMH and Power mixed AMH models AMH Power-AMH Parameter S ρ b θ log L ba log L 0 α = (0.0882) (0.0444) (0.1079) (0.1133) (0.1076) (0.1023) (0.0983) (0.0929) (0.0727) (0.0593) (0.0522) (0.0462) (0.0321) (0.0217) (0.0130) θ = (0.0868) (0.1076) (0.1103) (0.1026) (0.0947) (0.0840) (0.0649) (0.0368) (0.0245) (0.0581) (0.0857) (0.1174) (0.1448) (0.1637) (0.1914) (0.3722) (0.5984) (0.7958) (1.0601) (2.5647) (7.4427) ( ) (0.0432) (0.0572) (0.0836) (0.1241) (0.1662) (0.2339) (0.4535) (1.7771) (4.9038) Notes: (i) Sample size n =1000 (ii) Standard errors in braces (iii) Estimates are averaged over 200 replications (iv) Figures to 4dp

15 Table 2: NBA data: IFM estimates of the AMH and Power AMH models Margins APM, PPM (i) Copula Estimate log L Entire data: Beta( b θ = (0.0637) (0.8204) (0.0778) n =1988 N( ) (5) bα = (0.0027) (0.0020) (0.1618) Partition 1 (ii) : Beta r ( b θ = ( ) (4.5668) (0.0685) n + =1299 Beta( ) (8) bα = (0.3542) (0.5820) (0.1699) Partition 2 (ii) : Beta ( b θ = (0.7087) (3.3704) (0.1284) n =689 N( ) (9) bα = (0.0037) (0.0051) (1.2457) Notes: (i) Cells contain the fitted models for AP M above and PPM underneath (ii) Partition 1: AP M < , Partition 2: AP M > (iii) Jackknife standard errors in braces (iv) Figures to 4dp 15

16 Figure 1: Scatter plot of PPM against AP M, with histograms of each attached 16

17 Figure 2: Frequency polygon and preferred fit (dashed) for AP M-partitioned data 17