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Modelling with parameter- mixture copulas October 2006 Xiangyuan Tommy Chen Econometrics & Business Statistics The University of Sydney Supervisor: Murray D Smith

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I Introduction Copulas: A function that binds together univariate marginal distributions, to form a multivariate distribution. E.g. A bivariate copula C(u,v), with domain (0≤u ≤1) and (0≤v ≤1), binds two marginal distribution functions F (x) and G (y) to produce a bivariate distribution function:

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I Introduction Parameter mixing –A hierarchical model: the parent distribution function with parameter θ –Assume parameter not constant, but follows a distribution with pdf : –Then X has the following mixture distribution function: Famous example: the Beta-Binomial distribution: –In the Binomial(n,p) distribution, assume the success probability p has a Beta(a,b) distribution. –This mixing generates a 3-parameter distribution (n,a,b).

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II Past Research Copulas: –Large body of work on theory and application of copulas. –A flexible way of modelling correlation. Mixture distributions: –Used to generate many new models. Mixture copulas: –Small literature. –Nelsen (1999): copula after mixing is still a copula. –Mikusinski et al (1991): probabilistic interpretation; uniform mixture of “shuffles of C”. –Ferguson (1995): models from uniform mixtures of “shuffles of C”. Two major deficiencies: –Relationship between mixture copulas and parent copulas. –Modelling with mixture copulas other than uniform mixtures of “shuffles of C”.

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III. Properties of mixture copulas Question: Does mixing applied to dependence parameters introduce useful new copulas? –Do mixture copulas have desirable properties? –What is the relationship between mixture copulas and their parents?

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III. Properties of mixture copulas Mixing applied to several copula families which are useful in modelling: –Ali-Mikhail-Haq (AMH) –Farlie-Gumbel-Morgenstern (FGM) –Gumbel-Barnett These distributions were mixed with: –Beta distribution Other copulas were also mixed with: –Gamma –Exponential

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III. Examples of mixture copulas

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III. Properties of mixture copulas Equivalent functional form – If new copula functionally equivalent to old, nothing is gained. –E.g. Copula is linear in parameters. Dependence coverage – Mixture family can have up to the same coverage as old. –Each copula family can describe a range of dependence structures, indexed by a dependence parameter. –Since mixing averages across the parent family, coverage of the mixture family is the same as that of the parent family –Limiting forms of mixture family match limiting forms of parent family.

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III. Properties of mixture copulas Identification – If new parameters not identified, nothing is gained. –Parameter-mixing usually extends flexibility of model. But added flexibility comes not from parameter mixing itself; Added flexibility occurs only if parameter space is extended –Even if one dependence parameter becomes two through mixing, they will not be identified: Increasing/decreasing one has the same effect as decreasing/increasing the other. –If new parameters are not identified, then the model is not successfully extended from one parameter to two parameters through mixing.

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IV. Experiment An experiment to compare the modelling properties of mixed and unmixed copulas Data are generated from: 1.The AMH copula with uniform (0,1) margins 2.The AMH-Beta(a,1) mixture copula with uniform (0,1) margins For each set of data, fit: 1.The AMH copula with uniform (0,1) margins 2.The AMH-Beta(a,1) mixture copula with uniform (0,1) margins MC iteration: oThe experiment is conducted for a range of parameter values. oEach experiment is repeated 200 times at n=1000. Sample average results reported.

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IV. Experiment - Results The mixed model is not the generalisation of the unmixed model. –Each model performs better when it is the true model. –Mixing constructs a non-nested model. –Parameter-mixing adds flexibility only if parameter space is extended –Mixed model is unable to generate the product copula (independence case) Advantage disappears towards the limits of dependence –Models indistinguishable at limits. –Greatest advantage occurs near centre of dependence range.

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V. Application – NBA Basketball Data: US professional basketball player statistics. Investigate dependence between: –Assists per minute (APM) –Points per minute (PPM) Career averages for players from the season through to the season (n=1988) –Seasons before excluded. –Only players who played > 48 minutes. Simonoff (1996) examined the APM and PPM for NBA guards in the season. –Correlation is negative if APM<0.2 –Correlation is positive if APM>0.2

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V. Application - Method Partition –Partition 1: APM<0.097; Positive correlation. –Partition 2: APM>0.097; Negative correlation. Estimation: Inference Functions for Margins (IFM): –First estimate marginal distributions by MLE – Choose the best fit from a range of models. –Using marginal estimates, estimate copula parameter (dependence) by MLE. Godambe Information Matrix: –Standard error estimation by Jackknife method –Block size 50; 40 blocks for whole data set.

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VI. Application – Models fitted Whole dataset: –AMH copula –AMH-Beta(a,1) mixed copula Partition 1: APM<0.097 –AMH copula –AMH-Beta(a,1) mixed copula –AMH-Beta(a,1) (+) mixed* Partition 2: APM>0.097 –AMH copula –AMH-Beta(a,1) mixed copula –AMH-Beta(a,1) (-) mixed* * Informatively mixed copulas: instead of mixing into whole domain of θ, mix into the positive or negative domain only.

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IV. Application - Results All copula-based models far outperform the Bivariate Normal control. For full data set: –Mixture significantly outperforms parent. –Consistent with low correlation in data set. For Partitions: –Informative mixtures perform better than parent. –However difference smaller with larger correlation. Effect of Partitioning: –Together, partitioned models far outperform whole dataset estimation.

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V. Conclusion – Key results 1.Mixing does not generalise the model. 2.For copulas, new parameters introduced by mixing are not identified. Hence mixed and unmixed models compete on an equal footing. 3.Each model does better when it is (closer to) the true model. 4.Differences disappear as we approach the limits of dependence coverage. 5.Mixing can be used to effectively convey prior information.

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V. Further research Identification –Are all one-parameter to two-parameter mixtures unidentified? –Further research needed on identification for mixture copulas and parameter-mixing generally. Covariates –Assuming a variable parameter (as is done in parameter mixing) has important implications for inclusion of covariates –NBA example: APM-PPM correlation may vary by player position: Include “position” as a covariate? Finite mixture of copulas?

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