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Modelling with parameter-mixture copulas

Modelling with parameter-mixture copulas. October 2006 Xiangyuan Tommy Chen Econometrics & Business Statistics The University of Sydney xche9124@mail.usyd.edu.au Supervisor: Murray D Smith. I Introduction. Copulas:

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Modelling with parameter-mixture copulas

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  1. Modelling with parameter-mixture copulas October 2006 Xiangyuan Tommy Chen Econometrics & Business Statistics The University of Sydney xche9124@mail.usyd.edu.au Supervisor: Murray D Smith

  2. 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:

  3. I Introduction • Parameter mixing • A hierarchical model: the parent distribution functionwith 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).

  4. 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”.

  5. 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?

  6. 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

  7. III. Examples of mixture copulas

  8. 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.

  9. 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.

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

  11. 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.

  12. 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 1950-51 season through to the 1993-94 season (n=1988) • Seasons before 1950-51 excluded. • Only players who played > 48 minutes. • Simonoff (1996) examined the APM and PPM for NBA guards in the 1992-1993 season. • Correlation is negative if APM<0.2 • Correlation is positive if APM>0.2

  13. 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.

  14. 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.

  15. 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.

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

  17. 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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