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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. I Introduction. Copulas:

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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

Supervisor: Murray D Smith

i introduction
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:
i introduction1
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).
ii past research
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”.
iii properties of mixture copulas
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?
iii properties of mixture copulas1
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
iii properties of mixture copulas2
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.
iii properties of mixture copulas3
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.
iv experiment
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.
iv experiment results
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.
v application nba basketball
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
v application method
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.
vi application models fitted
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.

iv application results
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.
v conclusion key results
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.
v further research
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?