Modelling with parameter-mixture copulas

1 / 22

# Modelling with parameter-mixture copulas - PowerPoint PPT Presentation

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:

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## Modelling with parameter-mixture copulas

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Modelling with parameter-mixture copulas

October 2006

Xiangyuan Tommy Chen

The University of Sydney

xche9124@mail.usyd.edu.au

Supervisor: Murray D Smith

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 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
• 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
• 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 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 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 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
• 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
• 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.
• 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
• 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
• 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
• 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
• 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
• 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?