- 193 Views
- Uploaded on

Download Presentation
## Modelling with parameter-mixture copulas

**An Image/Link below is provided (as is) to download presentation**

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

Econometrics & Business Statistics

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.

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

- 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?

Download Presentation

Connecting to Server..