Copula Functions and Bivariate Distributions:

1
Copula Functions and Bivariate Distributions
Applications to Political Interdependence Alejan
dro Quiroz Flores, Wilf Department of Politics,
NYU
Motivation
Joint Political Survival
Bivariate Distribution Functions for Survival
Analysis
Consider a political system with n-components.
These components are subject to shocks that
increase their conditional probability of
failure. We want to estimate the impact of these
shocks both directly to the subjects and
indirectly through the failure of one of these
subjects. In other words, there might be
interdependence between the survival time of both
components. This interdependence can be modeled
as a full system of equations with non-normal
disturbances. In spite of the importance of these
events, political science has made little
progress in the development or application of the
methods to estimate this interdependence. What
needs to be done is to derive multivariate
distributions in order to estimate the parameters
that govern this interdependence. There are
several ways of deriving multivariate
distributions. However, copula functions seem to
be the best way to do so.
How to produce bivariate distributions? The key
resides in having nice marginal distributions and
a copula function. How do we find the copula?
There are several methods inversion, geometric,
and algebraic methods. The following are Weibull
distributions derived from different copula
functions. Exponentials are given by assuming
pi1. The parameter theta is an association
coefficient. Gumbel Distribution 1

Plackett Distribution
Gumbel
Distribution 2

Ali-Mikhail-Haq Distribution
It is not uncommon for governments to receive
external shocks. Indeed, international crises, as
well financial shocks, political scandals, labor
strikes, and transportation breakdowns, among
other problems, threaten the political survival
of leaders. Often times, some leaders manage to
reduce the impact of these shocks by firing the
minister responsible for the policy failure.
Yet, some leaders shamelessly keep those
ministers in spite of their obvious failure in
the area they are responsible for. Why do some
leaders fire some of their agents in some crises
but not in others? Can leaders effectively
respond to shocks by manipulating the composition
of their cabinet? Does this depend on the type of
political system? In order to answer this
question I collected data on the tenure in office
of 7,428 foreign ministers in 181 countries,
spanning the years 1696-2004. When combined with
data on leaders, we can answer some of the
questions presented above. Although I am working
on the theory, in this poster I have summarized
the necessary methods to empirically test
hypotheses on the joint political survival of
leaders and foreign ministers.
Copula Functions
Bivariate Exponential Family

Bivariate Weibull Family
Provisional Regression Results
Functions that join or couple multivariate
distribution functions to their one dimensional
marginal distribution functions. A
two-dimensional subcopula is a function C with
the following properties 1. Dom C S1 x S2,
where S1 and S2 are subsets of I containing 0 and
1. 2. C is grounded and 2-increasing. 3. For
every u in S1 and v in S2, then C(u,1)u and
C(1,v)v. Note that for every (u,v) in Dom C,
0ltC(u,v)lt1 so Ran C is also a subset of
I. A two dimensional copula is a subcopula C
whose domain is I2.
The dependent variable is a vector with two
elements the total survival time of a leader and
the median survival time of ministers that held
office with that particular leader. The mean
survival time for leaders is 3.83 years with a
standard deviation of .1316 whereas the mean
survival time for median ministers is 2.02 years
with a standard deviation of .0668. Sample size
is 1966. The correlation for the survival times
is .1802 Coefficient Weibull(1) Weibull(2)
B.Weibull(1) B.Weibull(2) Shape 1 .7452
.7452
.7433 (.0128)
(.0124) (.0126) Scale 1
3.173
3.173 3.073 (.1012)
(NA)
(.0984) Shape 2 .8819
.8818 .8776
(.0144)
(.0142) (.0141) Scale 2
1.882 1.882
1.824
(.0509) (NA) (.0489) Assoc

0 1
(NA)
(.0405)
Monte Carlo Simulations Full Maximum Likelihood
Estimates 1000 replications Scale(1)2,
Scale(2)3 N from 100 to 1000 in increments of
100 Independent Exponentials
Associated Exponentials (a.15)
Monte Carlo Simulations Full Maximum Likelihood
Estimates 1000 replications Scale(1)2,
Scale(2)3 N from 100 to 1000 in increments of
100 Independent Weibulls Associated
Weibulls (a.9) and (a.5)
Copula Functions and Random Variables
Future Research
Define a pair of random variables X and Y with
cumulative distribution functions F(x) and G(y),
respectively, and a joint cumulative distribution
function H(x,y). Sklars Theorem There exists a
copula C such that for all x,y in
R, H(x,y)CF(x),G(y) Frechet-Hoeffding
Bounds MaxF(x)G(y)-1,0ltH(x,y)ltMinF(x),G(y)
Scale Invariant Measures of Dependence Focus
not on correlation coefficients as linear
dependence between random variables, but on scale
invariant measures of association. Measures of
Association Kendalls Tau and Spearmans Rho
focus on estimating probability of concordance
(large values of one variable associated with
large values of another variable) and discordance
between random variables.

• Develop more PDFs to compare distributions
  through simulation.
• Check for better algorithms for maximization
  procedure. So far R does a good job, but we can
  do better.
• Incorporate time varying covariates in the
  likelihood.
• Bivariate censoring is not a problem. But how to
  deal with univariate censoring?
• The PDFs are unstable for certain values of the
  parameters. How to solve this problem?
• Compare Copula Functions with other procedures
  for the generation of multivariate distributions,
  such as the conditional distributions,
  transformation, and rejection approaches.
• Further develop theory for joint political
  survival. So far I am working on a nested
  principal-agent model.