222 Some Special Distributions
Example 3.7.4.In this example, we develop by compounding a heavy-tailed
skewed distribution. AssumeXhas a conditional gamma pdf with parameters
kandθ−^1. The weighting function forθis a gamma pdf with parametersαandβ.
Thus the unconditional (marginal or compounded) pdf ofXis
h(x)=
∫∞
0
[
θα−^1 e−θ/β
βαΓ(α)
][
θkxk−^1 e−θx
Γ(k)
]
dθ
=
∫∞
0
xk−^1 θα+k−^1
βαΓ(α)Γ(k)
e−θ(1+βx)/βdθ.
Comparing this integrand to the gamma pdf with parametersα+kandβ/(1 +βx),
we see that
h(x)=
Γ(α+k)βkxk−^1
Γ(α)Γ(k)(1 +βx)α+k
, 0 <x<∞,
which is the pdf of thegeneralized Pareto distribution(and a generalization of
theFdistribution). Of course, whenk=1(sothatXhas a conditional exponential
distribution), the pdf is
h(x)=αβ(1 +βx)−(α+1), 0 <x<∞,
which is thePareto pdf. Both of these compound pdfs have thicker tails than the
original (conditional) gamma distribution.
While the cdf of the generalized Pareto distribution cannot be expressed in a
simple closed form, that of the Pareto distribution is
H(x)=
∫x
0
αβ(1 +βt)−(α+1)dt=1−(1 +βx)−α, 0 ≤x<∞.
From this, we can create another useful long-tailed distribution by lettingX=Yτ,
0 <τ.ThusYhas the cdf
G(y)=P(Y≤y)=P[X^1 /τ≤y]=P[X≤yτ].
Hence, this probability is equal to
G(y)=H(yτ)=1−(1 +βyτ)−α, 0 <y<∞,
with corresponding pdf
G′(y)=g(y)=
αβτ yτ−^1
(1 +βyτ)α+1
, 0 <y<∞.
We call the associated distribution thetransformed Pareto distributionor the
Burr distribution(Burr, 1942), and it has proved to be a useful one in modeling
thicker-tailed distributions.