Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

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.