Selected Distributions and Their Relationships .1 Introduction

CONTINUOUS MODELS

5.3 Selected Distributions and Their Relationships .1 Introduction

(g) The modified Bessel function,𝐾_𝜆(𝑥)may be defined, for half-integer values of the index parameter𝜆, by𝐾−𝜆(𝑥) =𝐾_𝜆(𝑥), together with

𝐾_𝑚+¹

2

(𝑥) =

√𝜋 2𝑥𝑒^−𝑥

∑𝑚 𝑗=0

(𝑚+𝑗)!

(𝑚−𝑗)!𝑗! ( 1

2𝑥 )𝑗

, 𝑚= 0,1,….

Use part (f) to prove that, for𝛼 >0,𝜃 >0, and𝑚= 0,1,…,

∫

∞ 0

𝑥^𝑚−32𝑒^{−𝛼𝑥−2𝑥𝜃} 𝑑𝑥= 2 ( 𝜃

2𝛼 )^𝑚₂−¹₄

𝐾_𝑚−¹

2

(√2𝛼𝜃) .

5.3 Selected Distributions and Their Relationships

Transformed beta (α, θ, γ, τ)

Generalized Pareto (γ = 1) Burr

(τ = 1)

Inverse Burr (α = 1)

Pareto (γ = τ = 1)

Inverse Pareto (γ = α = 1) Loglogistic

(α = τ = 1) Paralogistic (α = γ, τ = 1)

Inverse paralogistic (τ = γ, α = 1)

Figure 5.2 The transformed beta family.

Gamma (τ =1)

Weibull (α = 1)

Inverse gamma (τ = 1)

Inverse Weibull (α= 1) Transformed gamma

(α, θ, τ)

Inverse transformed gamma (α, θ, τ)

Exponential (α = τ = 1)

Inverse exponential (α = τ = 1)

Figure 5.3 The transformed/inverse transformed gamma family.

The demonstration relies on two facts concerning limits:

𝛼→lim∞

𝑒^−𝛼𝛼^𝛼−1∕2(2𝜋)^1∕2

Γ(𝛼) = 1 (5.4)

and

𝑎→lim∞

( 1 +𝑥

𝑎 )_𝑎+𝑏

=𝑒^𝑥. (5.5)

The limit in (5.4) is known as Stirling’s formula and provides an approximation for the gamma function. The limit in (5.5) is a standard result found in most calculus texts.

To ensure that the ratio𝜃∕𝛼^1∕𝛾goes to a constant, it is sufficient to force it to be constant as𝛼and𝜃become larger and larger. This can be accomplished by substituting 𝜉𝛼^1∕𝛾for𝜃in the transformed beta pdf and then letting𝛼→∞. The first steps, which

also include using Stirling’s formula to replace two of the gamma function terms, are 𝑓(𝑥) = Γ(𝛼+𝜏)𝛾𝑥^𝛾𝜏−1

Γ(𝛼)Γ(𝜏)𝜃^𝛾𝜏(1 +𝑥^𝛾𝜃^−𝛾)^𝛼+𝜏

= 𝑒^{−𝛼−𝜏}(𝛼+𝜏)^{𝛼+𝜏−1∕2}(2𝜋)^1∕2𝛾𝑥^𝛾𝜏−1 𝑒^−𝛼𝛼^𝛼−1∕2(2𝜋)^1∕2Γ(𝜏)(𝜉𝛼^1∕𝛾)^𝛾𝜏(1 +𝑥^𝛾𝜉^−𝛾𝛼⁻¹)^𝛼+𝜏

= 𝑒^−𝜏[(𝛼+𝜏)∕𝛼]^{𝛼+𝜏−1∕2}𝛾𝑥^𝛾𝜏−1 Γ(𝜏)𝜉^𝛾𝜏[

1 + (𝑥∕𝜉)^𝛾∕𝛼]_𝛼+𝜏 . The two limits,

𝛼→lim∞

( 1 +𝜏

𝛼

)_𝛼+𝜏−1∕2

=𝑒^𝜏 and lim

𝛼→∞

[

1 +(𝑥∕𝜉)^𝛾 𝛼

]_𝛼+𝜏

=𝑒^{(𝑥∕𝜉)𝛾}, can be substituted to yield

𝛼→lim∞𝑓(𝑥) = 𝛾𝑥^𝛾𝜏−1𝑒^{−(𝑥∕𝜉)𝛾} Γ(𝜏)𝜉^𝛾𝜏 ,

which is the pdf of the transformed gamma distribution. □ With a similar argument, the inverse transformed gamma distribution is obtained by letting 𝜏go to infinity instead of𝛼(see Exercise 5.23).

Because the Burr distribution is a transformed beta distribution with𝜏= 1, its limiting case is the transformed gamma with𝜏 = 1 (using the parameterization in the previous example), which is the Weibull distribution. Similarly, the inverse Burr has the inverse Weibull as a limiting case. Finally, letting𝜏=𝛾 = 1shows that the limiting case for the Pareto distribution is the exponential (and similarly for their inverse distributions).

As a final illustration of a limiting case, consider the transformed gamma distribution as parameterized previously. Let𝛾⁻¹√

𝜉^𝛾 →𝜎and𝛾⁻¹(𝜉^𝛾𝜏− 1)→𝜇. If this is done by letting𝜏→∞(so both𝛾and𝜉must go to zero), the limiting distribution will be lognormal.

In Figure 5.4, some of the limiting and special case relationships are shown. Other interesting facts about the various distributions are also given.¹

5.3.4 Two Heavy-Tailed Distributions

With an increased emphasis on risk management, two distributions have received particular attention as beingextreme value distributions. This a well-developed area of study, but here we present the results without going into the theory. The first setting for developing such distributions is to examine the maximum observation from a sample of independently and identically distributed (i.i.d.) observations. In insurance applications, knowing some- thing about the largest claim that might be paid provides information about the risk. Note that this is not the same as measuring VaR because the quantile being examined depends on the sample size. The Fisher–Tippett theorem [40] states that in the limit the maximum (properly scaled) will have one of only three possible distributions. The only one that is interesting for actuarial applications is theFr´echet distribution, which is identical to

1Thanks to Dave Clark of Munich Reinsurance America, Inc. for creating this graphic.

“Transformed Beta” Family of Distributions Two parameters

Mode > 0 Mean and higher

Lognormal moments always

exist

Inverse gamma Gamma

Three parameters Inverse

transformed Transformed

gamma gamma

Four parameters

Inverse Weibull Transformed beta Weibull

Inverse Burr

Burr

Inverse Pareto Pareto

Loglogistic

Mean and higher Mode = 0

moments never exist

Special case Limiting case

(parameters approach zero or infinity)

Figure 5.4 Distributional relationships and characteristics.

what we have called the inverse Weibull distribution. Thus, if the goal is to model the maximum observation from a random sample, the inverse Weibull distribution is likely to be a good candidate. Intuitively, the maximum from a random sample (particularly one from a heavy-tailed distribution) will be heavy tailed and thus the inverse Weibull is also confirmed as a good choice when a heavy-tailed model is needed.

The second setting examines the excess loss random variable as the truncation point is increased. The Balkema–de Haan–Pickands theorem (see Balkema and de Haan [11]

and Pickands [102]) states that in the limit this distribution (properly scaled) converges to one of three distributions. Two are relevant for insurance observations. One is the exponential distribution, which is the limiting case for lighter-tailed distributions. The other is what we term the Pareto distribution (and is called the generalized Pareto distribution in extreme value theory, which is not the same as the distribution of that name in this book).

The insurance situation of interest is when there is a high deductible, as often occurs in reinsurance contracts. If there is no interest in the distribution for moderate losses, the Pareto distribution is likely to be a good model for large losses. Again, even if this is not the situation of interest, this development provides further confirmation of the Pareto distribution as a good model for heavy-tailed losses.

More detail about extreme value distribution can be found in Section 5.6 of the third edition of this book [73] and also appears in the companion book to the fourth edition on advanced loss models [74].

5.3.5 Exercises

5.21 For a Pareto distribution, let both 𝛼 and 𝜃 go to infinity with the ratio𝛼∕𝜃 held constant. Show that the result is an exponential distribution.

5.22 Determine the limiting distribution of the generalized Pareto distribution as𝛼and𝜃 both go to infinity.

5.23 Show that as𝜏 → ∞in the transformed beta distribution, the result is the inverse transformed gamma distribution.