ON THE TRUNCATED COMPOSITE LOGNORMAL-PARETO MODEL

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LOGNORMAL-PARETO MODEL

SANDRA TEODORESCU

The Lognormal and Pareto distributions are frequently used to model payments data. We consider here the truncated composite Lognormal-Pareto model that appears to be more adequate for modelling such data.

AMS 2000 Subject Classification: 60E05, 62F03, 46N30, 62P05.

Key words: Lognormal distribution, Pareto distribution, truncated composite Lognormal-Pareto model.

1. INTRODUCTION

The actuarial and insurance industries frequently use the Lognormal and Pareto distributions to model their payments data. Many authors use the generalized Pareto distribution to model the payments data, especially for large loss data or reinsurance payments [1]. This is done because insurance payments data are tipically highly positevely skewed and distributed with large upper tails.

A disadvantage of using Pareto model for actuarial data is that it covers well the behaviour of large losses, but fails to cover small losses, which are modeled quite well by Lognormal, Gamma or Weibull distributions.

Obviously, the Lognormal model, for instance, covers large data as well, but it fade away to zero more quickly than the Pareto model. Therefore, using Lognormal model for all data set results in underestimating payment losses.

Cooray and Ananda [1] proposed a model that combine Lognormal distribution for small losses and Pareto for large ones in a composite model. Starting from this work we study the truncated composite Lognormal-Pareto model.

In this paper we restrict to insurance claims that are both upper and lower limited for reasons such as:

– the introduction of deductibles in most insurance contracts;

– the fact that, in general, small costs are not reported to the insurer (they being directly paid by the insured) while on the other hand, there exists a natural upper limit for these costs (namely, the amount insured).

MATH. REPORTS12(62),1 (2010), 71–84

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In Section 2 we present the construction and some characteristics of the truncated composite model Lognormal-Pareto through its density, cumulative distribution function, and rth initial moments. Statistical inference for the truncated Lognormal-Pareto model is presented in Section 3. It is based on the method of moments and the maximum likelihood estimation. The estimation method suggested is ilustrated on generated data in Section 4.

2. CONSTRUCTION AND SOME CHARACTERISTICS

OF THE TRUNCATED COMPOSITE LOGNORMAL-PARETO MODEL

The Pareto model with a long and thick upper tail is used to model large loss data, while large data with low frequencies as well as small data with high frequencies are usually modeled by the Lognormal distribution density.

Compared with the Lognormal, the resulting composite Lognormal-Pareto has a similar shape but does not fade away to zero too quickly as the Lognormal [1].

In this section we introduce a truncated composite Lognormal-Pareto model and present its properties and parameter estimation techniques. Fol- lowing Cooray and Ananda [1], the truncated composite Lognormal-Pareto density is derived from the density

(2.1) f(x) =

cf1(x) ifa < x≤θ+b, cf2(x) ifθ+b≤x <∞,

wherecis a normalizing constant,f₁(x) is a two-parameter Lognormal density while f2(x) is a two-parameter Pareto density, i.e.,

f₁(x) = 1

√

2π(x−a)σ exp

"

−1 2

ln (x−a)−µ σ

2#

, x > a, σ >0, µ∈R, and

f₂(x) = αθ^α

(x−b)^α+1, x>θ+b,

where µ, α, θ, σ are unknown parameters such thatσ >0,α >0,θ+b >0.

Imposing the necessary conditions to have a smooth continuous density at θ+b, namely, R∞

a f(x)dx = 1, f1(θ+b) = f2(θ+b) and f₁⁰(θ+b) = f₂⁰(θ+b), we get the following result.

Proposition 2.1.The truncated composite Lognormal-Pareto density is given by

(2.2)

f_{LP t}^c (x) =







1 [1+Φ(k)]√

2π(x−a)σ exp

−¹₂_{ln(x−a)−µ}

σ

2

if a < x≤θ+b,

αθ^α

[1+Φ(k)](x−b)^α+1 if θ+b≤x <∞

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with α= _θ+b−a^θ h

1 +ln(θ+b−a)−µ σ²

i

−1 and k= ln(θ+b−a)−µ

σ .

Proof. The continuity and differentiability conditions at θ+blead to

(2.3) α+ 1 = θ

θ+b−a

1 +k σ

, where

(2.4) k^not= ln (θ+b−a)−µ

σ .

The normalizing constant c follows from the condition R∞

a f(x)dx = 1, that yields

(2.5) c= 1

Φ(k) + 1,

where Φ (·) is the cdf of the standard normal distribution.

Thus, the truncated composite Lognormal-Pareto density is given by (2.2).

Remark 2.1. Replacing α given by (2.3) in the continuity condition, we obtain

(2.6) θ= a−b

1 +A(k;σ), where A(k;σ)^not= _σ¹h

√1

2πexp

−^k₂²

−ki

−1.From (2.4) and (2.6) we have

(2.7) µ= lnA(k;σ) (b−a)

1 +A(k;σ) −kσ.

Inserting (2.6) into (2.3) yields

(2.8) α=− 1

A(k;σ)

1 + k σ

−1.

Remark 2.2. Initially, f_{LP t}^c (x) had four parameters, but imposing the continuity and differentiability conditions at θ+b, the number of parameters reduces to three: θ,µ and σ. The parameter α can be expressed in terms of these. With the notation above, we can reduce even more the number of free parameters of this model, from three to two (e.g., by expressingα, θ andµ in terms of kand σ).

Some illustrative density curves for the truncated Lognormal-Pareto model are presented in Figures 2.1 through 2.4.

Figure 2.1 shows how the shape of Lognormal-Pareto densities changes when keeping k= 1 forσ = 0.2,σ = 0.5 and σ = 1. One can notice that the higher value of σ, the more quickly the truncated Lognormal-Pareto density

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fade away to zero. Moreover, the higher value of σ, the smaller the mode of the truncated composite Lognormal-Pareto distribution.

Fig. 2.1. The truncated composite Lognormal-Pareto density curves fork= 1, a= 0.5, b= 1 and different values ofσ.

Figure 2.2 shows how the shape of Lognormal-Pareto densities changes when keepingσ = 0.5 fork= 0.5,k= 1 andk= 1.5. One can notice that the higher value of k, the more quickly the truncated Lognormal-Pareto density fade away to zero. Moreover, the higher value of k, the smaller the mode of the truncated composite Lognormal-Pareto distribution.

Fig. 2.2. The truncated composite Lognormal-Pareto density curves forσ= 0.5, a= 0.5, b= 1 and different values ofk.

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Fig. 2.3. The truncated composite Lognormal-Pareto density curves fork= 0.9, σ= 0.7 and different values ofaandb.

In Figures 2.3 and 2.4 we compare the densities of the truncated composite Lognormal-Pareto model plotted for the same values of k and σ and different values ofaandband, respectively, for the same values ofaandband different values of kand σ. Let us note that for this choice of parameters, as a orbdecreases, the truncated composite distributions are heavier tailed.

Fig. 2.4. The truncated composite Lognormal-Pareto density curves fora= 0.5, b= 1 and different values ofkandσ.

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Proposition 2.2. The cumulative distribution function of the truncated Lognormal-Pareto model is given by

(2.9) F_{LP t}^c (x) =









 1 1 + Φ(k)Φ

ln (x−a)−µ σ

if a < x≤θ+b,

1− 1

1 + Φ(k) θ

x−b α

if θ+b≤x≤ ∞.

Proof. Forx∈(a, θ+b] we have F(x) =

Z x a

cf1(t) dt=

= 1

1 + Φ(k) Z x

a

√ 1

2π(t−a)σ exp

"

−1 2

ln (t−a)−µ σ

2# dt=

= 1

1 + Φ(k)Φ

ln (x−a)−µ σ

. Similarly, for x∈[θ+b,∞), we get

F(x) = Z θ+b

a

cf₁(t) dt+ Z x

θ+b

cf₂(t) dt=I₁+I₂, where

I1= 1 1 + Φ(k)

Z ln(θ+b−a)−µ σ

−∞

√1 2π exp

−1 2z²

dz= Φ(k) 1 + Φ (k), and

I₂= 1 1 + Φ(k)

1− θ^α (x−b)^α

. Thus,

F(x) = 1− 1 1 + Φ(k)

θ^α (x−b)^α, so that (2.9) holds.

Proposition 2.3. The unique mode of the truncated composite Lognormal- Pareto distribution is

(2.10) x^{cLP t}_{mod e}= exp µ−σ² +a.

Proof. Solving the equation f⁰(x) = 0, where f is the density of the truncated composite Lognormal-Pareto distribution, we get the unique local maximum at x given by (2.10).

Other characteristics of the model we discuss here are the initial moments.

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Proposition 2.4. The initial rth moment of the truncated composite Lognormal-Pareto distribution is given by

E(X^r) = 1 1 + Φ(k)

r

X

j=0

r j

a^r−j

Φ

ln (θ+b)−µ

σ −jσ

(2.11) ×

×exp 1

2j²σ²+jµ

+αθ^α

j

X

l=0

j l

(b−a)^j−l (θ+a)^l−α α−l

)

for l < α.

Proof. We have E(X^r) =

r

X

j=0

r j

a^r−j

Z ∞ a

(x−a)^jf(x)dx.

Setting x−a=y we get E(X^r) =

r

X

j=0

r j

a^r−j

Z θ+b 0

y^jf(y+a) dy+ Z ∞

θ+b

y^jf(y+a) dy

. Let

I1j = 1 1 + Φ(k)

Z θ+b 0

y^j 1

√

2πyσexp

"

−1 2

lny−µ σ

2#

dy, j= 0, r.

Setting ^lny−µ_σ =z we have I_1j = 1

1 + Φ(k)

√1

2π exp (jµ)

Z ^{ln(θ+b)−µ}

σ

−∞

exp

−1 2 h

(z−jσ)²−j²σ²i dz.

Another change of variable z−jσ=u yields I1j = 1

1 + Φ(k)exp 1

2j²σ²+jµ

Φ

ln (θ+b)−µ

σ −jσ

and

I2j = 1 1 + Φ(k)

Z ∞ θ+b

y^jf(y+a) dy=

= αθ^α 1 + Φ(k)

Z ∞ θ+b

y^j

(y+a−b)^α+1dy, j = 0, r, and with y+a−b=twe get

I2j = αθ^α 1 + Φ(k)

j

X

l=0

j l

(b−a)^j−l(θ+a)^l−α

α−l , l−α <0.

Consequently, this yields (2.11).

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3. STATISTICAL INFERENCE FOR THE TRUNCATED COMPOSITE LOGNORMAL-PARETO MODEL

The truncated composite Lognormal-Pareto model, unlike the non-truncated model, could be used to introduce deductibles in most non-life insurance contracts, i.e., the data are left censored. The franchise can be set

– as a fixed value applicable to each damage;

– as a percentage of the sum insured, applicable to each damage;

– as a percentage of each damage value.

Also, in most cases of insurance payments, there is a limit for the maximum amount of payment, i.e., the data are right censored.

In this section we consider the estimation of model parameters of the truncated composite Lognormal-Pareto distribution for such data.

Let a random sample (x₁, x₂, . . . , x_n) from the two-parameter truncated composite Lognormal-Pareto model (see Remark 2.2) with density given by (2.2). Without any loss of generality, we assume that it is an ordered sample, i.e., x₁≤x₂ ≤ · · · ≤x_n.

The method of momentsconsist of equating the unobservable popu- lation moments with the sample moments:

(2.12) E(X) =x, E X²

=y, where x = _n¹

n

P

i=1

x_i , y = _n¹

n

P

i=1

x²_i, and E(X), E X²

are given by Proposi- tion 2.4. System (2.12) is equivalent to

aΦ

ln (θ+b)−µ σ

+ Φ

ln (θ+b)−µ

σ −σ

exp

σ² 2 +µ

+ +θ^α(b−a)

(θ+a)^α + αθ^α

(α−1) (θ+a)^α−1 = [1 + Φ(k)]1 n

n

X

i=1

xi

and a²Φ

ln (θ+b)−µ σ

+ 2aΦ

ln (θ+b)−µ

σ −σ

exp

σ² 2 +µ

+ +Φ

ln (θ+b)−µ

σ −2σ

exp 2σ²+ 2µ

+θ^α(b−a) (b−a+ 1) (θ+a)^α + +αθ^α(2b−2a+ 1)

(α−1) (θ+a)^α−1 + αθ^α

(α−2) (θ+a)^α−2 = [1 + Φ(k)]1 n

n

X

i=1

x²_i. It follows from (2.8), (2.6) and (2.7) that α, θ and µ can be expressed in terms of k and σ. The likelihood equations are too complicated to obtain an explicit solution, and the solving of such non-linear equations could be

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performed by a mathematics software. Solving these non-linear equations, we get the solutions ek and eσ.

Maximum Likelihood Estimation (MLE). Let a random sample from the two-parameter truncated composite Lognormal-Pareto model with density function given by (2.2). In order to evaluate the likelihood function, we must have an idea of where is situated the unknown parameterθ, so assume that xm ≤θ+b≤xm+1. Then the likelihood function is

L(x₁, x₂, . . . , x_n;α, θ, µ, σ) =

n

Y

i=1

f(x_i) =

m

Y

i=1

f(x_i)×

n

Y

i=m+1

f(x_i) =

(2.2)

= C^LPα^n−mσ^−mθ^(n−m)α[1 + Φ(k)]⁻ⁿ

n

Y

i=m+1

(x_i−b)^−α×

×exp

"

−1 2

m

X

i=1

ln (xi−a)−µ σ

2# , where

C^LP = 1

√

2πm m

Q

i=1

(xi−a)

n

Q

i=m+1

(xi−b) .

So, the loglikelihood function is

lnL(x₁, x₂, . . . , x_n;α, θ, µ, σ) = lnC^LP −mlnσ−1 2

m

X

i=1

ln (x_i−a)−µ σ

2

+

+(n−m) lnα+ (n−m)αlnθ−α

n

X

i=m+1

ln (x_i−b)−nln [1 + Φ (k)]. Replacing α, µ and θ by (2.8), (2.7) and (2.6) the loglikelihood function be- comes

lnL(x₁, x₂, . . . , x_n;k, σ) = lnC^LP −mlnσ−

−1 2

m

X

i=1

1

σ ln(x_i−a) (1 +A(k;σ)) A(k;σ)(b−a) +k

2

+ +(n−m) ln

− 1 A(k;σ)

1 + k

σ

−1

+ (n−m) ln

− 1 A(k;σ)

1 +k

σ

−1

×

×ln a−b 1 +A(k;σ) −

− 1 A(k;σ)

1 +k

σ

−1 n

X

i=m+1

ln (x_i−b)−nln [1+Φ(k)].

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In order to maximize the likelihood function we have to solve the system

∂lnL

∂k = ∂lnL

∂σ = 0.

Thus, the likelihood equations are 1

A(k;σ) (1+A(k;σ)) 1 σ

∂A(k;σ)

∂k −1 m

X

i=1

1

σln(x_i−a) (1+A(k;σ)) A(k;σ) (b−a) +k

+

+(n−m) 1

h

−_A(k;σ)¹ 1 +^k_σ

−1 i

1 A²(k;σ)

∂A(k;σ)

∂k

1 + k σ

− 1 A(k;σ)σ

+

+(n−m) 1

A²(k;σ)

∂A(k;σ)

∂k

1+ k σ

− 1 A(k;σ)σ

ln a−b

1+A(k;σ)−(n−m)×

×

− 1 A(k;σ)

1 +k

σ

−1

1 1+A(k;σ)

∂A(k;σ)

∂k −

− 1

A²(k;σ)

∂A(k;σ)

∂k

1 + k σ

− 1 A(k;σ)σ

ⁿ X

i=m+1

ln (xi−b)−nΦ⁰(k) Φ(k) = 0 and

−m σ + 1

σ

m

X

i=1

1

σ ln(x_i−a) (1 +A(k;σ))

A(k;σ) (b−a) +k 1

σln(x_i−a) (1 +A(k;σ)) A(k;σ) (b−a) +

+ 1

A(k;σ) (1 +A(k;σ))

∂A(k;σ)

∂σ

+ (n−m) 1

h−_A(k;σ)¹ 1 +_σ^k

−1i×

× 1

A²(k;σ)

∂A(k;σ)

∂σ

1 +k σ

+ k

A(k;σ)σ²

+ +(n−m)

1 A²(k;σ)

∂A(k;σ)

∂σ

1 +k σ

+ k

A(k;σ)σ²

×

×ln a−b

1 +A(k;σ) −(n−m)

− 1 A(k;σ)

1 +k

σ

−1

1 1 +A(k;σ)

∂A(k;σ)

∂σ −

− 1

A²(k;σ)

∂A(k;σ)

∂σ

1 + k σ

+ k

A(k;σ)σ² n

X

i=m+1

ln (xi−b) = 0.

But we can also express ^∂A(k,σ)_∂σ and ^∂A(k,σ)_∂k in terms ofk andσ as

∂A(k;σ)

∂σ = 1 σ²

"

k−exp −^k₂²

√2π

#

, ∂A(k;σ)

∂k =−1 σ

"

1 +kexp −^k₂²

√2π

# .

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Thus, the likelihood equations become

−





1 +^√¹

2πke⁻^k

2 2

√1 2πe⁻^k

2

2 −k−σ ^√¹

2πe⁻^k

2

2 −k

+ 1



×

×

m

X

i=1





 1 σ ln

(x_i−a)

√1 2πe⁻^k

2

2 −k

√1

2πe⁻^k

2

2 −k−σ

(b−a) +k







+ (n−m) k²+kσ+ 1 √1

2πe⁻^k

2

2 −k−σ+

+ (n−m)

√1 2πe⁻^k

2

2 k²+kσ+ 1 √1

2πe⁻^k

2

2 −k−σ

2 ln a−b

1 σ

√1 2πe⁻^k

2 2 −k

−

−(n−m)

√1 2πe⁻^k

2 2

1 +^√¹

2πke⁻^k

2 2

√1

2πe⁻^k

2

2 −k−σ ^√¹

2πe⁻^k

2 2 −k

+

√1 2πe⁻^k

2

2 k²+kσ+ 1 √1

2πe⁻^k

2

2 −k−σ 2

n

X

i=m+1

ln (x_i−b)−nΦ⁰(k) Φ(k) = 0 and

−m σ + 1

σ²

m

X

i=1





 1 σln

(x_i−a)

√1 2πe⁻^k

2

2 −k

√1

2πe⁻^k

2

2 −k−σ

(b−a) +k







×





 ln

(xi−a)

√1 2πe⁻^k

2 2 −k

√1

2πe⁻^k

2

2 −k−σ

(b−a)

− σ

√1 2πe⁻^k

2

2 −k−σ







−

−(n−m)

√1 2πe⁻^k

2 2 −σ σ

√1 2πe⁻^k

2

2 −k

−σ²

−(n−m)

√1 2πe⁻^k

2 2

√1 2πe⁻^k

2

2 −k−σ2×

×ln a−b

1 σ

√1

2πe⁻^k²² −k+

√1 2πe⁻^k

2 2

√1 2πe⁻^k

2

2 −k−σ2 n

X

i=m+1

ln (x_i−b) = 0.

The likelihood equations are too complicated to obtain an explicit solution. The solving of such non-linear equations might be performed by the

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Newton-Raphson method. The unknown parameters are k and σ while the starting values, calculated with the method of moments, are ek andσ.e

If we denote by ϕ(k;σ) the first likelihood equation and by ψ(k;σ) the second one, then the generalized Newton-Raphson mehod leads to the iterations











ki+1 =ki+ψ(ki;σi)ϕ⁰_σ(ki;σi)−ϕ(ki;σi)ψ⁰_σ(ki;σi) J

σi+1 =σi+ϕ(ki;σi)ψ_k⁰ (ki;σi)−ψ(ki;σi)ϕ⁰_k(ki;σi) J

k₀ =ek, σ₀ =eσ

for k = 0,1,2, . . . , where J = det ϕ⁰_k(k;σ) ϕ⁰_σ(k;σ) ψ_k⁰ (k;σ) ψ_σ⁰ (k;σ)

!

. The number of steps are fixed or STOP when the conditions|k_i+1−k_i|< εand|σ_i+1−σ_i|< ε are satisfied (for example, ε= 10⁻⁵).

Denote byk_{M L}andσ_{M L}the solutions of the equations above (the maximum likelihood estimators ofkandσ). Then, using (2.8), (2.7) and (2.6), the maximum likelihood estimators ofα,µandθdenoted byα_{M L},µ_{M L}andθ_{M L}, respectively, are

αM L=− 1 A(k_{M L};σ_{M L})

1 +kM L

σ_{M L}

−1, µ_{M L}= lnA(k_{M L};σ_{M L}) (b−a)

1 +A(k_{M L};σ_{M L}) −k_{M L}σ_{M L}, θ_{M L}= a−b

1 +A(k_{M L};σ_{M L}).

4. NUMERICAL EXAMPLE

In order to illustrate the algorithm presented in Section 3, we will consider a data set that was generated from the truncated composite Lognormal- Pareto model with k= 1, σ= 1, a= 0.5, andb= 1. The true value of m is, in this case, 63. See Table 1. The generating algorithm used is based on the inversion of the cumulative function distribution (2.9).

In Table 2, columns 2–3 result from the data sample while columns 4–5 are calculated using the truncated composite Lognormal-Pareto distribution function.

The estimated values of the parameters are k_{M L} = 1.052 and σ_{M L} = 0.875. We applied theχ² test to check the distribution fitting, and the results for k_{M L} = 1.052 andσ_{M L}= 0.875 are given in Table 2. Theχ² distance calculated for the estimated values of the parameters is d²(kM L;σM L) = 14.686

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while the 0.95 χ²(8) quantile is 15.507. This means that the χ² test accepts the truncated Lognormal-Pareto model, as expected.

Table1

100 truncated Lognormal-Pareto values fork= 1,σ= 1,a= 0.5,b= 1 0.540304 0.568442 0.572128 0.57233 0.574639

0.583038 0.589578 0.606612 0.617003 0.620313 0.640025 0.647126 0.670048 0.687739 0.690579 0.699163 0.713921 0.720001 0.728684 0.739623 0.765188 0.774864 0.77709 0.781919 0.789553 0.78996 0.791298 0.825368 0.82833 0.84066 0.846474 0.848826 0.849732 0.895717 0.928346 0.928666 0.939448 0.951401 0.963153 0.968491 0.97787 0.983999 1.004191 1.016115 1.033985 1.043214 1.044728 1.106784 1.107132 1.168907 1.236006 1.308571 1.359533 1.396383 1.404328 1.415723 1.47486 1.479591 1.483656 1.518889 1.539842 1.556266 1.639157 1.657395 1.784296 1.896014 1.998283 2.006224 3.80672 4.298746 4.910924 5.652674 6.466787 8.455752 8.515934 9.364622 9.381685 10.78291 15.50163 19.76519 40.21232 48.98251 70.35656 106.294 151.7704 177.4311 644.1338 2146.641 2238.48 10160.17 16580.82 57605.55 182093.5 490441 2211688 4678495 10836937 221727707 229434209 26885797131

Table2

Grouped data and theχ² test

Sample abs Trunc LP Trunc LP

Classes freq. ni rel freq,fi th freq,pi n(f_i−p_i)² p_i

0.540303–0.59999 7 0.07 0.0265 7.13481

0.6–0.9999 35 0.35 0.2888 1.29222

1–1.49999 17 0.17 0.1384 0.71969

1.5–4.99999 12 0.12 0.1332 0.13202

5–9.99999 6 0.06 0.0426 0.70757

10–99.99999 6 0.06 0.1018 1.72179

100–4999.99999 6 0.06 0.1095 2.23911

5000–499999.99999 5 0.05 0.0726 0.70375

500000–26885797132 6 0.06 0.0648 0.03560

P 100 1 χ² distance = 14.686

REFERENCES

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Received 9 February 2009 Ecological University of Bucharest Faculty of Economic Sciences

Bd. Vasile Milea nr. 1G 061341 Bucharest, Romania cezarina [email protected]