ON THE TRUNCATED COMPOSITE WEIBULL-PARETO MODEL

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WEIBULL-PARETO MODEL

SANDRA TEODORESCU and EUGENIA PANAITESCU

The composite Weibull-Pareto model [2] was introduced as an alternative to the composite Lognormal-Pareto [1] used to model the insurance payments data. We present some properties of the truncated composite Weibull-Pareto model.

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

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

1. INTRODUCTION

The insurance payments data are frequently modeled by the Lognormal and Pareto distributions. Many authors use the generalized Pareto distribution to model the payments data, especially for large loss data or reinsu- rance payments [1] because the insurance payments data are tipically highly positevely skewed and distributed with large upper tails.

The Pareto model is also used in the actuarial industry because it covers well the behaviour of large losses. Unlike the Lognormal, Gamma or Weibull distributions which model quite well small losses, the Pareto model fails. Ob- viously, the Weibull model, for instance, covers large data as well, but it fade away to zero more quickly than Pareto model. Therefore, using one of these models for all data set results in underestimating payment losses.

Frequently, in the actuarial and insurance industries, the Weibull distribution is used in general (non-life) insurance to model the size of reinsu- rance claims.

A composite model that combines the Lognormal distribution for small losses and the Pareto for large ones was introduced by Cooray and Ananda [1].

In the same manner, as an alternative model to composite Lognormal-Pareto proposed in [1], the composite Weibull-Pareto model [2] was built up and a comparison of these models was made in [8].

Starting from these papers, in this paper we study the truncated composite Weibull-Pareto model. In general, the truncated model is suitable for modelling payments data which appear in deductibles contracts. So, in this

MATH. REPORTS11(61),3 (2009), 259–273

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

– the introduction of deductibles in most non-life 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).

This new truncated composite Weibull-Pareto model, unlike the non- truncated model, could be also used to introduce deductibles in most non-life insurance contracts, i.e., the data are left censored. Also, in most cases of insurance payments, there is a limit for the maximum amount of a payment, i.e., the data are right censored.

In Section 2 we present the truncated composite Weibull-Pareto model through its density, cumulative distribution function, andrth initial moments, and we determine the likelihood function.

On account of the results obtained in Section 2, an algorithm for esti- mating the parameters of the truncated composite model is given in Section 3.

A numerical example based on simulated data set is presented in Section 4.

2. THE TRUNCATED COMPOSITE WEIBULL-PARETO MODEL

The composite Weibull-Pareto model is a combination of the Weibull distribution, which cover well the behaviour of small losses up to a threshold parameter, and of the Pareto distribution for the rest of the domain. The resulting composite Weibull-Pareto density has a larger tail than the Weibull density but a smaller tail than the Pareto density [9].

In this section we introduce a truncated composite Weibull-Pareto model and present its properties and parameter estimation techniques.

Following Cooray and Ananda [1] and Ciumara [2], the truncated composite Weibull-Pareto density is derived from the density

(2.1) f(x) =

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

where c is a normalizing constant, f₁(x) is a Weibull density whilef₂(x) is a two-parameter Pareto density, i.e.,

f₁(x) = β

γ^β(x−a)^β⁻¹exp −

x−a γ

β!

, x > a, γ >0, β >1, f₂(x) = αθ^α

(x−b)^α+1, x > θ+b, θ >0,α >1, where β >0,γ >0,α >0,θ >0 are unknown parameters.

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

f_{W P}^c _t(x) = (2.2)

=











1 2−exp

− θ+b−a

γ

β β

γ^β(x−a)^β⁻¹exp

−

x−a γ

β

if a < x≤θ+b,

1 2−exp

− θ+b−a

γ

β αθ^α

(x−b)^α+1 if θ+b≤x <∞, where α= _θ+b^θ₋_ah

β

θ+b−a γ

β

−β+ 1i

−1.

Proof.From the normalizing conditionR∞

a f(x)dx= 1 we get

(2.3) c= 2−e⁻^k⁻1

, where

(2.4) k^not=

θ+b−a γ

β

.

In order to obtain a smooth, continuous density at θ+b, we have to impose continuity and differentiability conditions at this point. These conditions are

(2.5) β

γ^β (θ+b−a)^β⁻¹exp −

θ+b−a γ

β!

= α θ and

(2.6) β

γ^β (θ+b−a)^β⁻²e⁻

θ+b−a γ

β"

β−1−β

θ+b−a γ

β#

=−α(α+1) θ² .

From the last two equations we get

(2.7) α= θ

θ+b−a

"

β

θ+b−a γ

β

−β+ 1

#

−1.

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

Remark 2.1. Replacingαgiven by (2.7) in the continuity condition (2.5), we obtain

(2.8) e⁻^k= 1− 1

k +(a−b) θβ

1 k, whence

(2.9) θ= a−b

A(k;β),

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where A(k;β)^not= β ke⁻^k−k+ 1

. From (2.4) we have

(2.10) γ = θ+b−a

k^1/β . Inserting (2.9) into (2.7) and (2.10) yields

(2.11) α= 1

1−A(k;β)(βk−β+ 1)−1 and

(2.12) γ = (a−b) (1−A(k;β)) A(k;β)k^1/β . Remark 2.2. Initially, f_{W P}^c

t(x) had four parameters, but imposing the continuity and differentiability conditions at θ+b, the number of parameters reduces to three: β, γ and θ. 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 expressing α,θ andγ in terms of kand β).

Five illustrative density curves for the truncated composite Weibull- Pareto model are presented in Figures 1 through 4.

Fig. 1. The truncated composite Weibull-Pareto density curves fork= 2, a= 0.5,b= 1 and different values ofβ.

In Figure 1, we display the density curves for the truncated composite Weibull-Pareto model. Notice that for this choice of parameters, asβincreases, the five densities approach zero faster.

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In Figure 2 we also display the density curves for the truncated composite Weibull-Pareto model and for this choice of parameters, askincreases, the five densities approach zero faster while the modes decrease.

Fig. 2. The truncated composite Weibull-Pareto density curves forβ= 3, a= 0.5,b= 1 and different values ofk.

In the next two diagrams we display the density curves of the truncated composite Weibull-Pareto model plotted for the same values of k and β and different values of aandbin Figure 3, and for the same values ofaand band different values of k and β in Figure 4.

Fig. 3. The truncated composite Weibull-Pareto density curves fork= 2, β= 3 and different values ofaandb.

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Fig. 4. The truncated composite Weibull-Pareto density curves plotted fora= 0.5,b= 1 and different values ofkandβ.

Let us note that for this choice of parameters, as b decreases, the truncated composite distributions are less heavy tailed.

Proposition2.2. The cumulative distribution function of the truncated composite Weibull-Pareto model is given by

(2.13) F_{W P t}^c (x) =











1−exp

− x−a

γ

β

2−exp

− θ+b−a

γ

β if a < x≤θ+b, 1− (_x−b^θ )^α

2−exp

− θ+b−a

γ

β if θ+b≤x ≤ ∞.

Proof. Forx∈(a, θ+b] we have

F(x) = Z _x

a

cf₁(y) dy=

1−exp

−

x−a γ

β

2−exp

−

θ+b−a γ

β.

Similarly, for x∈[θ+b,∞) we get

F(x) =c Z _θ+b

a

f₁(y) dy+c Z _x

θ+b

f₂(y) dy=I₁+I₂,

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where

I₁ =c Z θ+b

a

β

γ^β (y−a)^β⁻¹exp −

y−a γ

β! dy

=c

"

1−exp −

θ+b−a γ

β!#

and

I₂ =c Z _x

θ+b

αθ^α

(y−b)^α+1dy=c

1− θ

x−b α

.

Thus,

F(x) =I₁+I₂ =c

"

2−exp −

θ+b−a γ

β!

− θ

x−b α#

and this yields (2.13).

Proposition2.3. The unique mode of the truncated composite Weibull- Pareto distribution is

(2.14) x^{cW P t}_mode =γ

β−1 β

_β¹ +a.

Proof. Solving the equationf^′(x) = 0, wheref is the density of the truncated composite Weibull-Pareto distribution, we get the unique local maximum at x given by (2.14).

Other characteristics of the truncated composite Weibull-Pareto model are the initial moments.

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

E(X^r) = 1

2−exp

−

θ+b−a γ

β× (2.15)

× Xr

j=0

r j

a^r⁻^j

( γ^jΓ j

β +1;

θ+b γ

β! +αθ^α

Xj

l=0

j l

(b−a)^j⁻^l(θ+a)^l⁻^α α−l

)

for l < α, where Γ(s;z) =R_z

0 y^s⁻¹e⁻^ydy is the incomplete Gamma function.

Proof. If X is a random variable with truncated composite Weibull- Pareto distribution, then

E(X^r) = Z ∞

a

x^rf(x)dx,

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where f is the truncated composite Weibull-Pareto density given by (2.2).

Consequently, we have E(X^r) =

Xr

j=0

r j

a^r⁻^j

Z ∞ a

(x−a)^jf(x)dx.

Setting x−a=y we get E(X^r) =

Xr

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

I_1j =c Z _θ+b

0

y^j β

γ^βy^β⁻¹e⁻

y γ

β

dy, j= 0, r.

Setting

y γ

β

=z we get

I_1j =cγ^jΓ j β + 1;

θ+b γ

β! .

Let

I_2j =cαθ^α Z ∞

θ+b

y^j

(y+a−b)^α+1dy, j = 0, r.

Setting y+a−b=twe get I_2j =cαθ^α

Xj

l=0

j l

(b−a)^j⁻^l (θ+a)^l⁻^α α−l for l−α <0. Then

E(X^r) =c Xr

j=0

r j

a^r⁻^j

γ^jΓ j β + 1;

θ+b γ

β! +

+αθ^α Xj

l=0

j l

(b−a)^j⁻^l(θ+a)^l⁻^α α−l

for l−α <0.Replacing cgiven by (2.3), we obtain (2.15).

Proposition 2.5. Let (x₁, . . . , x_n) be a random sample from the two- parameter truncated composite Weibull-Pareto model. Assuming that x₁ ≤

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x₂ ≤ · · · ≤x_n and x_m≤θ+b≤x_m+1, the likelihood function is given by L(x₁, x₂, . . . , x_n;α, β, γ, θ) =

( 2−exp

"

−

θ+b−a γ

β#)⁻n

β^mγ⁻^βm× (2.16)

×αⁿ⁻^mθ^α(n⁻^m)exp

"

− 1 γ^β

Xm

i=1

(x_i−a)^β

# Q^m

i=1

(x_i−a)^β⁻¹ Qn

i=m+1

(x_i−b)^α+1 .

Moreover, if k∈R\ {k₀},where k₀ is the solution of the equation e⁻^k−1+_k¹ = 0, then the maximum likelihood estimate of k denoted k_{M L}, is the solution of the equation

(2.17) −n e⁻^k

2−e⁻^k + βm

(1−k) e⁻^k−1

(ke⁻^k−k+ 1) [1−β(ke⁻^k−k+ 1)] +m k−

−(n−m) βk²+ (β−1) (1−k) k[1−β(ke⁻^k−k+ 1)]−

−(n−m)βe⁻^k

βk²+ (β−1) (1−k)

[1−β(ke⁻^k−k+ 1)]² ln a−b

β(ke⁻^k−k+ 1)−

−(n−m) βke⁻^k

(1−k) e⁻^k−1

(ke⁻^k−k+ 1) [1−β(ke⁻^k−k+ 1)]−

−

"

βk

(1−k) e⁻^k−1

(ke⁻^k−k+1) [1−β(ke⁻^k−k+1)]+ 1

# β^β ke⁻^k−k+1β

[1−β(ke⁻^k−k+1)]^β(a−b)^β×

× Xm

i=1

(x_i−a)^β+βe⁻^k

βk²+ (β−1) (1−k) [1−β(ke⁻^k−k+ 1)]²

Xn

i=m+1

ln (x_i−b) = 0 while the maximum likelihood estimate of β, denoted β_{M L},is the solution of the equation

(2.18) m

β −mln

1−β ke⁻^k−k+ 1

(a−b)

β(ke⁻^k−k+ 1) + βm+n−m

β[1−β(ke⁻^k−k+ 1)]+ + (n−m) ke⁻^k

[1−β(ke⁻^k−k+ 1)]²ln a−b

β(ke⁻^k−k+ 1)−

−(n−m) ke⁻^k

1−β(ke⁻^k−k+ 1)+ Xm

i=1

ln (xi−a)−

− ke⁻^k

[1−β(ke⁻^k−k+ 1)]² Xn

i=m+1

ln (x_i−b) + kβ^β ke⁻^k−k+ 1β

[1−β(ke⁻^k−k+ 1)]^β(a−b)^β×

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× ("

1

1−β(ke⁻^k−k+ 1)+ ln β ke⁻^k−k+ 1 [1−β(ke⁻^k−k+ 1)] (a−b)

# _m X

i=1

(xi−a)^β+

+ Xm

i=1

(x_i−a)^βln (x_i−a) )

= 0.

Proof. Let be x₁ ≤ x₂ ≤ · · · ≤ x_n a random sample from the two- parameter truncated composite Weibull-Pareto model, with density function given by (2.2). In order to determine the likelihood function, we must have an idea of where is situated the unknown parameter θ, so assume, e.g., that x_m≤θ+b≤x_m+1. Then the likelihood function is

L(x₁, x₂, . . . , x_n;α, β, γ, θ) = Yn

i=1

f(x_i) = Ym

i=1

f(x_i)× Yn

i=m+1

f(x_i)^(2.2)=

= (

2−exp

"

−

θ+b−a γ

β#)⁻n

β^mγ⁻^βmαⁿ⁻^mθ^α(n⁻^m)×

×exp

"

− 1 γ^β

Xm

i=1

(x_i−a)^β

# Q^m

i=1

(x_i−a)^β⁻¹ Qn

i=m+1

(x_i−b)^α+1 .

Replacing c, θ, α and γ by (2.3), (2.9), (2.11) and (2.12), respectively, the loglikelihood function becomes

lnL(x₁, x₂, . . . , x_n;k, β) =−nln 2−e⁻^k

+mlnβ−

−βmln(a−b) (1−A(k;β))

A(k;β)k^1/β + (n−m) ln

βk−β+ 1 1−A(k;β) −1

+

+ (n−m)

βk−β+ 1 1−A(k;β) −1

ln a−b A(k;β)−

− kA^β(k;β) (a−b)^β(1−A(k;β))^β

Xm

i=1

(x_i−a)^β+ (β−1) Xm

i=1

ln (x_i−a)−

−βk−β+ 1 1−A(k;β)

Xn

i=m+1

ln (x_i−b).

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

∂lnL

∂k = ∂lnL

∂β = 0.

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Thus, the likelihood equations are

−n e⁻^k

2−e⁻^k + βm

A(k;β) (1−A(k;β))

∂A(k;β)

∂k +m k+

+ (n−m) 1

βk−β+1 1−A(k;β)−1

β

1−A(k;β) + βk−β+1 (1−A(k;β))²

∂A(k;β)

∂k

+

+ (n−m)

β

1−A(k;β) + βk−β+ 1 (1−A(k;β))²

∂A(k;β)

∂k

ln a−b A(k;β)+ + (n−m)

βk−β+ 1 1−A(k;β) −1

1 A(k;β)

∂A(k;β)

∂k −

−

βk

A(k;β) (1−A(k;β))

∂A(k;β)

∂k +1

A^β(k;β) (a−b)^β(1−A(k;β))^β

Xm

i=1

(xi−a)^β−

−

β

1−A(k;β) + βk−β+ 1 (1−A(k;β))²

∂A(k;β)

∂k

Xn

i=m+1

ln (x_i−b) = 0 and

m

β −mln(a−b) (1−A(k;β))

A(k;β) + βm

A(k;β) (1−A(k;β))

∂A(k;β)

∂β +

+ (n−m) 1

βk−β+1 1−A(k;β)−1

βk−β+ 1 (1−A(k;β))²

∂A(k;β)

∂β + k−1 1−A(k;β)

+

+ (n−m)

βk−β+ 1 (1−A(k;β))²

∂A(k;β)

∂β + k−1 1−A(k;β)

ln a−b A(k;β)−

−(n−m)

βk−β+ 1 1−A(k;β) −1

1 A(k;β)

∂A(k;β)

∂β +

Xm

i=1

ln (x_i−a)−

−

βk−β+ 1 (1−A(k;β))²

∂A(k;β)

∂β + k−1 1−A(k;β)

Xn

i=m+1

ln (x_i−b)−

− kA^β(k;β) (a−b)^β(1−A(k;β))^β

β

A(k;β) (1−A(k;β))

∂A(k;β)

∂β + + ln A(k;β)

(a−b) (1−A(k;β)) Xm

i=1

(x_i−a)^β+ Xm

i=1

(x_i−a)^βln (x_i−a) )

= 0.

We have to impose the conditions A(k;β) 6= 0 and 1−A(k;β) 6= 0, k >0, β >1. These conditions are equivalent to k6=k₀, where k₀ is the solution of the equation e⁻^k−1 + _k¹ = 0.

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We can also express ^∂A(k,σ)_∂β and ^∂A(k,σ)_∂k in terms ofkand β as

∂A(k;β)

∂β =ke⁻^k−k+ 1, ∂A(k;β)

∂k =β

(1−k) e⁻^k−1 .

Thus, the likelihood equations follow as (2.17) and (2.18).

3. PARAMETER ESTIMATION FOR THE TRUNCATED COMPOSITE WEIBULL-PARETO MODEL

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

Let (x₁, . . . , x_n) be a random sample from the truncated composite Wei- bull-Pareto model with density function given by (2.2). From Remark 2.2, the number of unknown parameters is two, let them bekandβ. Without any loss of generality, we assume that the sample is ordered, i.e.,

x₁ ≤x₂≤ · · · ≤x_n.

The method of moments consist of equating the unobservable popu- lation moments with the sample moments, i.e.,

(2.19) E(X) =x, E X²

=y, where x= ¹_nP_n

i=1x_i,y= _n¹P_n

i=1x²_i, and E(X),E X²

are given by Propo- sition 2.4. System (2.19) is equivalent to

aΓ 1;

θ+b γ

β!

+γΓ 1 β + 1;

θ+b γ

β!

+(b−a)θ^α (θ+a)^α +

+ αθ^α

(α−1) (θ+a)^α⁻¹ = (

2−exp

"

−

θ+b−a γ

β#) 1 n

Xn

i=1

x_i

and

a²Γ 1;

θ+b γ

β! + 2a

"

γΓ 1 β + 1;

θ+b γ

β!

+ (b−a)θ^α (θ+a)^α +

+ αθ^α

(α−1) (θ+a)^α⁻¹

+γ²Γ 2 β + 1;

θ+b γ

β!

+(b−a)²θ^α (θ+a)^α + + 2(b−a)αθ^α

(α−1) (θ+a)^α⁻¹ + αθ^α

(α−2) (θ+a)^α⁻² =

= (

2−exp

"

−

θ+b−a γ

β#) 1 n

Xn

i=1

x²_i.

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It follows from (2.9), (2.11) and (2.12) that θ, α and γ can be expressed in terms of kand β. The likelihood equations are too complicated to obtain an explicit solution. The solving of such non-linear equations could be performed by a mathematics software. Solving these non-linear equations, we get the solutions keand β.e

Maximum Likelihood Estimation (MLE).The likelihood equations (2.17) and (2.18) can be also solved by the Newton-Raphson method. The unknown parameters arek and β while the initial values, calculated with the method of moments, are keandβ. If we denote bye ϕ(k;β) the first likelihood equation and byψ(k;β) the second one, then the generalized Newton-Raphson mehod leads to the iterations











k_i+1 =k_i+ψ(k_i;β_i)ϕ^′_β(k_i;β_i)−ϕ(k_i;β_i)ψ^′_β(k_i;β_i) J

β_i+1 =β_i+ϕ(k_i;β_i)ψ^′_k(k_i^′β_i)−ψ(k;β_i)ϕ^′_k(k_i;β_i) J

k₀ =ek, β₀ =βe for k = 0,1,2, . . . , where J = det

ϕ^′_k(k;β) ϕ^′_β(k;β) ψ_k^′ (k;β) ψ_β^′ (k;β)

. The number of steps are fixed or STOP when the conditions|k_i+1−k_i|< εand|β_i+1−β_i|< ε are satisfied (for example, ε= 10⁻⁵). Denote by k_{M L} and β_{M L} the solutions of the equations above (the maximum likelihood estimators ofkandβ). Then, using (2.11), (2.9) and (2.12), the maximum likelihood estimators of α,θ and γ denoted by α_{M L},θ_{M L} and γ_{M L}, respectively, are

α_{M L}= β_{M L}k_{M L}−β_{M L}+ 1

1−A(k_{M L};β_{M L}) −1, θ_{M L} = a−b A(k_{M L};β_{M L}), γ_{M L} = (a−b) [1−A(k_{M L};β_{M L})]

A(k_{M L};β_{M L})k_{M L}^1/β^{M L} .

4. A NUMERICAL EXAMPLE

As an example, we estimate the two parameters of the truncated composite Weibull-Pareto model using a data set consisting of 100 values that was sampled from this model with k= 2,β = 3,a= 0.5,b= 1. The true value of m is, in this case, 64. See Table 1.

The generating algorithm used is based on the inversion of the cumulative function distribution (2.13).

To estimate the parameters, we apply the algorithm presented in Sec- tion 3. The estimated values of the parameters are k_{M L} = 2.023 andβ_{M L} = 3.467. At 95% level of significance, the χ² goodness-of-fit with 8 degrees of

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Table 1. 100 truncated Weibull-Pareto values fork= 2,β= 3,a= 0.5,b= 1 0.6216 0.6885 0.6960 0.6964 0.7010 0.7170 0.7287 0.7567 0.7722 0.7769 0.8032 0.8119 0.8380 0.8562 0.8590 0.8672 0.8806 0.8858 0.8932 0.9021 0.9215 0.9284 0.9300 0.9333 0.9385 0.9388 0.9397 0.9613 0.9631 0.9703 0.9736 0.9750 0.9755 0.9999 1.0156 1.0158 1.0207 1.0261 1.0312 1.0335 1.0374 1.0400 1.0482 1.0528 1.0597 1.0631 1.0637 1.0854 1.0855 1.1051 1.1244 1.1435 1.1559 1.1645 1.1663 1.1689 1.1817 1.1827 1.1835 1.1908 1.1949 1.1982 1.2138 1.2170 1.2388 1.2566 1.2720 1.2732 1.4756 1.5190 1.5690 1.6250 1.6819 1.8064 1.8099 1.8581 1.8591 1.9339 2.1553 2.3280 2.9776 3.2056 3.6914 4.3726 5.0946 5.4576 9.9668 18.194 18.588 40.838 52.910 102.74 190.50 324.69 731.49 1096.0 1725.1 8810.4 8974.6 117740

Table 2. Grouped data and theχ²test (columns 2-3 results from the data sample while columns 4-5 are calculated using the truncated

composite Weibull-Pareto distribution function) Sample abs. Trunc W-P Trunc W-P

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

0.6216–0.79999 10 0.1 0.05419 3.87148

0.8000–0.89999 9 0.09 0.08359 0.04912

0.9000–0.94999 8 0.08 0.05617 1.01087

0.9500–0.99999 7 0.07 0.06190 0.10597

1.0000–1.09999 15 0.15 0.12206 0.63948

1.1000–1.19999 13 0.13 0.09040 1.73429

1.2000–1.99999 16 0.16 0.17883 0.19829

2.0000–99.9999 13 0.13 0.24236 5.20979

100.00–117740 9 0.09 0.09029 0.00009

P 100 1 χ²distance=12.81942

freedom is 15.507. From Table 2 one can notice that theχ²distance calculated for the estimated values of the parameters is d²(k_{M L};β_{M L}) = 12.81942. On account of the values obtained we decide that theχ²test accepts the truncated composite Weibull-Pareto model, as expected.

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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]

and

“Carol Davila” University of Medicine and Pharmacy Faculty of Medicine

Departament of Medical Informatics and Biostatistics Bd. Eroilor Sanitari nr. 8

050474 Bucharest, Romania [email protected]