ON THE TRUNCATED COMPOSITE WEIBULL-PARETO MODEL

Transcription

1 ON THE TRUNCATED COMPOSITE WEIBULL-PARETO MODEL SANDRA TEODORESCU and EUGENIA PANAITESCU The composite Weibull-Pareto model 2] was introduced as an alternative to the composite Lognormal-Pareto ] 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.. 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 reinsurance payments ] 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. Obviously, 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 reinsurance claims. A composite model that combines the Lognormal distribution for small losses and the Pareto for large ones was introduced by Cooray and Ananda ]. In the same manner, as an alternative model to composite Lognormal-Pareto proposed in ], 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. REPORTS 6), ),

2 260 Sandra Teodorescu and Eugenia Panaitescu 2 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 nontruncated 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, and rth initial moments, and we determine the likelihood function. On account of the results obtained in Section 2, an algorithm for estimating the parameters of the truncated composite model is given in Section 3. A numerical example based on simulated data set is presented in Section 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 ] and Ciumara 2], the truncated composite Weibull-Pareto density is derived from the density { cf x) if a < x θ b, 2.) fx) = cf 2 x) if θ b x <, where c is a normalizing constant, f x) is a Weibull density while f 2 x) is a two-parameter Pareto density, i.e., f x) = ) ) x a x a) exp, x > a, > 0, >, f 2 x) = αθ α x b) α, x > θ b, θ > 0, α >, where > 0, > 0, α > 0, θ > 0 are unknown parameters.

3 3 On truncated composite Weibull-Pareto model 26 Proposition 2.. The truncated composite Weibull-Pareto density is given by 2.2) = where α = 2exp 2exp θba θba θ θba fwp c t x) = ) ) x a) exp xa ) ) if a < x θ b, ) ) αθ α xb) α if θ b x <, θba ) ]. Proof. From the normalizing condition a fx)dx = we get 2.3) c = 2 e k), where 2.4) k not = ) θ b a. 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 ) ) θ b a 2.5) θ b a) exp = α θ and 2.6) θ b a)2 e ) θba ) ] θ b a = αα) θ 2. From the last two equations we get ) θ θ b a 2.7) α = ]. θ b a Thus, the truncated composite Weibull-Pareto density is given by 2.2). Remark 2.. Replacing α given by 2.7) in the continuity condition 2.5), we obtain 2.8) e k = a b) k θ k, whence 2.9) θ = a b Ak;),

4 262 Sandra Teodorescu and Eugenia Panaitescu 4 where Ak;) not = ke k k ). From 2.4) we have 2.0) = θ b a k /. Inserting 2.9) into 2.7) and 2.0) yields 2.) α = and 2.2) = k ) Ak;) a b) Ak;)) Ak;) k /. Remark 2.2. Initially, f c WP t x) had four parameters, but imposing the continuity and differentiability conditions at θ b, the number of parameters reduces to three:, 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 and in terms of k and ). Five illustrative density curves for the truncated composite Weibull- Pareto model are presented in Figures through 4. Fig.. The truncated composite Weibull-Pareto density curves for k = 2, a = 0.5, b = and different values of. In Figure, we display the density curves for the truncated composite Weibull-Pareto model. Notice that for this choice of parameters, as increases, the five densities approach zero faster.

5 5 On truncated composite Weibull-Pareto model 263 In Figure 2 we also display the density curves for the truncated composite Weibull-Pareto model and for this choice of parameters, as k increases, the five densities approach zero faster while the modes decrease. Fig. 2. The truncated composite Weibull-Pareto density curves for = 3, a = 0.5, b = and different values of k. 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 and different values of a and b in Figure 3, and for the same values of a and b and different values of k and in Figure 4. Fig. 3. The truncated composite Weibull-Pareto density curves for k = 2, = 3 and different values of a and b.

6 264 Sandra Teodorescu and Eugenia Panaitescu 6 Fig. 4. The truncated composite Weibull-Pareto density curves plotted for a = 0.5, b = and different values of k and. Let us note that for this choice of parameters, as b decreases, the truncated composite distributions are less heavy tailed. Proposition 2.2. The cumulative distribution function of the truncated composite Weibull-Pareto model is given by 2.3) FWPtx) c = ) ) exp xa 2exp θba xb) θ α 2exp θba ) ) if a < x θ b, ) ) if θ b x. Proof. For x a,θ b] we have Fx) = x a cf y)dy = ) ) exp xa ) ). 2 exp θba Similarly, for x θ b, ) we get Fx) = c θb a x f y)dy c f 2 y)dy = I I 2, θb

7 7 On truncated composite Weibull-Pareto model 265 where and Thus, θb ) ) y a I = c a y a) exp dy ) )] θ b a = c exp x αθ α ) θ α ] I 2 = c θb y b) αdy = c. x b ) ) θ b a Fx) = I I 2 = c 2 exp and this yields 2.3). ) ] θ α x b Proposition 2.3. The unique mode of the truncated composite Weibull- Pareto distribution is ) 2.4) x cwpt mode = a. Proof. Solving the equation f x) = 0, where f is the density of the truncated composite Weibull-Pareto distribution, we get the unique local maximum at x given by 2.4). 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 2.5) ) = ) ) 2 exp θba r { ) ) r )a rj j j θ b j } j Γ j ; αθ )ba) α jlθa)lα l αl j=0 for l < α, where Γs;z) = z 0 ys e y dy is the incomplete Gamma function. Proof. If X is a random variable with truncated composite Weibull- Pareto distribution, then E X r ) = a l=0 x r fx)dx,

8 266 Sandra Teodorescu and Eugenia Panaitescu 8 where f is the truncated composite Weibull-Pareto density given by 2.2). Consequently, we have E X r ) = Setting x a = y we get E X r ) = r j=0 r j=0 r )a rj x a) j fx)dx. j a r θb ] )a rj y j f y a) dy y j f y a)dy. j 0 θb Let I j = c θb 0 y j y e y ) dy, j = 0,r. Setting Let y ) = z we get ) ) I j = c j j θ b Γ ;. I 2j = cαθ α Setting y a b = t we get for l α < 0. Then E X r ) = c I 2j = cαθ α j=0 αθ α θb j l=0 y j y a b) αdy, j = 0,r. j )b a) jl θ a) lα l α l r { ) ) r )a rj j j θ b Γ j ; j ) j b a) jl θ a) lα } l α l l=0 for l α < 0. Replacing c given by 2.3), we obtain 2.5). Proposition 2.5. Let x,...,x n ) be a random sample from the twoparameter truncated composite Weibull-Pareto model. Assuming that x

9 9 On truncated composite Weibull-Pareto model 267 x 2 x n and x m θ b x m, the likelihood function is given by { ) ]} θba n 2.6) Lx,x 2,...,x n ;α,,,θ)= 2exp m m ] m α nm θ αnm) exp x i a) x i a). n x i b) α i=m Moreover, if k R\ {k 0 }, where k 0 is the solution of the equation e k k = 0, then the maximum likelihood estimate of k denoted k ML, is the solution of the equation 2.7) n ek 2 e k m k) e k ] ke k k ) ke k k )] m k n m) k2 ) k) k ke k k )] n m) ek k 2 ) k) ] a b ke k k )] 2 ln ke k k ) ke k k)e k ] n m) ke k k ) ke k k )] k k) e k ] ] ke k k) ke k k)] ke k k ) ke k k)] a b) x i a) ek k 2 ) k) ] n ke k k )] 2 ln x i b) = 0 i=m while the maximum likelihood estimate of, denoted ML, is the solution of the equation 2.8) m m ln ke k k )] a b) ke k k ) n m) ke k ke k k )] 2 m n m ke k k )] ke k ke k k )] 2 ln a b ke k k ) ke k m n m) ke k k ) ln x i a) n i=m k ke k k ) ln x i b) ke k k )] a b)

10 268 Sandra Teodorescu and Eugenia Panaitescu 0 { ke k k ) ln ke k k ) ] m ke k x i a) k )] a b) } x i a) ln x i a) = 0. Proof. Let be x x 2 x n a random sample from the twoparameter 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. Then the likelihood function is Lx,x 2,...,x n ;α,,,θ) = n m f x i ) = f x i ) n i=m f x i ) 2.2) = { ) ]} θ b a n = 2 exp m m α nm θ αnm) exp ] x i a) m x i a) n i=m x i b) α. Replacing c, θ,α and by 2.3), 2.9), 2.) and 2.2), respectively, the loglikelihood function becomes ln Lx,x 2,...,x n ;k,) = n ln 2 e k) m ln ] a b) Ak;)) k m ln Ak;) k / n m) ln Ak;) ] k n m) Ak;) ln a b Ak;) ka k;) a b) Ak;)) x i a) ) ln x i a) k Ak;) n i=m ln x i b). In order to maximize the likelihood function we have to solve the system ln L k = ln L = 0.

11 On truncated composite Weibull-Pareto model 269 Thus, the likelihood equations are and n ek 2 e k m Ak;) Ak;) Ak;)) k n m) ) k Ak;) n m) Ak;) k Ak;) Ak;)) 2 k ) k n m) Ak;) Ak;) ] k Ak; ) Ak; )Ak; )) k Ak;) k Ak;) Ak;)) 2 k m a b) Ak;)) m ln Ak;) m k Ak;) k Ak;) Ak;)) 2 k ] ln a b Ak;) Ak;) k A k;) ab) Ak;)) ] n i=m m Ak;) Ak;)) k Ak;) Ak;)) 2 ] x i a) ln x i b) = 0 Ak;) ] k n m) ) k Ak;) Ak;) ] k Ak;) k n m) Ak;)) 2 ln a b Ak;) Ak;) ) k n m) Ak;) Ak;) ln x i a) Ak;) ] k Ak;) k n Ak;)) 2 ln x i b) Ak;) i=m ka { k;) Ak;) a b) Ak;)) Ak;) Ak;)) ] m } Ak;) ln x i a) x i a) ln x i a) = 0. a b) Ak;)) We have to impose the conditions Ak;) 0 and Ak;) 0, k > 0, >. These conditions are equivalent to k k 0, where k 0 is the solution of the equation e k k = 0.

12 270 Sandra Teodorescu and Eugenia Panaitescu 2 We can also express Ak,σ) Ak;) = ke k k, and Ak,σ) k Ak;) k in terms of k and as = k)e k ]. Thus, the likelihood equations follow as 2.7) and 2.8). 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 x,...,x n ) be a random sample from the truncated composite Weibull-Pareto model with density function given by 2.2). From Remark 2.2, the number of unknown parameters is two, let them be k and. Without any loss of generality, we assume that the sample is ordered, i.e., x x 2 x n. The method of moments consist of equating the unobservable population moments with the sample moments, i.e., 2.9) EX) = x, E X 2) = y, where x = n n x i, y = n n x2 i, and EX), E X 2) are given by Proposition 2.4. System 2.9) is equivalent to ) ) θ b ) ) θ b aγ ; Γ ; b a)θα θ a) α { αθ α ) ]} θ b a α ) θ a) α = n 2 exp x i n and ) ) θ b ) ) a 2 θ b Γ ; 2a Γ ; αθ α ] α ) θ a) α 2 2 Γ ; θ b 2b a)αθα α )θ a) α αθ α α 2) θ a) α2 = { ) ]} θ b a n = 2 exp x 2 i n. b a)θα θ a) α ) ) b a)2 θ α θ a) α

13 3 On truncated composite Weibull-Pareto model 27 It follows from 2.9), 2.) and 2.2) that θ, α and can be expressed in terms of k and. 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 k and. Maximum Likelihood Estimation MLE). The likelihood equations 2.7) and 2.8) can be also solved by the Newton-Raphson method. The unknown parameters are k and while the initial values, calculated with the method of moments, are k and. If we denote by ϕk;) the first likelihood equation and by ψ k; ) the second one, then the generalized Newton-Raphson mehod leads to the iterations k i = k i ψ k i; i ) ϕ k i; i ) ϕk i ; i ) ψ k i; i ) J i = i ϕk i; i ) ψ k k i i) ψ k; i )ϕ k k i; i ) J k 0 = k, 0 = ϕ for k = 0,,2,..., where J = det k k;) ϕ k;) ) ψ k k;) ψ k;). The number of steps are fixed or STOP when the conditions k i k i < ε and i i < ε are satisfied for example, ε = 0 5 ). Denote by k ML and ML the solutions of the equations above the maximum likelihood estimators of k and ). Then, using 2.), 2.9) and 2.2), the maximum likelihood estimators of α, θ and denoted by α ML, θ ML and ML, respectively, are α ML = MLk ML ML Ak ML ; ML ), θ ML = ML = a b) Ak ML; ML )] Ak ML ; ML )k /. ML ML a b Ak ML ; ML ), 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 00 values that was sampled from this model with k = 2, = 3, a = 0.5, b =. The true value of m is, in this case, 64. See Table. The generating algorithm used is based on the inversion of the cumulative function distribution 2.3). To estimate the parameters, we apply the algorithm presented in Section 3. The estimated values of the parameters are k ML = and ML = At 95% level of significance, the χ 2 goodness-of-fit with 8 degrees of

14 272 Sandra Teodorescu and Eugenia Panaitescu 4 Table. 00 truncated Weibull-Pareto values for k = 2, = 3, a = 0.5, b = Table 2. Grouped data and the χ 2 test 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. n i rel. freq., f i th. freq., p i nf i p i ) 2 p i χ 2 distance= freedom is From Table 2 one can notice that the χ 2 distance calculated for the estimated values of the parameters is d 2 k ML ; ML ) = On account of the values obtained we decide that the χ 2 test accepts the truncated composite Weibull-Pareto model, as expected. REFERENCES ] M. Cooray and M.M.A. Ananda, Modeling actuarial data with a composite lognormal- Pareto model. Scand. Actuar. J ), ] Roxana Ciumara, An actuarial model based on the composite Weibull-Pareto distribution. Math. Rep. Bucur.) 858) 2006),

15 5 On truncated composite Weibull-Pareto model 273 3] M. Iosifescu, G. Mihoc and R. Theodorescu, Teoria probabilităţilor şi statistica matematică. Ed. Tehnică, Bucureşti, ] N. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions. Willey, New York, ] R. Kaas, M. Goovaerts, M. Denuit and J. Dhaene, Modern Actuarial Risk Theory. Kluwer, Boston, ] S.A. Klugman, H.H. Panjer and G.E. Willmot, Loss Models: from Data to Decisions. 2nd Ed. Wiley, New York, ] J.M. Ortega, Numerical Analysis. Academic Press, New York, ] V. Preda, Teoria deciziilor statistice. Ed. Academiei Române, Bucureşti, ] V. Preda and Roxana Ciumara, On composite models: Weibull-Pareto and Lognormal- Pareto. A comparative study. Romanian J. Econom. Forecasting ), ] Sandra Teodorescu and Raluca Vernic, A composite Exponential Pareto distribution. An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat ),, ] Sandra Teodorescu and Raluca Vernic, On the truncated composite Exponential-Pareto distributions. Bul. Ştiinţ. Univ. Piteşti Ser. Mat. Inform ), 2. 2] Sandra Teodorescu and Raluca Vernic, Some composite Exponential-Pareto models for actuarial prediction. Submitted to Romanian J. Econom. Forecasting. Received 9 February 2009 Ecological University of Bucharest Faculty of Economic Sciences Bd. Vasile Milea nr. G 0634 Bucharest, Romania cezarina teodorescu@yahoo.com and Carol Davila University of Medicine and Pharmacy Faculty of Medicine Departament of Medical Informatics and Biostatistics Bd. Eroilor Sanitari nr Bucharest, Romania e.panaitescu@yahoo.com