PARAMETRIC DISTRIBUTION FAMILIES USEFUL FOR MODELLING NON-LIFE INSURANCE PAYMENTS DATA. TAIL BEHAVIOUR. Sandra Teodorescu 1

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1 PARAMETRIC DISTRIBUTION AMILIES USEUL OR MODELLING NON-LIE INSURANCE PAYMENTS DATA. TAIL BEHAVIOUR Sandra Teodorescu Abstract The present paper describes a series o parametric distributions used or modeling non-lie insurance payments data. O those listed special attention is paid to the transormed Beta distribution amily. This distribution as well as those which are obtained rom it (special cases o our-parameter transormed Beta distribution) are used in the modeling o high costs or even etreme ones. In the literature it ollows the tail behaviour o distributions depending on the parameters because the insurance payments data are tipically highly positively skewed and distributed with large upper tails. In the paper is described the tail behavior o the distribution in the let and right side respectively and deduced rom it a general case. There are also some graphs o probability density unction or one o the transormed Beta amily members which comes to reinorce the comments made. Keywords: non-lie insurance payments data actuary parametric distribution transormed Beta distribution tail distribution behaviour. Introduction The insurance companies are responsible or covering claims which are aleatory in character. The quantum o claims can hardly be known it can only be assessed through calculations based on the probability theory. Hence the signiicance o a comprehensive inormation system comprising data and long-term assessments o the requency and degree o risks has emerged. Based on such a system one can approimate the evolution o potential payments to cover the value o claims and o insured accounts as well as o the level premium system being also able to adopt decisions linked to the initiation o new supplemental insurance programs to protect assets people and liability insurance or to the unctioning and organizational structure o the insurance company. An insurance company resembles a living organism comprising several departments that must perorm vital unctions or the insured person to successully survive. One o the main departments o such a company is the claims control department designed to prevent claims whenever possible and to diminish those that cannot be avoided. The claims control has always been an important unction ever since the early days o the insurance system which is gaining increasing importance nowadays. In this paper we study a ew parameter distribution models o insurance costs. There are a number o characteristics that our collection o distributions should have. Through those it is another one describeing that some o them should have moderate tails some should have heavy tails (see [4]). The issue o the shape o the right-hand tail is a critical one or the actuary. I the possibility o large losses were remote there might be little interest in the purchase o insurance. On the other hand many loss processes produce large claims with enough Associate proessor Ph.D. aculty o Economic Sciences Nicolae Titulescu University o Bucharest Romania tsandra@net.ro 04

2 regularity to inspire people and companies to purchase insurance and to create problems or the provider o insurance. or any distribution that etends probability to ininity the issue is one o how quickly the density unction approaches zero as the loss approaches ininity. The slower it happens the more probability is pushed onto higher values and we then say the distribution has a heavy thick or at tails. We avoid the adjective long because in insurance a long tail usually means that it takes a long time rom the issuance o the policy until all claims are settled. Classical Eponential and Pareto distributions used or insurance payments data also the composite Eponential-Pareto has been studied in detail in [0]. Also the comparative studies between Eponential Pareto and Eponential-Pareto distributions has been made in [8] and [9]. Results regarding some characteristics o shited composite Lognormal-Pareto and Weibull-Pareto as well the truncated ones the probability density unctions has been presented in [7]. The composite Lognormal-Pareto and Weibull-Pareto models has been studied in the literature and used to model insurance payments data (see [2] [] and [5]).. Transormed Beta amily Many o the distributions presented in this section are speciic cases o others. So there is a amily o distributions named transormed Beta amily who are useul or modeling large claims. or eample a Burr distribution with gamma= and and arbitrary is an Pareto distribution. Through this process our distributions can be organized into one grouping as illustrated in igure. The transormed Beta amiliy indicates two special cases o a dierent nature. The paralogistic and inverse paralogistic distributions are created by setting the two nonscale parameters ao the burr and inverse burr distributions equal to each other rather than to a speciic value.. our-parameter transormed Beta distribution: ; u u 2. Three-parameter distribution 2.. Tree parameter generalized Pareto distribution: It is obtained rom transormed Beta distribution or. Thus: ; u u 2.2. Tree-parameter Burr distribution: It is obtained rom transormed Beta distribution or. Thus: 05

3 u u 2.3. Tree-parameter inverese Burr distribution: It is obtained rom transormed Beta distribution or. Thus: u u 3. Two-parameter distributions 3.. Two-parameter Pareto distribution: It is obtained rom transormed Beta distribution or or rom generalized Pareto or or rom Burr distribution or. Thus: 3.2. Two-parameter inverese Pareto distribution: It is obtained rom transormed Beta distribution or or rom generalized Pareto distribution or or rom inverse Burr distribution or. Thus: 3.3. Two-parameter Loglogistic distribution: It is obtained rom transormed Beta distribution or or rom inverse Burr distribution or rom Burr distribution or. Thus: 06

4 u Two-parameter Paralogistic distribution: u It is obtained rom transormed Beta distribution or or rom Burr distribution or. Thus: u 2 u 3.5. Two-parameter inverese Paralogistic distribution: It is obtained rom transormed Beta distribution or or rom inverse Burr distribution or. Thus: u 2 2 u In conclusion the distributions are represented as it ollows: Paralogistic Transormed Beta Inverse Paralogistic Burr Generalized Pareto Inverse Burr Loglogistic Pareto Inverse Pareto 07

5 ig.. Transormed Beta amily 3. Tail behavior in the transormed Beta amily It is to be epected that with our parameters the transormed Beta distribution can assume a variety o shapes which can be investigated by determining the relationships between the parameters and the tail behavior. or the right-hand tail we look at the eistence o positive moments as well as the behavior o or. We can provide the same analysis or the let-hand tail (that is behavior or small claims). 3..Right tail behavior or the transormed Beta distribution the k-th moment eist or positive values o k provided k. Thus or all members o the transormed Beta amily there are only a inite number o moments. However it is clear that only the parameters and are involved in determining the behavior at the right end. To urther investigate this behavior consider the. ratio When both the numerator and denominator go to zero thus we can apply L Hôpital: or lim lim lim lim. the term in brackets goes to and thereore: lim which is a constant and so the probability in the right-hand tail is approimately proportional to. As a result the behavior in the right tail is tied to the product. So as the product gets larger the tail gets shorter (becomes zero aster). our illustrative density curves or the Burr distribution are presented in igure. 08

6 ig. The Burr density curves plotted or. In igure we display ew density curves or the Burr distribution. Notice that or this choice o parameters as increases the our densities approach zero aster. The general case Let be a unction so that lim t 0 and lim t0 t 0 t Then t R. 0 0 lim lim lim. We can see that there are unctions that have the above-mentioned properties. or eample t sin t 0 t t t tg t. 0 0 So the probability in the right-hand tail is approimately proportional to and the product determines the right-hand tail behavior. 09

7 With regard to the limiting distributions as goes to ininity we obtained the transormed Gamma distribution. It has all the positive moments and so all o its members have lighter tails than members o the transormed Beta amily. As goes to ininity we obtain the inverse transormed Gamma distribution. The same limit ( ) applied to the eistence o moments and so the right tail behavior is similar to that or the transormed Beta distribution. The inverse transormed Gamma distribution is obtained as goes to ininity and so all members o this amily have heavy right tails whose behavior is governed by the product o two nonscale parameters. 3.2.Let tail behavior The let tail behavior can be investigated by looking at the eistence o negative moments. or the transormed Beta distributions more negative moments eist as the product gets larger so it appears these parameters control the shape o the distribution or small P X the requisite limit is: loss values. With regard to probability in the let tail lim 0 lim 0 lim 0 lim 0. The probability in the let-hand tail is approimately proportional to. Again the thickness o the tail is governed by the product and larger values indicate a thinner tail. Conclusions As a result the behavior in the right tail o density in the transormed Beta amily is tied to the product. So as the product gets larger the tail gets shorter (becomes zero aster). In the let hand the thickness o the tail is governed by the product and larger values indicate a thinner tail. It is interesting to note that the parameter aects the behavior o both tails (let and right). As increases both tails get lighter and lighter. So in some way this parameter controls the spread around the middle while the parameters and control the right and let ends respectively. As a scale parameter plays no role in setting the shape o the distribution. Reerences Ciumara R. An actuarial model based on composite Weibull-Pareto distribution 2006 Mathematical Reports Vol 8 (58) nr.4/2006 0

8 Cooray M. and Ananda M.M.A. Modeling actuarial data with a composite lognormal-pareto model. Scand. Actuar. J. 5 (2005) Kaas R. Goovaerts M. Denuit M. Dhaene J. - Modern Actuarial Risk Theory- Kluwer Academic Publishers Boston 200 Klugmann S.A. Panjer H.N. Willmot G.E. - Loss models. rom data to decisions 998 Wiley Preda V. Ciumara R. On composite models: Weibull-Pareto and Lognormal-Pareto. A comparative study 2006 Romanian Journal o Economic orecasting 2 Teodorescu S. Metode statistice pentru estimarea repartiţiei daunelor din asigurări Editura Universităţii din Bucureşti 2009 Teodorescu S. On the truncated composite Lognormal-Pareto model Mathematical Reports Academia Română vol 2 (62) no. 200 Teodorescu S. Vernic R. Some composite Eponential-Pareto models or actuarial prediction Romanian Journal o Economic orecasting Academia Română vol. 2(4) 2009 Teodorescu S. Vernic R. A composite Eponential Pareto distribution Analele Ştiinţiice ale Universităţii Ovidius din Constanţa Seria Matematică vol. nr.xiv ascicol Teodorescu S. Vernic R. The truncated composite Eponential Pareto distribution Buletinul Ştiinţiic al Universităţii din Piteşti Seria Matematică şi Inormatică nr.2 pag