Advances in Fuzzy Systems

Volume 2019, Article ID 1393946, 14 pages

https://doi.org/10.1155/2019/1393946

## An Estimation of a Hybrid Log-Poisson Regression Using a Quadratic Optimization Program for Optimal Loss Reserving in Insurance

^{1}Institute of Basic Sciences Technology and Innovation, Pan African University-Jomo Kenyatta University of Agriculture and Technology, P.O. Box 62000-00200, Nairobi, Kenya^{2}National Advanced School of Engineering, University of Maroua, P.O. Box 46 Maroua, Cameroon^{3}Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon^{4}School of Computing and Information Technology, Jomo Kenyatta University of Agriculture and Technology, P.O. Box 62000-00200, Nairobi, Kenya

Correspondence should be addressed to Apollinaire Woundjiagué; rf.oohay@erianiloppaoppa

Received 15 December 2018; Revised 7 March 2019; Accepted 20 May 2019; Published 16 June 2019

Academic Editor: Patricia Melin

Copyright © 2019 Apollinaire Woundjiagué et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this article, we are interested in developing an alternative estimation method of the parameters of the hybrid log-Poisson regression model. In our previous paper, we have proposed a hybrid log-Poisson regression model where we have derived the analytical expression of the fuzzy parameters. We found that the hybrid model provide better results than the classical log-Poisson regression model according to the mean square error prediction and the goodness of fit index. However, nowhere we have taken into account the optimal value of (cut) which is of greatest importance in fuzzy regressions literature. In this paper, we provide an alternative estimation method of our hybrid model using a quadratic optimization program and the optimized value (cut). The expected value of fuzzy number is used as a defuzzification procedure to move from fuzzy values to crisp values. We perform the hybrid model with the alternative estimation we are suggesting on two different numerical data to predict incremental payments in loss reserving. From the mean square error prediction, we prove that the alternative estimation of the new hybrid model with an optimized value predicts incremental payments better than the classical log-Poisson regression model as well as the same hybrid model with analytical estimation of parameters. Hence we have optimized the outstanding loss reserves.

#### 1. Introduction

“An important role of a non-life actuary is the calculation of provisions, mainly IBNR (incurred but not reported) reserve. Then, finding the fair value of loss reserve is a relevant topic for non-life actuaries. Indeed, insurance companies must simultaneously have enough reserves to meet their commitment to policyholders and have enough funds for their investments. Therefore several methods have been proposed in actuarial science literature to capture this fair value” [1].

“In one hand, we distinguish deterministic methods [2–4]. They provide crisp predictions for reserves. In [*sic*] the other hand, [5–8] present stochastic methods. Those methods don’t give only a crisp value of the reserves but provide also their variability. For more details on existing loss reserving methods, one can consult [9–19]” [1].

But in [20], there are some experiences where stochastic methods can give unrealistic estimates. For example, when the claims are related to body injures, the future losses for the company will depend on the growth of the wage index that helps to determine the amount of indemnity and depends also on changes in court practices and public awareness of liability matters. Then the information is vague. Therefore the use of Fuzzy Set Theory becomes very attractive when the information is vague as in this case. Hence one should think about models that can handle both fuzziness and randomness, namely, hybrid models.

In [1], we have proved that we can improve the classical log-Poisson regression model through a hybrid one where the parameters are derived and have an analytical form. Although the new model provides better results than the classical log-Poisson regression model (according to mean square error prediction (MSEP) and goodness of fit index), we still have a large value of the MSEP. This could be due to the choice of value which is important in fuzzy regression framework. The purpose of this paper is to provide an alternative estimation which is taking into account the optimized value. We will prove in this paper that the optimized value is of the greatest importance because the value of the MSEP will be very low compared to the MSEP we are getting from the analytical estimation of the hybrid model.

In this paper, we investigate the possibilistic approach to estimate the hybrid model, i.e., to estimate the asymmetric triangular fuzzy coefficients (ATFC) of the model through a quadratic optimization program and by taking into account an optimized value of (cut) in loss reserving framework. We have investigated the fuzzy least-squares approach in our previous paper [1]. To move from FN to crisp values, we shall use the expected value of FN.

Our objective therefore is to come with a new estimation method of fuzzy parameters in the hybrid log-Poisson model where the optimized value will be taken into account. From two different data, we prove that our hybrid model the new estimation method provides best predictions of reserves compared to the classical log-Poisson model according to the MSEP criterion.

The structure of the paper is as follows: We present in the first section the preliminaries on fuzzy sets and their properties. In the second section, we shall review some models and results on FRM. In the third section, the framework of estimation of loss reserve with log-Poisson regression will be introduced. Our contribution starts from Section 5 where we propose a new estimation method of the hybrid log-Poisson regression model [1] for loss reserving through a quadratic optimization program by taking into account the optimized value and we prove its relevance from two datasets. Then we conclude the article.

#### 2. Preliminaries on Fuzzy Sets and Their Properties

In this section, we review some concepts related to our research. That is the concept of fuzzy set, membership function, FN, FRM.

##### 2.1. Review of Some Definitions and Properties of Fuzzy Sets

*Definition 1 (from [21]). *Let be a nonempty set and . In classical set theory, a subset of can be defined by its characteristic function as a mapping from the elements of to the elements of the set ,This mapping may be represented as a set of ordered pairs, with exactly one ordered pair present for each element of . The first element of the ordered pair is an element of the set , and the second element is an element of the set . The value zero is used to represent nonmembership, and the value one is used to represent membership. The truth or falsity of the statement “ is in ” is determined by the ordered pair . The statement is true if the second element of the ordered pair is 1, and the statement is false if it is 0.

Similarly, a fuzzy subset (also called fuzzy set) of a set can be defined as a set of ordered pairs, each with the first element from and the second element from the interval , with exactly one ordered pair present for each element of . This defines a mapping called membership function.

*Definition 2 (from [21]). *The membership function of a fuzzy set , denoted by , is defined bywhere is typically interpreted as the membership degree of element in the fuzzy set .

The degree to which the statement “ is in ” is true is determined by finding the ordered pair . The degree of truth of the statement is the second element of the ordered pair. A fuzzy set on can also be defined as a set of tuplesand could be represented by a graphic.

*Definition 3 (from [22]). *Let be the set of objects and The cut of is the set defined by

*Definition 4 (from [23]). *(1)A FN is a fuzzy set of a universe (the real line ) such that(a)all its cut are convex which is equivalent to the fact that is convex, that is, and ;(b) is normalized, that is, such that (c) is continued membership function of bounded support, where and are equipped with the natural topology.(2)A triangular fuzzy number (TFN) is a FN denoted by ; , such that and with the centre of , its left spread and its right spread [24]. A TFN could be defined with its membership degree function or, with its level ( cut () (see [23]), i.e., or(i)If , then define a STFN(ii)Otherwise ; then define an ATFN (see Figure 1).*(**1) Notes and Comments*. It is well know that if is a FN, then , the level (cut) of , is a compact set of , for all

Let us present some properties on TFN.