Insurance: Mathematics and Economics 51 (2012) 216–221 Contents lists available at SciVerse ScienceDirect Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime A maximum-entropy approach to the linear credibility formula Amir T. Payandeh Najafabadi a,∗ , Hamid Hatami b , Maryam Omidi Najafabadi c a Mathematical Sciences Department, Shahid Beheshti University, G.C. Evin, 1983963113, Tehran, Iran b E.C.O College of Insurance, Allameh Tabatabái University, Tehran, Iran c Department of Agricultural Extension and Education, Science and Research Branch, Islamic Azad University, Tehran, Iran article info Article history: Received May 2010 Received in revised form December 2010 Accepted 28 August 2011 JEL classification: C11 C16 Keywords: Linear credibility formula Bayes’ estimator Mean square-error technique Maximum-entropy technique Panel data Time series data Crop insurance abstract Payandeh [Payandeh Najafabadi, A.T., 2010. A new approach to credibility formula. Insurance: Mathematics and Economy 46, 334–338] introduced a new technique to approximate a Bayes’ estimator with the exact credibility’s form. This article employs a well known and powerful maximum-entropy method (MEM) to extend results of Payandeh Najafabadi (2010) to a class of linear credibility, whenever claim sizes have been distributed according to the logconcave distributions. Namely, (i) it employs the maximum-entropy method to approximate an appropriate Bayes’ estimator (with respect to either the square-error or the Linex loss functions and general increasing and bounded prior distribution) by a linear combination of claim sizes; (ii) it establishes that such an approximation coincides with the exact credibility formula whenever the require conditions for the exact credibility (see below) are held. Some properties of such an approximation are discussed. Application to crop insurance has been given. © 2011 Elsevier B.V. All rights reserved. 1. Introduction Classical credibility theory is a linearized Bayesian forecasting method, which combines different collections of data to obtain an accurate overall estimator. In elementary cases where all observa- tions are assumed to be independent and identically distributed, Payandeh Najafabadi (2010) established a new approach to ap- proximate an appropriate Bayes’ estimator by a convex combi- nation of a prior’s and current observations’ means. Payandeh’s (2010) method is no longer valid whenever observations are de- pendent or have to have different weights. Credibility theory began with papers by Mowbray (1914) and Whitney (1918). Bailey (1950) showed that the formula P = z µ + (1 − z ) ¯ X may be derived from Bayes’ theorem, either by using a Bernoulli(p)-Beta or by using Poisson(λ)-Gamma models. Bailey’s work led to the application of Bayesian methodology to credibility theory. An excellent introduction to credibility theory can be found, e.g., in Goovaerts and Hoogstad (1987), Herzog (1996), Kaas et al. (1996), Klugman et al. (2004) and Bühlmann and Gisler (2005). See also Norberg (2004) for an overview with useful references and links to Bayesian statistics and linear estimation. ∗ Corresponding author. E-mail address: amirtpayandeh@sbu.ac.ir (A.T. Payandeh Najafabadi). However, the credibility is restricted by a family of distribu- tions, conjugate prior, and the square-error loss function. Neither the claim distribution which are not members of the exponential fam- ily of distributions nor the non-conjugate prior, the predicted mean (Bayes’ estimator with respect to the square-error loss) is no longer linear with respect to the data (see Diaconis and Ylvisaker, 1979) and the credibility formula is no longer true. On the other hand, whenever the policyholder is undercharged (and the insurance company loses its money) or the insured is overcharged (and the insurer is at risk of losing the policy), the square-error loss as- signs a similar penalty to over- and undercharge. In order to assign more (or less) penalty to overcharged, one has to consider a loss function rather than the square-error loss. A loss function rather than the square-error loss (and Entropy loss) usually leads to a Bayes’ estimator which cannot be a linear combination of obser- vations’ and prior’s means. Therefore, in such cases, the credibil- ity formula no longer holds. Bühlmann (1967) overcame the prior limitation and proved that in a class of linear estimators with form δ Lin (X 1 ,..., X n ) = c 0 +  n j=1 c j X j an estimator P = z µ + (1 − z ) ¯ X , is also a distribution free credibility formula, which minimizes E {µ(θ) − δ Lin (X 1 ,..., X n )} 2 , whenever µ(θ) stands for the mean of an individual risk (i.e., µ(θ) = E (X |θ)), characterized by risk parameter θ , and ¯ X = (X 1 + X 2 +···+ X n )/n. Bühlmann and Straub (1970) then formalized the least square derivation of z = k/(n + k), where n is the number of trials or exposure units and 0167-6687/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2011.08.010