View source for Credibility theory

A branch of actuarial mathematics which explores principles for individual experience rating of insurance contracts by certain linear formulas. It emanated in 1918 from a work of A.W. Whitney [[#References|[a1]]], who proposed that the rate of premium currently charged for an individual insurance contract be a weighted average of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110490/c1104901.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110490/c1104902.png" /> is the observed mean claim amount per unit of risk exposed for the individual and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110490/c1104903.png" /> is the corresponding overall mean in the insurance portfolio (cf. also [[Risk theory|Risk theory]]). The weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110490/c1104904.png" />, which is a number between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110490/c1104905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110490/c1104906.png" />, was soon named credibility (or credibility factor), since it measures the  "amount of credence attached to the individual experience" , and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110490/c1104907.png" /> was called the credibility premium (or credibility premium rate).

Attempts to establish a theoretical basis for rating by credibility formulas bifurcated into two branches. First, the so-called limited fluctuation credibility theory took the point of view that the underlying true individual premium, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110490/c1104908.png" />, say, is a fixed, unknown parameter to be estimated. It focused in particular on developing criteria for  "full credibility" , <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110490/c1104909.png" />. This branch of the theory faded around 1960. For an overview see [[#References|[a2]]].

The other branch of credibility theory stuck to Whitney's original view of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110490/c11049010.png" /> as a [[Random variable|random variable]] and developed into what, in the wider perspective of contemporary (1996) [[Statistical decision theory|statistical decision theory]], is linear Bayes and linear empirical Bayes estimation applied to insurance problems (cf. also [[Bayesian approach|Bayesian approach]]). The basic paradigm was already set up in Robbins' empirical Bayes theory and Kalman's linear filtering theory when, at the end of the 1960s, H. Bühlmann in two seminal papers [[#References|[a6]]], [[#References|[a7]]] formulated the credibility problem as that of minimizing the mean-squared error <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110490/c11049011.png" /> over all inhomogeneous linear functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110490/c11049012.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110490/c11049013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110490/c11049014.png" /> constants. Under the assumption of conditional unbiasedness of the individual claims ratio, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110490/c11049015.png" />, the solution is the credibility premium <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110490/c11049016.png" /> above, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110490/c11049017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110490/c11049018.png" />. The mean-squared error criterion applied to linear formulas amounts to minimizing a positive-definite quadratic form in the coefficients. This simple optimization problem could readily be carried over to more complex models and wider classes of linear premium formulas, and a rapid development now followed in this branch of the theory, which received the name greatest accuracy credibility theory. Its evolutionary history paralleled, and partly preceded, that of its statistics counterpart, notably as regards Hilbert space methods [[#References|[a3]]], hierarchical and dynamical models, and schemes for recursive computation.

A survey of the history of credibility theory up to 1979 is given in [[#References|[a4]]]. An up-to-date introduction to the field is [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.W. Whitney,   "The theory of experience rating"  ''Proc. Casualty Actuarial Soc.'' , '''4'''  (1918)  pp. 274–292</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.H. Longley-Cook,   "An introduction to credibility theory"  ''Proc. Casualty Actuarial Soc.'' , '''49'''  (1962)  pp. 194–221</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. de Vylder,   "Geometrical credibility"  ''Scand. Actuarial J.''  (1976)  pp. 121–149</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Norberg,   "The credibility approach to experience rating"  ''Scand. Actuarial J.''  (1979)  pp. 181–221</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B. Sundt,   "An introduction to non-life insurance mathematics" , VVW e. V. Karlsruhe  (1993)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H. Bühlmann,   "Experience rating and credibility"  ''ASTIN Bull.'' , '''4'''  (1967)  pp. 199–207</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  H. Bühlmann,   "Experience rating and credibility"  ''ASTIN Bull.'' , '''5'''  (1969)  pp. 157–165</TD></TR></table>