PROCEEDINGS May 15, 16, 17, 18, 2005 - Casualty Actuarial Society

466 A NEW METHOD OF ESTIMATING LOSS RESERVESc ij >c i,j+1 for j =0,1,:::,N ¡ 1. (The case where the incrementalincurred losses form an arbitrary sequence isconsidered later in Section 5.2.)Clearly, from the definitions of N, S i ,andthec ij ’s,S i = c i0 + c i1 + ¢¢¢+ c iN : (3)where N =1,2,:::: is known. Equation (3) shows that S i canbe viewed as being split at random into N + 1 pieces of lossc i0 ,c i1 ,:::,c iN with the jth piece of loss being revealed (i.e., madeknown) at the end of the jth development year. On the otherhand, Assumption 3 implies that the sequence c i0 ,c i1 ,:::,c iN isan ordered sequence. It is unlikely that a purely random split willlead to an ordered sequence. Thus the precise nature of the splitmust be specified.Suppose the total unknown incurred S i is split at randomunder a uniform distribution into N + 1 pieces of loss labeledX i1 ,X i2 ,:::,X i,N+1 such thatS i = X i1 + X i2 + ¢¢¢+ X i,N+1 : (4)We further assume that these pieces of loss are ordered and relabeledso thatX i(1) · X i(2) ·¢¢¢·X i(N+1) :By Assumption 3 the incremental incurred loss is a realizationof the ordered pieces of loss, i.e.,for j =0,1,:::,N.c ij = X i(N+1¡j) and C ij =jXX i(N+1¡k) (5)At this point, it is important to clarify what is meant by thestatement “S i is split at random under a uniform distribution.”Suppose we have N independent and identically distributed ran-k=0