A comparison of the ordinary and a varying parameter exponential smoothing

An adaptive-type exponential smoothing, motivated by an insurance tariff problem, is treated. We consider the processZn=ß(Zn –1)Xn+(1 –ß(Zn–1))Zn–1, whereXnare i.i.d. taking values in the interval [0,M],M≦ ∞ andßis a monotonically increasing function [0,M] → [c,d], 0 <c<d< 1. Together with (Zn), we consider the ordinary exponential smoothingYn=αXn+ (1 –α)Yn –1whereαis a constant, 0 <α< 1. We show that (Yn) and (Zn) are geometrically ergodic Markov chains (in the case of finite interval we even have uniform ergodicity) and thatEYn, EZnconverge to limitsEY, EZ,respectively, with a geometric convergence rate. Moreover, we show thatEzis strictly less thanEY=EXn.