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A comparison of the ordinary and a varying parameter exponential smoothing

Published online by Cambridge University Press:  14 July 2016

Heikki Bonsdorff
Heikki Bonsdorff*
Affiliation:
Pohjola Insurance Company Ltd
*
∗ Postal address: Pohjola Insurance Company Ltd, Lapinmäentie 1, 00300 Helsinki, Finland.
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Abstract

An adaptive-type exponential smoothing, motivated by an insurance tariff problem, is treated. We consider the process Zn = ß(Zn –1)Xn +(1 – ß (Zn–1))Zn–1, where Xn are i.i.d. taking values in the interval [0, M], M ≦ ∞ and ß is a monotonically increasing function [0, M] → [c, d], 0 < c < d < 1.

Together with (Zn), we consider the ordinary exponential smoothing Yn = αXn + (1 – α)Yn –1 where α is 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 that EYn, EZn converge to limits EY, EZ, respectively, with a geometric convergence rate. Moreover, we show that Ez is strictly less than EY = EXn.

Keywords

MARKOV CHAINSGEOMETRIC ERGODICITYUNIFORM ERGODICITYADAPTIVE SMOOTHINGEXPONENTIAL SMOOTHINGINSURANCE
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 4 , December 1989 , pp. 784 - 792
Copyright
Copyright © Applied Probability Trust 1989 

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