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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

Information

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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