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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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References

Bonsdorff, H. (1989) On changing the parameter of exponential smoothing in experience rating. To appear.CrossRefGoogle Scholar
Brown, R. G. (1963) Smoothing, Forecasting and Prediction of Discrete Time Series. Prentice-Hall, Englewood Cliffs, N.J. Google Scholar
Doukhan, P. and Ghindes, M. (1980) Estimations dans le processus “Xn + 1 = f (Xn) + en”. C. R. Acad. Sci. Paris 291, série A, 61–64.Google Scholar
Feigin, P. D. and Tweedie, R. L. (1985) Random coefficient autoregressive processes: A Markov chain analysis of stationarity and finiteness of moments. J. Time Series Analysis 6, 1–14.Google Scholar
Gardner, E. S. Jr. (1985) Exponential smoothing: The state of the art. J. Forecasting 4, 1–28.CrossRefGoogle Scholar
HögnäS, G. (1986) Comparison of some non-linear autoregressive processes. J. Time Series Anal. 7, 205–211.Google Scholar
Kremer, E. (1982) Exponential smoothing and credibility theory. Insurance: Math. Econom. 1, 213–217.Google Scholar
Makridakis, S. et al. (1982) The accuracy of extrapolation (time series) methods: results of a forecasting competition. J. Forecasting 1, 111–153.Google Scholar
Mokkadem, A. (1987) Sur un modèle autorégressif non linéaire, ergodicité et ergodicité géométrique. J. Time Series Analysis 8, 195–204.Google Scholar
Nummelin, E. (1984) General Irreducible Markov Chains and Non-Negative Operators. Cambridge Tracts in Maths 83, Cambridge University Press, Cambridge.Google Scholar
Nummelin, E. and Tuominen, P. (1982) Geometric ergodicity of Harris recurrent Markov chains with applications to renewal theory. Stoch. Proc. Appl. 12, 187–202.Google Scholar
Orey, S. (1971) Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand Reinhold, London.Google Scholar
Rantala, J. (1984) An application of stochastic control theory to insurance business. Acta Universitatis Tamperensis A 164. Tampere, Finland.Google Scholar
Simberg, H. (1964) Individuelle Prämienregelung, eine Art des ‘Experience Rating’. Trans. 17th Internat. Congr. Actuaries, London Edinburgh 3, 650–659.Google Scholar
Trigg, D. W. and Leach, A. G. (1967) Exponential smoothing with an adaptive response rate. Operat. Res. Quart. 18, 53–59.Google Scholar
Tweedie, R. L. (1983) The existence of moments for stationary Markov chains. J. Appl. Prob. 20, 191–196.Google Scholar