Models Associated with Extended Exponential Smoothing

Denis Bosq

Communications in Statistics - Theory and Methods, 2015, vol. 44, issue 3, 468-475

Abstract: We study an extended form of exponential smoothing which is more flexible than the original one: let (Xt,t∈Z)$(X_{t},\: t\in \mathbb {Z})$ be a real stochastic process, observed until time n, consider the probabilistic predictor of Xn + 1 defined as Xn+1*=α∑j=0∞βjXn-j,(α∈R,β∈R),\[ X_{n+1}^{*}=\alpha \sum _{j=0}^{\infty }\beta ^{j}X_{n-j},\;\;\;(\alpha \in \mathbb {R},\:\beta \in \mathbb {R}), \] where the series is supposed to be convergent in mean square. We look for stochastic models such that X*n + 1 is the best linear predictor of Xn + 1, given Xt, t ⩽ n. We obtain various ARIMA models depending on (α, β). In this context we study estimation of (α, β) and give some indications concerning hypotheses testing. Finally, extension to functional stochastic processes is considered.

Date: 2015
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DOI: 10.1080/03610926.2012.748916

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