Aggregation of predictors for non stationary sub-linear processes and online adaptive forecasting of time varying autoregressive processes
Résumé
In this work, we study the problem of aggregating a finite number of
predictors for non stationary sub-linear processes. We provide oracle
inequalities relying essentially on three ingredients: 1) a uniform bound of
the $\ell^1$ norm of the time-varying sub-linear coefficients, 2) a Lipschitz
assumption on the predictors and 3) moment conditions on the noise appearing
in the linear representation. Two kinds of aggregations are considered giving
rise to different moment conditions on the noise and more or less sharp
oracle inequalities. We apply
this approach for deriving an adaptive predictor for locally stationary time
varying autoregressive (TVAR) processes. It is obtained by aggregating a
finite number of well chosen predictors, each of them enjoying an optimal
minimax convergence rate under specific smoothness conditions on the TVAR
coefficients. We show that the obtained aggregated predictor achieves a
minimax rate while adapting to the unknown smoothness. To prove
this result, a lower bound is established for the minimax rate of the
prediction risk for the TVAR process. Numerical experiments complete this
study. An important feature of this approach is that the aggregated predictor
can be computed recursively and is thus applicable in an online prediction
context.
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