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Mathematical Statistics and Stochastic Processes by Denis Bosq

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Chapter 15

Prediction

15.1. Generalities

Let images be a real, square-integrableprocess with basis space images, images be the sub-σ-algebra generated by Xnj, j ≥ 0, and images be the closed linear subspace of images generated by the same variables and the constant 1.

We wish to predict Xn+h from the observed variables X1,…,Xn. The strictly positive integer h is called the horizon of the prediction.

With respect to the quadratic error, the best predictor given Xnj, j ≥ 0, is the conditional expectation

images

This is the orthogonal projection in images of Xn+h onto images.

The best linear predictor of Xn+h is its orthogonal projection onto . If (Xt) is Gaussian, it coincides with .

A statistical predictor is a known function of the data: ...

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