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Erschienen in:
Prediction: Some Heuristic Notions Kapitel lesen Erstes Kapitel lesen

2015 | OriginalPaper | Buchkapitel

3. Model-Based Prediction in Regression

verfasst von : Dimitris N. Politis

Erschienen in: Model-Free Prediction and Regression

Verlag: Springer International Publishing

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Abstract

In applying the Model-free Prediction Principle, the problem of nonparametric regression is addressed in detail; the model-free predictors are worked out, and shown to be applicable under minimal assumptions. Interestingly, model-free prediction in regression is a totally automatic technique that does not necessitate the search for an optimal data transformation before model fitting. The resulting model-free predictive distributions and intervals are compared to their corresponding model-based analogs, and the use of cross-validation is extensively discussed. As an aside, improved prediction intervals in linear regression are also obtained.

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Vorheriges Kapitel The Model-Free Prediction Principle
Nächstes Kapitel Model-Free Prediction in Regression
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Fußnoten
1
In general, the L 2–optimal predictor of Y f would be given by the conditional expectation of Y f given \(Y _{1},\ldots,Y _{n}\) as well as x f. However, under model (3.1), the Y data are independent, and \(E(Y _{\mathrm{f}}\vert Y _{1},\ldots,Y _{n},x_{\mathrm{f}})\) simplifies to just E(Y f | x f); the same will be true under the Model-free regression setting of Chap. 4 The study of dependent data will be undertaken in Part III of the book.
 
2
If σ 2(x) is not assumed constant, then \(\tilde{e}_{t} = e_{t}C_{t}/(1 -\delta _{x_{t}})\) where \(C_{t} = s_{x_{t}}/s_{x_{t}}^{(t)}\).
 
3
mx ∗ and s x ∗ can use the same bandwidth as the original estimators m x and s x provided these are slightly undersmoothed; otherwise, a two bandwidth trick is recommended—see Remark  3.5.2.
 
4
Efron (1983) proposed an iterated bootstrap method in order to correct the downward bias of the bootstrap estimate of variance of prediction error; notably, his method involved the use of predictive residuals albeit at the 2nd bootstrap tier—see also Efron and Tibshirani (1993, Chap. 17.7).
 
5
If c = 0, the bound (3.34) is trivial: | W t | < ∞.
 
6
In the rare case of non-unique minima in PRESAR cross-validation, the dilemma may be resolved by picking the result closest to one given by PRESS.
 
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Metadaten
Titel
Model-Based Prediction in Regression
verfasst von
Dimitris N. Politis
Copyright-Jahr
2015
Buch
Model-Free Prediction and Regression

Print ISBN: 978-3-319-21346-0

Electronic ISBN: 978-3-319-21347-7

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-3-319-21347-7_3