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Frequentist Model Averaging

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Model Averaging

Part of the book series: SpringerBriefs in Statistics ((BRIEFSSTATIST))

Abstract

We provide an overview of frequentist model averaging. For point estimation, we consider different methods for selecting the model weights, including those based on AIC, bagging, weighted AIC, stacking and focussed methods. For interval estimation, we consider Wald, MATA and percentile-bootstrap intervals. Use of the methods are illustrated by examples involving real data.

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Notes

  1. 1.

    This can come as a surprise; see [159] for a useful discussion of the assumptions underlying AIC.

  2. 2.

    As discussed in Sect. 2.2, when counting the number of parameters in a model we include any scale parameters, such as the error variance in a normal linear model.

  3. 3.

    See [24] for a discussion of the connection between the model-selection probabilities \(p \left( S = m \right) \) \((m=1,\dots ,M)\) and AIC weights.

  4. 4.

    Throughout the rest of the chapter, it will be implicit that constrained-optimisation is used whenever we determine the weights by minimising an objective function.

  5. 5.

    For normal linear models, AIC(w) is equivalent to Mallows model averaging (MMA) [79, 121, 143, 190, 220, 223]. Although MMA was developed without the assumption of normal errors, for simplicity we use the more general name AIC(w) when referring to MMA.

  6. 6.

    As with AIC(w), the choice of estimate of any scale parameter will not affect the weights.

  7. 7.

    This name is potentially confusing as the original jackknife is somewhat different, involving the use of pseudo-values to reduce the bias of an estimate obtained from a single model [48, 154, 175].

  8. 8.

    Other modifications to AIC(w) in this setting have been proposed [129, 220, 226].

  9. 9.

    This assumption has also been used in interval estimation (Sect. 3.4.1).

  10. 10.

    An alternative derivation avoids the notion of selecting a random sample from a population of models [24]. However this involves regarding \(\theta \) as a weighted mean of least-false values of \(\theta \).

  11. 11.

    It has been wrongly claimed that \(\widehat{\theta }\) is often assumed to be unbiased [48]. The only theory that involves this assumption (asymptotically) is the local misspecification framework (Sect. 3.2.3).

  12. 12.

    Even if \(\widehat{b}_m\) is unbiased, \(\widehat{b}_m^{\,2}\) will be biased as an estimate of \(b_m^{2}\), but analytical bias-adjustment would involve estimation of the correlation between \(\widehat{\theta }_{m_1}\) and \(\widehat{\theta }_{m_2}\) \((m_1 \ne m_2)\), and any decrease in bias might be offset by an increase in variance.

  13. 13.

    There is also a logical problem associated with use of (3.18) [24].

  14. 14.

    Both [21] and [24] wanted to avoid assuming that the true model is in the model set.

  15. 15.

    This estimate is not simply the standard deviation of the \(\widehat{\theta }_{\left( b \right) }\) in (3.8) [21].

  16. 16.

    It has been wrongly claimed that use of this interval involves assuming that the largest model is not in the model set [48].

  17. 17.

    Unfortunately, the work of [106] has led to the impression that the MATA interval will not perform well in general [48].

  18. 18.

    A similar issue arises when using a technique such as RJMCMC in the Bayesian setting (Sect. 2.2.1), where a large number of iterations may be required in order to visit each model often enough to obtain reliable estimates of both the posterior model probabilities and the posteriors for those parameters in models with low posterior model probabilities.

  19. 19.

    This example also provides evidence that the Wald interval can perform well, despite the issues raised in Sect. 3.4.1.

  20. 20.

    Unless we use DIC weights, which can depend on the paramtetrisation (Sect. 2.5).

  21. 21.

    Conversely, we could adjust the nominal confidence level for each interval until they all have the same width, and then choose the one with the highest true coverage rate [150].

  22. 22.

    This procedure is similar to use of all possible singleton models in the context of focussed model averaging (Sect. 3.6.2) [32]. In order for \(\widehat{\theta }\) to be consistent, however, [32] require the weights to sum to one, as each \(\widehat{\theta }_m\) is consistent [88].

  23. 23.

    This constraint can also be useful for generalisation of the conclusions [15].

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Fletcher, D. (2018). Frequentist Model Averaging. In: Model Averaging. SpringerBriefs in Statistics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-58541-2_3

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