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In several instances of statistical practice, it is not uncommon to use the same data for both model selection and inference, without taking account of the variability induced by model selection step. This is usually referred to as post-model selection inference. The shortcomings of such practice are widely recognized, finding a general solution is extremely challenging. We propose a model averaging alternative consisting on taking into account model selection probability and the like-lihood in assigning the weights. The approach is applied to Bernoulli trials and outperforms Akaike weights model averaging and post-model selection estimators.

In statistical modeling practice, it is typical to ignore the variability of the model selection step, which can result in inaccurate post-selection inference (Berk et al. ( [

References using Akaike’s information criterion, AIC (Akaike [

Let

In model selection, the model selection criterion determines which model is to be assigned weight one, i.e. which model is selected and subsequently used to estimate the parameter of interest. We note that, since the value of the selection criterion depends on the data, the index,

Clearly, the selected model depends on the set of candidate models,

Some classical model averaging weights base the weights on penalized likelihood values. Let

where

“Akaike weights” (denoted by

Since the selection procedure (S) and likelihood are important for model selection, we therefore suggest estimating

The likelihoods are taken into account because they quantify the relative plausibility of the data under each competing model; the estimated selection probability

If the selection probabilities depend on some parameter for which a closed form expression exists, and if one can find an estimator of the parameter, then it is possible to obtain estimators for these probabilities.

Let

Y-binomial (n, q), q unknown. Inference will be based on Y, since the likelihood function of the X_{i}’s is

bility mass function (PMF) of Y; the quantity of interest is

(a) Consider the choice between the 2 models:

Let

where

The estimated probabilities are given by

The Akaike weights are defined by

The adjusted likelihood weights are defined by

The weighted estimators are

(b) Consider now a choice between the following two models:

AIC is used to select a model,

Model 1 is chosen if

The PMSE

The Akaike weights are defined by

and the adjusted weights is defined by

Consider also a choice between the following models:

PMSE

model

The estimated model selection probabilities

Numerical computations of the properties for these estimators are for

regions of the parameter space, but the adjusted likelihood weights are better than both.

In this paper, we have considered model averaging in frequentist perspective; and proposed an approach of assigning weights to competing models taking account model selection probability and likelihood. The method appears to perform well for Bernoulli trials. The method needs to be applied in variety of situations before it can be adopted.

We Thank the Editor and the referee for their comments on earlier versions of this paper.

Georges Nguefack-Tsague,Walter Zucchini,Siméon Fotso, (2016) Frequentist Model Averaging and Applications to Bernoulli Trials. Open Journal of Statistics,06,545-553. doi: 10.4236/ojs.2016.63046