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. ( [1] [2] ), Belloni et al. ( [3] [4] ), Tibshirani et al. [5] , and Chernozhukov et al. [6] ). The model selection step is often a complex decision process and can involve collecting expert opinions, preprocessing, applying a variable selection rule, data-driven choice of one or more tuning parameters, among others. Except in simple cases, it is hard to explicitly characterize the form of the post-selection of interest while incorporating the variability of model selection. References for model selection include e.g. Zucchini [7] and Zucchini et al. [8] . An alternative to selecting a single model for estimation purposes is to use a weighted average of the estimates resulting from each of the models under consideration. This leads to the class of model averaging estimators. Model averaging can be done either in Bayesian and frequentist approaches. The most common Bayesian approach is Bayesian model averaging (BMA) and its variants, using Bayesian information criterion (BIC) as approximation (Schwarz [9] ). The seminal paper of Hoeting et al. [10] fully describes the basic of BMA. BMA and its applications can be found in Nguefack- Tsague ( [11] [12] ), Nguefack-Tsague and Ingo [13] , Nguefack-Tsague and Zucchini ( [14] [15] ). Several options are available for specifying the weights in frequentist approaches; references on least squares regression types and like include Hansen ( [16] - [21] ), Hansen and Racine [22] , Cheng and Hansen [23] , Charkhi et al. [24] , and Wan et al. [25] . The aforementioned weighting schemes perform model averaging on a set of nested candidate models with the weights vector chosen such that a specific criterion is minimized.

References using Akaike’s information criterion, AIC (Akaike [26] ) include Burnham and Anderson [27] , Nguefack-Zucchini [28] , Nguefack-Tsague ( [29] - [32] ). The R package [33] MuMIn is used to perform model averaging based on Burnham and Anderson [27] . Schomaker and Heumann [34] , and Schomaker [35] developes model averaging schemes based on multiple imputation and shrinkage; the R package MAMI is used for practical implementations. This paper is organized as follows: In Section 2, we develop model averaging based on information criterion while, in Section 3, we propose a new approach for computing the weights for the competing models, one that takes both account the selection probability and the likelihood of each model. Section 4 illustrates with applications to Bernoulli trials. The paper ends with concluding remarks.

Let be a set of K plausible models to estimate, the quantity of interest. Denote by the estimator of obtained when using model. Model averaging involves finding non-negative weights, , that sum to one, and then estimating by

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, , of the selected model is a random variable. We therefore denote the selected model by, and the corresponding estimator of the quantity of interest, , by. In terms of the notation introduced above, we may write

Clearly, the selected model depends on the set of candidate models, , and on the selection procedure, which we denote by S. However, it is important to realize that, even if the same and S, are used, different samples can lead to different models being selected; is a “randomly selected model”. In this section we focus attention on post-model selection estimators (PMSEs), which is the special case of model averaging estimators with zero/one weights only.

Some classical model averaging weights base the weights on penalized likelihood values. Let denote an “information criterion”of the form

where is a penalty term, and is the maximized likelihood value for the model. The Akaike infor- mation criterion (AIC, Akaike [26] ) is the special case with, where is the number of parameters of model. Buckland et al. [36] proposed using weights of the form:

“Akaike weights” (denoted by) refer to the case with. Numerous applications of Akaike weights are given in Burnham and Anderson [27] .

Since the selection procedure (S) and likelihood are important for model selection, we therefore suggest estimating by a weighted average of the in which the weights take account of S, specifically where they depend on estimators.

The likelihoods are taken into account because they quantify the relative plausibility of the data under each competing model; the estimated selection probability adjusts the weights for the selection pro- cedure. Both of these components are required. If one were to use only the likelihoods to determine the weights then complex models (i.e. models having many parameters) would automatically be assigned larger weights. The weights are similar to the weights defined in (3) but they differ in the way the likelihood is adjusted. With the proposed method a “bad” model will be penalized by any reasonable selection procedure through the probability, even if it is complex in terms of the number of parameters. We let the selection procedure determine in how far a model is penalized.

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 be n independent Bernouilli trials, that is, is the number of successes;

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

and involves the sufficient statistic Y., is the proba-

bility mass function (PMF) of Y; the quantity of interest is. Sensitivity analyses showed that the finding obtained here are insensitive to parameter choice, irrespective of the sample size n.

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