where σ2 =2nR K2(u)du — R £R K(u)K(u + v)du] 2 dvθ
Proof: See the Appendix.
To better understand the implication of A6 for the determination of the combination weights pj (KI), it
is helpful to rewrite the previous result as follows:
nh1r2(KI + 1 [2) -A N(m, σ2) (25)
where m = KI(g, fθ* ) = αC, from (21) and (23). This implies that to estimate the mean of the distribution
it is necessary to pin down the α, whose estimation is based on the ‘plausibility’ of the candidate models.
Assumption A6 elicits the following definition of plausible model:
Def : Mj = fj (θ, x) is plausible, thus will be included in the set M, if the expected value of its approxi-
mation error is equal to αC
In other words, according to A6, all the competing models are on average expected to have the same
distance from the true model g . Subsequently, as suggested by the definition of m, α could be estimated by
a suitably normalized average of all models’ misspecification:
1
b = -τ Σ,KIj /C = KI (g,fθt )/C
(26)
J
j
Therefore, to obtain pj (KI) we have to employ the c.d.f. of a Normal with mean E(KIj) and variance
σ2 . This entails that, if a model performs better than the average performance of all plausible models, that
is 0 <kij < mb n , then it receives an high weight in the models combination. On the other hand, if the model
performs poorly relative to all other models, that is kij > mb n, then its probability of being correct ( pj (KI))
will be low.
5 Finite sample performance of the NPQMLE estimator
In order to analyze the behavior of the parameter estimator in finite sample, we provide the results of a set
of Montecarlo experiments, where we use the Kullback-Leibler distance between the true and the estimated
model (KI(g, fj (θ))) to judge the goodness of the estimation methodology, and to compare it to QMLE.
We use 1000 iterations for each experiment. At each iteration, I first generate the data according to some
distribution g that represents the true model; second, estimate the nonparametric density fbn using a second
order Gaussian kernel; third, determine the optimal value of the parameters minimizing the KI between fbn
and each candidate model fj (θ, x); fourth, evaluate KI between g and fj (θ, x) at θNPQMLE and θQM LE
respectively. Finally, to evaluate the performance of NPQMLE and to compare it to QMLE we compute
the average of KI(g, fj (θNPQMLE)) and of KI(g, fj (θQM LE)) out of 1000 stored values. The smoothing
parameter h is chosen according to h = 1.06σbxn-β where 0 <β<1. Further, in accordance with Theorem
3, β must satisfy ∣ < β < 1.
The basic design is as follows:
Xi - G(9, 3)
that is, the true model is a family of univariate Gamma distributions G(ς, λ) with parameters ς =9and
λ =3. We choose a set of three candidate models in which we also include the true ones. In particular, the
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