Abstract
This work is concerned with a semiparametric associated kernel estimator for count explanatory variables. The proposed semiparametric estimator is a multiplicative combination between a parametric model and a discrete nonparametric kernel estimator of Nadaraya–Watson type. In this semiparametric approach, the parametric model plays the role of the start function and the nonparametric kernel estimator is a correction factor of the parametric estimate. Some asymptotic properties of the discrete semiparametric kernel regression estimator are pointed out; in particular, we show its asymptotic normality and the order of the optimal bandwidth. The parametric part is illustrated by some nonlinear and generalised linear models; for the nonparametric estimator, we apply the discrete general triangular associated kernel providing bias reduction. The usefulness of the discrete semiparametric kernel regression estimator is shown on three practical examples in comparison with logistic, generalised linear and additive models.
Highlights
IntroductionThe discrete semiparametric associated kernel regression estimator results from a parametric estimation l(x) ≡ l(x; ˆ ) of l multiplied by a nonparametric Nadaraya–Watson estimation ωn of ω as follows: n ωn(x) =
Let (x1, y1), (x2, y2), . . . , be the observations of the variables (X, Y ) in S × R connected through the model yi = m(xi) + ei, where m : S → R is an unknown regression function to be estimated and ei is assumed to be the residual from the real random variable ǫi with mean E(ǫi) = 0 and variance Var(ǫi) = σ 2 < ∞
We demonstrate its asymptotic normality; one can refer to Martins-Filho et al(2008) for the asymptotic normality of the semiparametric estimator proposed by Glad (1998)
Summary
The discrete semiparametric associated kernel regression estimator results from a parametric estimation l(x) ≡ l(x; ˆ ) of l multiplied by a nonparametric Nadaraya–Watson estimation ωn of ω as follows: n ωn(x) =. We use some parametric (logistic and generalised linear) models as start functions and a discrete associated kernel that provides bias reduction. We apply the GLM and GAM in comparison with the semiparametric model for fitting the sales data (Figure 2). The nonparametric correction in all the three examples is available only for discrete predictors even if the parametric models treat predictors as continuous variables Through these three applications, we point out that the discrete semiparametric associated kernel approach may produce better explanations of real data with both satisfying amounts of smoothing and goodness of fit.
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