12.4 Kernel Estimator

To develop a test about $ H_0$ we first introduce a nonparametric kernel estimator for $ m$. For an introduction into kernel estimation see Härdle (1990), Wand and Jones (1995) and (Härdle et al.; 2000) . Without loss of generality we assume that we are only interested in $ m(x)$ for $ x \in [0,1]$ and that $ f(x) \ge C_1$ $ \forall x \in [0,1]$ with a positive constant $ C_1$. If in a particular problem the data are supported by another closed interval, this problem can be transformed by rescaling into an equivalent problem with data support $ [0,1]$.

Let $ K$ be a bounded probability density function with a compact support on $ [-1,1]$ that satisfies the moment conditions:

$\displaystyle \int u K(u) du = 0,\ \int u^2 K(u) du =\sigma^2_K $

where $ \sigma_K^2$ is a positive constant. Let $ h$ be a positive smoothing bandwidth which will be used to smooth $ (X,Y)$.

The nonparametric estimator considered is the Nadaraya-Watson (NW) estimator

$\displaystyle \hat{m}(x) = { \sum_{i =1}^{n} Y_{i} K_{h}( x-X_i) \over \sum_{i=1}^{n} K_{h}( x-X_i) }$ (12.7)

with $ K_h(u) = h^{-1} K( h^{-1} u)$. This estimator is calculated in XploRe by the quantlets 23995 regest or 23998 regxest .

The parameter estimation of $ \theta $ depends on the null hypothesis. We assume here, that the parameter $ \theta $ is estimated by a $ \sqrt{n}$-consistent estimator. Let

$\displaystyle \tilde{m}_{\hat{\theta}}(x) = {\sum K_h(x-X_i) m_{\hat{\theta}}(X_i) \over\sum_{i=1}^n K_h(x-X_i)}$

be the smoothed parametric model. The test statistic we are going to consider is based on the difference between $ \tilde{m}_{\hat{\theta}}$ and $ \hat{m}$, rather than directly between $ \hat{m}$ and $ {m}_{\hat{\theta}}$, in order to avoid the issue of bias associated with the nonparametric fit.

The local linear estimator can be used to replace the NW estimator in estimating $ m$. However, as we compare $ \hat{m}$ with $ \tilde{m}_{\hat{\theta}}$ in formulating the Goodness-of-Fit test, the possible bias associated with the NW estimator is not an issue here. In addition, the NW estimator has a simpler analytic form.