We considered model (1) with the regression function being
and being distributed as . The covariate is generated according to with and the correlation coefficient between X and Z being 0.6, and , . Results for , and are reported. Simulations were run with different validation and primary data sizes ranging from to according to the ratio and , respectively. We generate 500 datasets for each sample size of .
To calculate , we used the normalized Legendre polynomials as basis and the standard normal kernel (denote ). For , we used an product kernel , and the bandwidth was selected by generalized cross-validation approach (GCV). For our estimator , we used the cross-validation approach to choosing the four parameters , , and . For this purpose, , and are selected separately as follows.
Here, we adopt the cross-validation (CV) approach to estimate by
where the subscript denotes the estimator being constructed without using the jth observation. Similarly, we get . After obtaining and , we then select by
where the subscript denotes the estimator being constructed without using the ith observation .
We compute the RASE at grid points of . Table 1 presents
Table 1. The RASE ( ) comparison for the estimators and .
the RASE for estimating curves when , and for various sample sizes. It is obvious that our proposed estimator has much smaller RASE than . As is expected, our proposed estimating method produces more accurate estimators than the Nadaraya-Watson estimators. Moreover, there is a drastic improvement in accuracy by using our estimator over the Nadaraya-Watson estimator; this improvement increases with .
This work was supported by GJJ160927 and Natural Science Foundation of Jiangxi Province of China under grant number 20142BAB211018.
Proof of Theorem 3.1:
Lemma 6.1. Suppose Assumptions (A1), (A2) (i) and (A4) hold. For each , we have
Proof of Lemma 6.1.. For each , by Assumptions (A2) (i) and (A4), we have
Note that are orthonormal and complete basis functions on . Under Assumptions (A2) (i), for each , we have . Then, using Cauchy-Schwarz inequality, is bounded in absolute value for each . Hence, we obtain that
Moreover, for each , we have
where we have used the fact that is uniformly bounded on .
We conclude that
By the triangle inequality and Jensen inequality, we have
Under Assumption (A2) (i), we can show that (see Lemma A1 of  ).
By construction of the estimator, we have
where the last equality is due to (10). The desired result follows immediately.,
Proof of Theorem 3.1. Define and . Notice that where . Then we have
It follows from Lemma 6.1 that or are . Under Assumptions (A1), we have and . Moreover, and . The main task remained is to establish the order of the term . By the triangle inequality and Jensen inequality, we have
Similar to the proof of Lemma 6.1, under Assumptions (A2)(ii), (A3) and (A4), it is easy to show that
Then, according to the Lemma 6.1, we have
Let , , and , if or , combining all these results, we complete the proof.,
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