suppose I'm interested in estimating $C=\int_{a}^{b}g(x)dx$, where $a$ and $b$ are known, and $g(x)=E(Y|X=x)$ is an unknown function of $x$. The data I have is $\{Y_{i},X_{i}\}_{i=1}^{n}$, then a natural estimator for $C$ could be obtained as follows:

First, obtain a nonparametric estimate for $g(x)$: for example, the local constant estimator $\widehat{g}(x)=\frac{\sum_{i=1}^{n}Y_{i}k(\frac{X_{i}-x}{h})}{\sum_{i=1}^{n}k(\frac{X_{i}-x}{h})}$ (we can also use local polynomial estimate).

Second, we do numerical integration over $[a,b]$ using $\widehat{g}(x)$ and get the estimate: $\widehat{C}=\frac{1}{M}\sum_{m=0}^{M}\widehat{g}(x_{m})$ with $\{x_{0},...,x_{M}\}$ be an equally distanced grid from $a$ to $b$ satisfying $x_{0}=a$ and $x_{M}=b$.

Are there any existing estimators like this or in the same spirit? Examples, papers or book chapters are welcome. Thanks!


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