Suppose I'm interested in estimating the unknown smooth density of $X$ denoted by $f(\cdot)$ using data $\{X_i\}_{i=1}^{n}$. Suppose I also know that $f(\cdot)$ is symmetric about 0 in the sense that $f(-x)=f(x)$ for any $x$ in the support. My questions are
1.How to impose or incorporate this symmetry restriction in the usual kernel density estimator defined as
$\widehat{f}(x)=\frac{1}{nh}\sum_{i=1}^{n}k(\frac{X_i-x}{h})$, where $k(\cdot)$ is the kernel function.
2.How does the symmetry-restricted kernel density estimator improve upon the naive kernel estimator defined above?
Intuitively, the symmetry-restricted kernel density estimator should be better because it used more information, But I don't know how to show or quantify such improvement. For example, does it converge faster?