Question: Let (hat{f}_{n}(t)) denote a kernel density estimator with kernel function (k) computed on a sample (X_{1}, ldots, X_{n}). Prove that, [Eleft[int_{-infty}^{infty} hat{f}_{h}(t) f(t) d tight]=h^{-1}

Let \(\hat{f}_{n}(t)\) denote a kernel density estimator with kernel function \(k\) computed on a sample \(X_{1}, \ldots, X_{n}\). Prove that,

\[E\left[\int_{-\infty}^{\infty} \hat{f}_{h}(t) f(t) d tight]=h^{-1} E\left[\int_{-\infty}^{\infty} k\left(\frac{t-X}{h}ight) f(t) d tight]\]

where the expectation on the right hand side of the equation is taken with respect to \(X\).

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