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Let ($x_1, ..., x_n$) be i.i.d. samples drawn from some distribution $P$ with an unknown probability density function $f$. Its kernel density estimator is \begin{align} \hat{f}_h(x) = \frac{1}{n}\sum_{i=1}^n K_h (x - x_i) \quad = \frac{1}{nh} \sum_{i=1}^n K\Big(\frac{x-x_i}{h}\Big), \end{align} where $K$ a symmetric non-negative function that integrates to one.

All consistency proofs (1,2) require that $h(n) \to 0$ for $n \to \infty$ and I think remember that this is a necessary condition for strong and even weak consistency. Are there any analysis about what happens when the bandwidth $h$ is a constant? Maybe error some bounds on the convergence to $f$?

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  • $\begingroup$ Isn't this a duplicate of the question you asked earlier: math.stackexchange.com/questions/914789/… $\endgroup$ – user76844 Sep 2 '14 at 13:21
  • $\begingroup$ They are similar. The question earlier was about consistency in general and the got correctly answered. Here I am explicitly interested in a constant bandwidth. $\endgroup$ – Manuel Schmidt Sep 2 '14 at 19:32
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Its not a question of error bounds (a so-called second-order condition), but of first order convergence. If an estimator is not consistent, then it does not converge to the true value (i.e., bias), its not that it has a high degree of inter sample variability (i.e., sampling variance). See this paper, especially remark 6 on page 4.

As a simple example, what is the bound on the error of the following estimator:

Let $X_i \sim \mathcal{N}(\mu,1),\;\;\; \hat \mu = \frac{1}{n}\sum_{i=1}^N X_i + 50$

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