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Let $X$ be a random variable with real values and with density $f$. Assume the support $f$ is bounded with supremum $m$ and has a positive value at that supremum: $$\forall x > m, f(x) = 0 \text{ and } f(m)> 0.$$ Given an I.I.D. sample $X_1, \cdots, X_n$ of this random variable, I'd like to estimate both $m$ and $f(m)$. What would be an good way to proceed ?

I think estimating $m$ by the sample maximum $\max(X_1, \cdots, X_n)$ yields a consistent estimator of $m$ with rate $\frac{1}{n}$, which satisfies me.

But I'm not sure on how to estimate $f(m)$. I tried to count the number of sample points falling in the interval $$\left[ \max(X_1, \cdots, X_n) - \frac{S}{\sqrt{n}}, \max(X_1, \cdots, X_n) \right]$$ where $S$ is a consistent estimator of the standard deviation of $X$. Hoping this would give a $\frac{1}{\sqrt{n}}$ consistent estimator of $f(m)$, but this seems to converge very slowly, and I don't think this is the best option.

Any input on that question would be greatly appreciated.

Pohoua
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  • How do you intend to distinguish between a case where the density falls rapidly to $0$ and one where it falls rapidly to a positive value, except with an enormous sample size? – Henry Oct 14 '21 at 16:50
  • Sorry, I'm not sure I understood your comment. You mean what if the slope of the density is very negative at $m$ ? – Pohoua Oct 14 '21 at 16:54
  • Does https://stats.stackexchange.com/questions/65866 answer your question? – whuber Oct 14 '21 at 19:07
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    @whuber : Thanks for the link to this nice question and dicussion. I guess I could use the proposed technique by replacing 0 by the maximum of the data. However, the cutting/ reweighting, while giving a good estimate of the density, has a bias at the edge, which is the value I'm interested in. For the logspline density estimation, I couldn't find any guarantee that the estimation is pointwise consistent (which would be what I need). So I'm affraid this doesn't exactly answer my question. – Pohoua Oct 19 '21 at 13:06

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