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Given a random variable X~B(n, p) I would like to estimate p using the mode. In this particular case, I cannot use $\hat{p}=\frac{X}{n}$ as an estimator. I have found that in general, that for a Binomial distribution: $$mode=\lfloor{(n+1)p}\rfloor$$I wish to take $$\hat{p}=\frac{\widehat{mode}}{n}$$ and now I want to check if this is a biased estimator, which comes to my problem...how do I go about this? $$\mathbb{E}[\hat{p}]=\frac{\mathbb{E}[\lfloor{(n+1)p}\rfloor]}{n}$$ How can I work with the floor function? And is this even possible?

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The OP confuses the mode of the distribution $mode=\lfloor{(n+1)p}\rfloor$ with the mode of the sample, which is the most frequent value in the sample. When equating the mode of the sample, e.g., $5$, with the theoretical mode, $\lfloor{(n+1)p}\rfloor$, there is not a one-to-one correspondence. Which shows that the (rational) estimate cannot be unbiased for all values of $p$.

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