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Let $x$ be an observation from $X\sim Bin(n,p)$. I want to estimate $p$ and use ML estimator, $\widehat{p}=\frac{x}{n}$. I also want to estimate the variance of the estimator $\widehat{p}$. It equals: $Var(\widehat{p})=Var(\frac{X}{n})=\frac{Var(X)}{n^2}=\frac{p(1-p)}{n}$. As we don't know $p$, it is often replaced by $\widehat{p}$ in the formula. We obtain: $$Var(\widehat{p})=\frac{\widehat{p}(1-\widehat{p})}{n}.$$ In my case, as $x=0$, $\widehat{p}=0$ and thus $Var(\widehat{p})=0$. We thus see that by approximating $p$ with $\widehat{p}$, we obtain a null variance. I will explain now why it's wrongly estimated. The reason why I get $x=0$ is because $p$ is low (but not null) and because the sample size $n$ is low. But when estimating $Var(\widehat{p})$, we approximate $p$ with $\widehat{p}=0$, which gives us a null variance.

How can we correct for that?

kjetil b halvorsen
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Anthony
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1 Answers1

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Why do you need an estimator of the variance $np(1-p)$? Recommendations would probably depend on context. An easier problem is to construct a confidence interval, see Confidence interval around binomial estimate of 0 or 1. A simple ad-hoc solution to get a variance estimator when $X=0$ is to take the upper limit of that confidence interval, and base the variance estimate on that.

To find an unbiased variance estimate which is non-zero when $X=0$ is probably hopeless (unless it is sometimes negative!) Proving that would be an interesting exercise.

If these adhockeries are not good enough, then you could go for the Bayesian ideas mentioned in comments.

kjetil b halvorsen
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