Imagine that I have an estimator which has positive support (like standard deviation). Is it a normal thing that I have an interval estimator which contains negative values?
any help will be appreciated.
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2$\begingroup$ Related: Can a mathematically sound prediction interval have a negative lower bound? $\endgroup$– Stephan KolassaCommented Jan 24, 2018 at 12:03
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1 Answer
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This happens in many corners of statistics. One case: normal approximation is applied to a bounded random variable, in which case the confidence interval may well go outside the range. Example: true proportion $p\in[0,1]$ is estimated by the sample proportion $\hat{p}\in[0,1]$. The 95% confidence interval based on normality is given by
$ [\hat{p} - 1.96 \sqrt{\hat{p}(1 - \hat{p})/n},\ \ \hat{p} + 1.96 \sqrt{\hat{p}(1 - \hat{p})/n}]. $
For $n = 10$ and $\hat{p} = 0.01$ the lower bound equals $-0.05166996 < 0$. Likewise, for $n = 10$ and $\hat{p} = 0.99$ the upper bound equals $1.05166996 > 1$.
