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If you know that this next observation comes from a specific group $i$, then the best prediction would be $$\exp(\hat \beta_0 + \hat b_i).$$ However, if you do not know from which group it comes from, then it would make more sense to use as prediction the average over the groups, which in your specific random intercepts model would be \exp \Bigl ( \hat \...

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Different point estimates are bound to happen given different data. The entire concept of the sampling distribution for the statistic is one way to resolve this. Given your first example, the confidence interval for the mean would be between approximately 4 and 16. This means that true incidence of lung cancer between approximately 4 and 16 is consistent ...

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It's very plainly biased because: (a) it's quite easy for the number of distinct values observed in the sample to be lower than the corresponding number in the population (it has positive probability), but (b) it's impossible for the number of distinct values in the sample to exceed the corresponding number in the population Consequently the expected ...

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To address the "when is this appropriate" part of your question: If you fit a mixed model, it may still be appropriate to do inference for the grand mean, provided you interpret it correctly. Specifically, prediction intervals around the grand mean that include the relevant variance components are useful tools for inference. But, given that each observation ...

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Recently we have published a paper suggesting a fast consistent mode estimator. P.S. Ruzankin and A.V. Logachov (2019). A fast mode estimator in multidimensional space. Statistics & Probability Letters However, our method is mostly aimed at large sample sizes as well. For smal sample sizes I would recommend fraction of range mode (FRM) approach. ...

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Your problem can be generalized as follows: The parameter $p$ is generated according to an unknown probability density. You can measure $p$ only with a measurement error $\sigma$, which yields an estimator $\hat{p}$ for $p$ that has a known distribution (binomial distribution). A simple approach is suggested by Glen_b in this answer to a similar question: ...

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