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I am struggling with the following exercise from Stapleton's book (it's exercise 7.4.1).

A box contains eight eggs, of which an unknown number $R$ are rotten. You take a simple random sample of three eggs. Let $X$ be the number of rotten eggs in the sample.

  • Find the MLE of $R$ (Warning: don't use calculus.)
  • What is the MLE for $E(X) = 3(R/8)$?

For the first question, we can compute the likelihood of having $X$ rotten samples given $R$ as: $$ p(X|R) = {R \choose X}{8 - R \choose 3 - X}/{8\choose 3}. $$ This leads to the following table ($X$ changes along a column, $R$ along a row): $$ \begin{array}{lccccccccc} X|R & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline 0 & 1 & 0.625 & 0.357 & 0.179 & 0.0714 & 0.0179 & 0 & 0 & 0\\ 1 & 0 & 0.375 & 0.536 & 0.536 & 0.429 & 0.268 & 0.107 & 0 & 0\\ 2 & 0 & 0 & 0.107 & 0.268 & 0.429 & 0.536 & 0.536 & 0.375 & 0\\ 3 & 0 & 0 & 0 & 0.0179 & 0.0714 & 0.179 & 0.357 & 0.625 & 1 \end{array} $$ In this case the maximum likelihood estimator would be $R=0$ for $X=0$, $R=8$ for $X=3$, but I do not understand how to choose one between $R=2$ and $R=3$ for $X=1$, given that both answers have the same likelihood.

For the second question, I am not sure. I suspect I should compute the likelihood of $E(X)$ like this:

$$ L(E(X)=\mu|R) = \sum_{i=0}^3 i\cdot p(i|R), $$ where $p(i|R)$ can be read on the table above. But maybe I am misunderstanding the question.

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    $\begingroup$ If likelihood is maximized for two values of $R$, then both are MLE. For the second question, use invariance/equivariance of MLE. $\endgroup$
    – StubbornAtom
    Feb 6, 2021 at 14:57
  • $\begingroup$ Please add self-study as a tag. $\endgroup$
    – Xi'an
    Feb 6, 2021 at 16:54
  • $\begingroup$ @StubbornAtom good point, so I should just say that the MLE estimate of $E(X)$ is just $3/8$ times whatever is the MLE estimate of $R$? This follows from the theorem you mention and a quick check that $E(X)$ is a one-to-one function of $R$. $\endgroup$
    – J. D.
    Feb 6, 2021 at 18:41
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    $\begingroup$ Yes, but invariance property holds for any function of MLE. $\endgroup$
    – StubbornAtom
    Feb 6, 2021 at 19:00

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