The Student Puzzle Corner contains one or two problems in statistics or probability. Sometimes, solving the problems may require a literature search. Current student members of the IMS are invited to submit solutions electronically (to bulletin@imstat.org with subject “Student Puzzle Corner”). The deadline is March 5, 2015. The names and affiliations of (up to) the first 10 student members to submit correct solutions, and the answer to the problem, will be published in the next issue of the Bulletin.
The Editor’s decision is final.

Student Puzzle Corner 8

In the previous issue, we considered a problem on random matrices. (The solution is here). This time, we will look at a somewhat unconventional problem on statistical inference.

We want to experimentally measure a physical constant, say $\mu$, that equals some positive integer $k$. We can repeat the experiment, but our measurements always have some experimental error. Suppose we model the sequence of experimental measurements $X_1,\cdots X_n$ as $X_i=\mu +{ϵ}_i,i=1,\cdots ,n$, where ${ϵ}_i\stackrel{\mathit{i}\mathit{i}\mathit{d}}{\sim }N(0,{\sigma }^2),\sigma >0$. The primary parameter $\mu$ as well as the nuisance parameter $\sigma$ are unknown; we need to estimate them. The parameter space is $\mathrm{\Theta }={\mathcal{Z}}_{+}\otimes (0,\mathrm{\infty })$.

Here is this issue’s exact problem:

(a) Prove that for all $n$ and all sample data $(x_1,\cdots ,x_n)$, there exist unique MLEs of $\mu ,{\sigma }^2$, and write them explicitly (that is, in completely closed form).

(b) The possible values of the primary parameter $\mu$ are separated; thus, intuitively, it is very hard for the MLE to get confused between the true value of $\mu$ and its illegitimate competitors. Heuristically, one would expect that the MLE $\hat{\mu }$ converges to the true $\mu$ very rapidly. To make it precise, calculate exactly $P_{\mu }(\hat{\mu }\ne \mu )$ and show that it converges to zero exponentially fast.

(c) Give a proof that $\widehat{{\sigma }^2}$, the MLE of ${\sigma }^2$ is also consistent in this scenario.

And now, a little background and discussion. Once you work the problem out, you will appreciate and enjoy the intuitive appeal of maximum likelihood. What comes naturally will stare you at the eye. If you are curious, change the model from Gaussian to something else, and see what happens then. Also, this is a simply stated problem in which estimates of different components of the parameter vector converge at different rates; this also happens in problems which are partly regular and partly nonregular. The part that’s regular will converge at the conventional $\sqrt{n}$ rate, while the part that’s irregular, perhaps asymptotically exponential, will converge at a faster rate.

To some decimals, certain physical constants are indeed positive integers, perhaps on scaling. One instance is the reciprocal of the fine structure constant in cosmology; more common examples are the proton-neutron mass ratio and Avogadro’s number. An interesting fact about Avogadro’s number is that Einstein presented a novel statistical estimate of it in his PhD thesis, and it is the first well known use of moment estimates. Einstein was not the first person to estimate Avogadro’s number; actually, in his thesis, he first made a calculation error and got an estimate inconsistent with estimates made by Jean Perrin. Einstein asked Ludwig Hopf to find the error, which he did find, and corrected Einstein’s estimate of the Avogadro number.