Now my grading is over, I can reflect on the unexpected difficulties in the mathematical statistics exam. I knew that the first question in the multiple choice exercise, borrowed from Cross Validation, was going to  be quasi-impossible and indeed only one student out of 118 managed to find the right solution. More surprisingly, most students did not manage to solve the (absence of) MLE when observing that n unobserved exponential Exp(λ) were larger than a fixed bound δ. I was also amazed that they did poorly on a N(0,σ²) setup, failing to see that

$\mathbb{E}[\mathbb{I}(X_1\le -1)] = \Phi(-1/\sigma)$

and determine an unbiased estimator that can be improved by Rao-Blackwellisation. No student reached the conditioning part. And a rather frequent mistake more understandable due to the limited exposure they had to Bayesian statistics: many confused parameter λ with observation x in the prior, writing

$\pi(\lambda|x) \propto \lambda \exp\{-\lambda x\} \times x^{a-1} \exp\{-bx\}$

$\pi(\lambda|x) \propto \lambda \exp\{-\lambda x\} \times \lambda^{a-1} \exp\{-b\lambda\}$

hence could not derive a proper posterior.

### 3 Responses to “post-grading weekend”

1. With regard to the cross validated post, at the end ” the log-likelihood goes to infinity” Cox and Hinkley argued that “if we take account of the inevitably discrete nature of observations” it does not go to infinity.

The reference is given on page 25 here – http://andrewgelman.com/wp-content/uploads/2011/05/plot13.pdf and a example by Radford Neal disucussed on page 35 and 36.

• Thanks Keith but do not tell my students!!!

2. Ugh. This reminds me that I need to get better at statistics before I start teaching it. I really want to believe that “Je ne parle pas français” is a good enough reason to not be able to eyeball the solutions (or paths to solutions) to these, but I fear that French isn’t really that hard… I’ve just never done undergrad stats.

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