estimation exam [best of]

Yesterday, I received a few copies of our CRC Press Handbook of Mixture Analysis, while grading my mathematical statistics exam 160 copies. Among the few goodies, I noticed the always popular magical equality

E[1/T]=1/E[T]

that must have been used in so many homeworks and exam handouts by now that it should become a folk theorem. More innovative is the argument that E[1/min{X¹,X²,…}] does not exist for iid U(0,θ) because it is the minimum and thus is the only one among the order statistics with the ability to touch zero. Another universal shortcut was the completeness conclusion that when the integral

was zero for all θ’s then φ had to be equal to zero with no further argument (only one student thought to take the derivative). Plus a growing inability in the cohort to differentiate even simple functions… (At least, most students got the bootstrap right, as exemplified by their R code.) And three stars to the student who thought of completely gluing his anonymisation tag, on every one of his five sheets!, making identification indeed impossible, except by elimination of the 159 other names.

almost uniform but far from straightforward

A question on X validated about a [not exactly trivial] maximum likelihood for a triangular function led me to a fascinating case, as exposed by Olver in 1972 in The American Statistician. When considering an asymmetric triangle distribution on (0,þ), þ being fixed, the MLE for the location of the tip of the triangle is necessarily one of the observations [which was not the case in the original question on X validated ]. And not in an order statistic of rank j that does not stand in the j-th uniform partition of (0,þ). Furthermore there are opportunities for observing several global modes… In the X validated case of the symmetric triangular distribution over (0,θ), with ½θ as tip of the triangle, I could not figure an alternative to the pedestrian solution of looking separately at each of the (n+1) intervals where θ can stand and returning the associated maximum on that interval. Definitely a good (counter-)example about (in)sufficiency for class or exam!

done! [#1]

After spending a few hours grading my 127 exams for most nights of this week, I am finally done with it! One of the exam questions was the simulation of XY when (X,Y) is a bivariate normal vector with correlation ρ, following the trick described in a X validated question asked a few months ago, namely that

XY≡R{cos(πU)+ρ}

but no one managed to establish this representation. And, as usual, some students got confused between parameters θ and observations x when writing a posterior density, since the density of the prior was defined in the exam with the dummy x, thereby recovering the prior as the posterior. Nothing terrible and nothing exceptional with this cohort of undergraduates. And now I still have to go through my second pile of exams for the graduate course I taught on Bayesian computational tools…