estimation exam [best of]

Yesterday, I received a few copies of our CRC Press Handbook of Mixture Analysis, while grading my mathematical statistics exam 135 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

\int_0^\theta \varphi(x) x^k \text{d}x

was zero then φ had to be equal to zero with no further argument (only one student thought to take the derivative). Plus a growing inability to differentiate even simple functions… (At least, most students got the bootstrap right, as exemplified by their R code.)