max vs. min

Another intriguing question on X validated (about an exercise in Jun Shao’s book) that made me realise a basic fact about exponential distributions. When considering two Exponential random variables X and Y with possibly different parameters λ and μ,  Z⁺=max{X,Y} is dependent on the event X>Y while Z⁻=min{X,Y} is not (and distributed as an Exponential variate with parameter λ+μ.) Furthermore, Z⁺ is distributed from a signed mixture

conditionally on the event X>Y, meaning that there is no sufficient statistic of fixed dimension when given a sample of n realisations of Z⁺’s along with the indicators of the events X>Y…. This may explain why there exists an unbiased estimator of λ⁻¹-μ⁻¹ in this case and (apparently) not when replacing Z⁺ by Z⁻. (Even though the exercise asks for the UMVUE.)

I thought I did make a mistake but I was wrong…

One of my students in my MCMC course at ENSAE seems to specialise into spotting typos in the Monte Carlo Statistical Methods book as he found an issue in every problem he solved! He even went back to a 1991 paper of mine on Inverse Normal distributions, inspired from a discussion with an astronomer, Caroline Soubiran, and my two colleagues, Gilles Celeux and Jean Diebolt. The above derivation from the massive Gradsteyn and Ryzhik (which I discovered thanks to Mary Ellen Bock when arriving in Purdue) is indeed incorrect as the final term should be the square root of 2β rather than 8β. However, this typo does not impact the normalising constant of the density, K(α,μ,τ), unless I am further confused.

back to the Bayesian Choice

Surprisingly (or not?!), I received two requests about some exercises from The Bayesian Choice, one from a group of students from McGill having difficulties solving the above, wondering about the properness of the posterior (but missing the integration of x), to whom I sent back this correction. And another one from the Czech Republic about a difficulty with the term “evaluation” by which I meant (pardon my French!) estimation.

done! [#2]

Phew! I just finished my enormous pile of homeworks for the computational statistics course… This massive pile is due to an unexpected number of students registering for the Data Science Master at ENSAE and Paris-Dauphine. As I was not aware of this surge, I kept to my practice of asking students to hand back solved exercises from Monte Carlo Statistical Methods at the beginning of each class. And could not change the rules of the game once the course had started! Next year, I’ll make sure to get some backup for grading those exercises. Or go for group projects instead…

triste célébration for World Statistics Day

As I was discussing yesterday night with my daughter about a practice stats exam she had just taken in medical school, I came upon the following question:

What is the probability that women have the same risk of cancer as men in the entire population given that the selected sample concluded against equality?

Which just means nothing, since conditioning on the observed event, say |X|>1.96, cancels any probabilistic structure in the problem. Worse, I have no idea what is the expected answer to this question!