## estimation exam [best of]

Posted in Books, Kids, Statistics with tags , , , , , , , , on January 29, 2019 by xi'an

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

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

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.

## bootstrap in Nature

Posted in Statistics with tags , , , , , , , , , , on December 29, 2018 by xi'an

A news item in the latest issue of Nature I received about Brad Efron winning the “Nobel Prize of Statistics” this year. The bootstrap is certainly an invention worth the recognition, not to mention Efron’s contribution to empirical Bayes analysis,, even though I remain overall reserved about the very notion of a Nobel prize in any field… With an appropriate XXL quote, who called the bootstrap method the ‘best statistical pain reliever ever produced’!

## exams

Posted in Kids, Statistics, University life with tags , , , , , , , on February 7, 2018 by xi'an
As in every term, here comes the painful week of grading hundreds of exams! My mathematical statistics exam was highly traditional and did not even involve Bayesian material, as the few students who attended the lectures were so eager to discuss sufficiency and ancilarity, that I decided to spend an extra lecture on these notions rather than rushing though conjugate priors. Highly traditional indeed with an inverse Gaussian model and a few basic consequences of Basu’s theorem. actually exposed during this lecture. Plus mostly standard multiple choices about maximum likelihood estimation and R programming… Among the major trends this year, I spotted out the widespread use of strange derivatives of negative powers, the simultaneous derivation of two incompatible convergent estimates, the common mixup between the inverse of a sum and the sum of the inverses, the inability to produce the MLE of a constant transform of the parameter, the choice of estimators depending on the parameter, and a lack of concern for Fisher informations equal to zero.

## what is your favorite teacher?

Posted in Kids, Statistics, University life with tags , , , , , , , , on October 14, 2017 by xi'an

When Jean-Louis Foulley pointed out to me this page in the September issue of Amstat News, about nominating a favourite teacher, I told him it had to be an homonym statistician! Or a practical joke! After enquiry, it dawned on me that this completely underserved inclusion came from a former student in my undergraduate Estimation course, who was very enthusiastic about statistics and my insistence on modelling rather than mathematical validation. He may have been the only one in the class, as my students always complain about not seeing the point in slides with no mathematical result. Like earlier this week when after 90mn on introducing the bootstrap method, a student asked me what was new compared with the Glivenko-Cantelli theorem I had presented the week before… (Thanks anyway to David for his vote and his kind words!)

## Nonparametric applications of Bayesian inference

Posted in Books, Statistics, University life with tags , , , , , , on April 22, 2016 by xi'an

Gary Chamberlain and Guido Imbens published this paper in the Journal of Business & Economic Statistics in 2003. I just came to read it in connection with the paper by Luke Bornn, Niel Shephard and Reza Solgi that I commented a few months ago. The setting is somewhat similar: given a finite support distribution with associated probability parameter θ, a natural prior on θ is a Dirichlet prior. This prior induces a prior on transforms of θ, whether or not they are in close form (for instance as the solution of a moment equation E[F(X,β)]=0. As in Bornn et al. In this paper, Chamberlain and Imbens argue in favour of the limiting Dirichlet with all coefficients equal to zero as a way to avoid prior dominating influence when the number of classes J goes to infinity and the data size remains fixed. But they fail to address the issue that the posterior is no longer defined since some classes get unobserved. They consider instead that the parameters corresponding to those classes are equal to zero with probability one, a convention and not a result. (The computational advantage in using the improper prior sounds at best incremental.) The notion of letting some Dirichlet hyper-parameters going to zero is somewhat foreign to a Bayesian perspective as those quantities should be either fixed or distributed according to an hyper-prior, rather than set to converge according to a certain topology that has nothing to do with prior modelling. (Another reason why setting those quantities to zero does not have the same meaning as picking a Dirac mass at zero.)

“To allow for the possibility of an improper posterior distribution…” (p.4)

This is a weird beginning of a sentence, especially when followed by a concept of expected posterior distribution, which is actually a bootstrap expectation. Not as in Bayesian bootstrap, mind. And thus this feels quite orthogonal to the Bayesian approach. I do however find most interesting this notion of constructing a true expected posterior by imposing samples that ensure properness as it reminds me of our approach to mixtures with Jean Diebolt, where (latent) allocations were prohibited to induce improper priors. The bootstrapped posterior distribution seems to be proposed mostly for assessing the impact of the prior modelling, albeit in an non-quantitative manner. (I fail to understand how the very small bootstrap sample sizes are chosen.)

Obviously, there is a massive difference between this paper and Bornn et al, where the authors use two competing priors in parallel, one on θ and one on β, which induces difficulties in setting priors since the parameter space is concentrated upon a manifold. (In which case I wonder what would happen if one implemented the preposterior idea of Berger and Pérez, 2002, to derive a fixed point solution. That we implemented recently with Diego Salmerón and Juan Antonio Caño in a paper published in Statistica Sinica.. This exhibits a similarity with the above bootstrap proposal in that the posterior gets averaged wrt another posterior.)