Another occurrence, while building my final math stat exam for my (quarantined!) third year students, of a question on X validated that led me to write down more precisely an argument for the decomposition of densities in exponential families. Albeit the decomposition is somewhat moot *(and lost on the initiator of the question since this person later posted an answer ignoring measures)*, as it all depends on the choice of the dominating measures over X, T(X), and the slices {x; T(x)=t}. The fact that the slice does depend on t requires the measure to accept a potential dependence on t, in which case the conditional density wrt this measure can as well be constant.

## Archive for mathematical statistics

## factorisation theorem on densities

Posted in Statistics with tags cross validated, dominating measure, exponential families, factorisation, final exam, mathematical statistics, sufficient statistics on December 23, 2020 by xi'an## sans sérif & sans chevron

Posted in Books, R, Statistics, University life with tags bicycle, book publishing, chevron, final exam, LaTeX, mathematical statistics, multiple answer test, R, R code, sans-sérif, simulation, vélo, Zoom on June 17, 2020 by xi'an{\sf df=function(x)2*pi*x-4*(x>1)*acos(1/(x+(1-x)*(x<1)))}

**A**s I was LaTeXing a remote exam for next week, including some R code questions, I came across the apparent impossibility to use < and > symbols in the sans-sérif “\sf” font… Which is a surprise, given the ubiquity of the symbols in R and my LaTeXing books over the years. Must have always used “\tt” and “\verb” then! On the side, I tried to work with the automultiplechoice LaTeX package [which should be renamed velomultiplechoice!] of Alexis Bienvenüe, which proved a bit of a challenge as the downloadable version contained a flawed file of automultiplechoice.sty! Still managed to produce a 400 question exam with random permutations of questions and potential answers. But not looking forward the 4 or 5 hours of delivering the test on Zoom…

## PhD position for research in ABC in Chalmers University

Posted in Statistics with tags ABC, Approximate Bayesian computation, Chalmers University, Gothenburg, likelihood-free methods, mathematical statistics, PhD position, simulation-based inference, Sweden, vacancy on May 27, 2020 by xi'an*[Posting a call for PhD candidates from Umberto Piccini as the deadline is June 1, next Monday!]*

A PhD student position in mathematical statistics on simulation-based inference methods for models with an “intractable” likelihood is available at the Dept. Mathematical Sciences, Chalmers University, Gothenburg (Sweden).

You will be part of an international collaboration to create new methodology bridging between simulation-based inference (such as approximate Bayesian computation and other likelihood-free methods) and deep neuronal networks. The goal is to ease inference for stochastic modelling.

Details on the project and the essential requirements are at https://www.chalmers.se/en/departments/math/research/research-groups/AIMS/Pages/ai-project-5.aspx

The PhD student position is fully funded and is up to 5 years, in the dynamic and international city of Gothenburg, the second largest city in Sweden, https://www.goteborg.com/en/ As a PhD student in Mathematical Sciences you will have opportunities for many inspiring conversations, a lot of autonomous work and some travel.

The position will be supervised by Assoc. Prof. Umberto Picchini.

Apply by **01 June 2020** following the instructions at

https://www.chalmers.se/en/about-chalmers/Working-at-Chalmers/Vacancies/Pages/default.aspx?rmpage=job&rmjob=8556

For informal enquiries, please get in touch with Umberto Picchini

## PhD studenships at Warwick

Posted in Kids, pictures, Statistics, University life with tags Brexit, CDT, Centre for Doctoral Training in Mathematics and Statistics, computational statistics, European Union, mathematical statistics, OxWaSP, PhD fellowship, University of Warwick on May 2, 2019 by xi'an

**T**here is an exciting opening for several PhD positions at Warwick, in the departments of Statistics and of Mathematics, as part of the Centre for Doctoral Training in Mathematics and Statistics newly created by the University. CDT studentships are funded for four years and funding is open to students from the European Union without restrictions. (No Brexit!) Funding includes a stipend at UK/RI rates and tuition fees at UK/EU rates. Applications are made via the University of Warwick Online Application Portal and should be made as quickly as possible since the funding will be allocated on a first come first serve basis. For more details, contact the CDT director, Martyn Plummer. I cannot but strongly encourage interested students to apply as this is a great opportunity to start a research career in a fantastic department!

## efficiency and the Fréchet-Darmois-Cramèr-Rao bound

Posted in Books, Kids, Statistics with tags Académie des Sciences, best unbiased estimator, Canada, Canadian Journal of Statistics, Cramer-Rao lower bound, cross validated, efficiency, Fréchet-Darmois-Cramèr-Rao bound, George Darmois, James-Stein estimator, mathematical statistics, Maurice Fréchet on February 4, 2019 by xi'an**F**ollowing some entries on X validated, and after grading a mathematical statistics exam involving Cramèr-Rao, or Fréchet-Darmois-Cramèr-Rao to include both French contributors pictured above, I wonder as usual at the relevance of a concept of *efficiency* outside [and even inside] the restricted case of unbiased estimators. The general (frequentist) version is that the variance of an estimator δ of [any transform of] θ with bias b(θ) is

I(θ)⁻¹ (1+b'(θ))²

while a Bayesian version is the van Trees inequality on the integrated squared error loss

(E(I(θ))+I(π))⁻¹

where I(θ) and I(π) are the Fisher information and the prior entropy, respectively. But this opens a whole can of worms, in my opinion since

- establishing that a given estimator is efficient requires computing both the bias and the variance of that estimator, not an easy task when considering a Bayes estimator or even the James-Stein estimator. I actually do not know if any of the estimators dominating the standard Normal mean estimator has been shown to be efficient (although there exist results for closed form expressions of the James-Stein estimator quadratic risk, including one of mine the Canadian Journal of Statistics published verbatim in 1988). Or is there a result that a Bayes estimator associated with the quadratic loss is by default efficient in either the first or second sense?
- while the initial Fréchet-Darmois-Cramèr-Rao bound is restricted to unbiased estimators (i.e., b(θ)≡0) and unable to produce efficient estimators in all settings but for the natural parameter in the setting of exponential families, moving to the general case means there exists one efficiency notion for every bias function b(θ), which makes the notion quite weak, while not necessarily producing efficient estimators anyway, the major impediment to taking this notion seriously;
- moving from the variance to the squared error loss is not more “natural” than using any [other] convex combination of variance and squared bias, creating a whole new class of optimalities (a grocery of cans of worms!);
- I never got into the van Trees inequality so cannot say much, except that the comparison between various priors is delicate since the integrated risks are against different parameter measures.