Archive for Master program
MASH in Le Monde
Posted in Statistics with tags big data, Ecole Normal Supérieure, ENS, interview, Le Monde, MASH, Master program, Université de Montpellier, Université Paris Dauphine on January 25, 2019 by xi'an
variance of an exponential order statistics
Posted in Books, Kids, pictures, R, Statistics, University life with tags climate simulation, ecdf, empirical cdf, ENSAE, George Casella, Luc Devroye, Malakoff, Master program, Monte Carlo Statistical Methods, order statistics, spacings on November 10, 2016 by xi'anThis afternoon, one of my Monte Carlo students at ENSAE came to me with an exercise from Monte Carlo Statistical Methods that I did not remember having written. And I thus “charged” George Casella with authorship for that exercise!
Exercise 3.3 starts with the usual question (a) about the (Binomial) precision of a tail probability estimator, which is easy to answer by iterating simulation batches. Expressed via the empirical cdf, it is concerned with the vertical variability of this empirical cdf. The second part (b) is more unusual in that the first part is again an evaluation of a tail probability, but then it switches to find the .995 quantile by simulation and produce a precise enough [to three digits] estimate. Which amounts to assess the horizontal variability of this empirical cdf.
As we discussed about this question, my first suggestion was to aim at a value of N, number of Monte Carlo simulations, such that the .995 x N-th spacing had a length of less than one thousandth of the .995 x N-th order statistic. In the case of the Exponential distribution suggested in the exercise, generating order statistics is straightforward, since, as suggested by Devroye, see Section V.3.3, the i-th spacing is an Exponential variate with rate (N-i+1). This is so fast that Devroye suggests simulating Uniform order statistics by inverting Exponential order statistics (p.220)!
However, while still discussing the problem with my student, I came to a better expression of the question, which was to figure out the variance of the .995 x N-th order statistic in the Exponential case. Working with the density of this order statistic however led nowhere useful. A bit later, after Google-ing the problem, I came upon this Stack Exchange solution that made use of the spacing result mentioned above, namely that the expectation and variance of the k-th order statistic are
![\mathbb{E}[X_{(k)}]=\sum\limits_{i=N-k+1}^N\frac1i,\qquad \mbox{Var}(X_{(k)})=\sum\limits_{i=N-k+1}^N\frac1{i^2}](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BE%7D%5BX_%7B%28k%29%7D%5D%3D%5Csum%5Climits_%7Bi%3DN-k%2B1%7D%5EN%5Cfrac1i%2C%5Cqquad+%5Cmbox%7BVar%7D%28X_%7B%28k%29%7D%29%3D%5Csum%5Climits_%7Bi%3DN-k%2B1%7D%5EN%5Cfrac1%7Bi%5E2%7D&bg=000000&fg=B0B0B0&s=0&c=20201002)
which leads to the proper condition on N when imposing the variability constraint.
I am cold all over…
Posted in Books, Kids, Statistics, University life with tags computational statistics, ENSAE, Master program, MCMC algorithms, Monte Carlo Statistical Methods, statistical computing, Université Paris Dauphine, Winter is coming on October 29, 2014 by xi'an
An email from one of my Master students who sent his problem sheet (taken from Monte Carlo Statistical Methods) late:
Bonsoir Professeur
Je « suis » votre cours du mercredi dont le formalisme mathématique me fait froid partout
Avec beaucoup de difficulté je vous envoie mes exercices du premier chapitre de votre livre.
which translates as
Good evening Professor,
I “follow” your Wednesday class which mathematical formalism makes me cold all over. With much hardship, I send you the first batch of problems from your book.
I know that winter is coming, but, still, making students shudder from mathematical cold is not my primary goal when teaching Monte Carlo methods!
[not] reading classics (#7)
Posted in Books, Kids, Statistics, University life with tags classics, Kaplan-Meier estimator, Master program, seminar, Université Paris Dauphine on December 15, 2013 by xi'an
This week, I decided not to report on the paper read at the Reading Classics student seminar, as it did not work out well-enough. The paper was the “Regression models and life-table” published in 1972 by David Cox… A classic if any! Indeed, I do not think posting a severe criticism of the presentation or the presentation itself would be of much use to anyone. It is rather sad as (a) the student clearly put some effort in the presentation, including a reproduction of an R execution, and (b) this was an entry on semi-parametrics, Kaplan-Meyer, truncated longitudinal data, and more, that could have benefited the class immensely. Alas, the talk did not take any distance from the paper, did not exploit the following discussion, and exceeded by far the allocated time, without delivering a comprehensible message. It is a complex paper with concise explanations, granted, but there were ways to find easier introductions to its contents in the more recent literature… It is possible that a second student takes over and presents her analysis of the paper next January. Unless she got so scared with this presentation that she will switch to another paper… [Season wishes to Classics Readers!]
reading classics (#4,5,6)
Posted in Books, Kids, Statistics, University life with tags AIC, Akaike's criterion, ARMA models, Benjamini, classics, FAR, generalised linear models, GLMs, Hochberg, John Nelder, Master program, multiple comparisons, seminar, Université Paris Dauphine, Valencia conferences on December 9, 2013 by xi'anThis week, thanks to a lack of clear instructions (from me) to my students in the Reading Classics student seminar, four students showed up with a presentation! Since I had planned for two teaching blocks, three of them managed to fit within the three hours, while the last one nicely accepted to wait till next week to present a paper by David Cox…
The first paper discussed therein was A new look at the statistical model identification, written in 1974 by Hirotugu Akaike. And presenting the AIC criterion. My student Rozan asked to give the presentation in French as he struggled with English, but it was still a challenge for him and he ended up being too close to the paper to provide a proper perspective on why AIC is written the way it is and why it is (potentially) relevant for model selection. And why it is not such a definitive answer to the model selection problem. This is not the simplest paper in the list, to be sure, but some intuition could have been built from the linear model, rather than producing the case of an ARMA(p,q) model without much explanation. (I actually wonder why the penalty for this model is (p+q)/T, rather than (p+q+1)/T for the additional variance parameter.) Or simulation ran on the performances of AIC versus other xIC’s…
[speakerdeck https://speakerdeck.com/xianblog/akaikes-aic-presentation-by-rozan-karagoz%5D
The second paper was another classic, the original GLM paper by John Nelder and his coauthor Wedderburn, published in 1972 in Series B. A slightly easier paper, in that the notion of a generalised linear model is presented therein, with mathematical properties linking the (conditional) mean of the observation with the parameters and several examples that could be discussed. Plus having the book as a backup. My student Ysé did a reasonable job in presenting the concepts, but she would have benefited from this extra-week in including properly the computations she ran in R around the glm() function… (The definition of the deviance was somehow deficient, although this led to a small discussion during the class as to how the analysis of deviance was extending the then flourishing analysis of variance.) In the generic definition of the generalised linear models, I was also reminded of the
generality of the nuisance parameter modelling, which made the part of interest appear as an exponential shift on the original (nuisance) density.
[speakerdeck https://speakerdeck.com/xianblog/glms-by-nelder-and-wedderburn-discussed-by-yse-wanono%5D
The third paper, presented by Bong, was yet another classic, namely the FDR paper, Controlling the false discovery rate, of Benjamini and Hochberg in Series B (which was recently promoted to the should-have-been-a-Read-Paper category by the RSS Research Committee and discussed at the Annual RSS Conference in Edinburgh four years ago, as well as published in Series B). This 2010 discussion would actually have been a good start to discuss the paper in class, but Bong was not aware of it and mentioned earlier papers extending the 1995 classic. She gave a decent presentation of the problem and of the solution of Benjamini and Hochberg but I wonder how much of the novelty of the concept the class grasped. (I presume everyone was getting tired by then as I was the only one asking questions.) The slides somewhat made it look too much like a simulation experiment… (Unsurprisingly, the presentation did not include any Bayesian perspective on the approach, even though they are quite natural and emerged very quickly once the paper was published. I remember for instance the Valencia 7 meeting in Teneriffe where Larry Wasserman discussed about the Bayesian-frequentist agreement in multiple testing.)
[speakerdeck https://speakerdeck.com/xianblog/controlling-the-false-discovery-rate-by-bing-wong%5D
