Méthodes de Monte-Carlo avec R

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R exam

Posted in Kids, pictures, Statistics, University life with tags exam, harmonic mean estimator, Introduction to Monte Carlo Methods with R, Méthodes de Monte-Carlo avec R, Monte Carlo methods, R, simulation, Université Paris Dauphine on November 28, 2011 by xi'an

Following a long tradition (!) of changing the modus vivendi of each exam in our exploratory statistics with R class, we decided this year to give the students a large collection of exercises prior to the exam and to pick five among them to the exam, the students having to solve two and only two of them. (The exercises are available in French on my webpage.) This worked beyond our expectations in that the overwhelming majority of students went over all the exercises and did really (too) well at the exam! Next year, we will hopefully increase the collection of exercises and also prohibit written notes during the exam (to avoid a possible division of labour among the students).

Incidentally, we found a few (true) gems in the solutions, incl. an harmonic mean resolution of the approximation of the integral

$\int_2^\infty x^4 e^{-x}\,\text{d}x=\Gamma(5,2)$

since some students generated from the distribution with density f proportional to the integrand over [2,∞) [a truncated gamma] and then took the estimator

$\dfrac{1-e^{-2}}{\frac{1}{n}\,\sum_{i=1}^n y_i^{-4}}\approx\dfrac{\int_2^\infty e^{-x}\,\text{d}x}{\mathbb{E}[X^{-4}]}\quad\text{when}\quad X\sim f$

although we expected them to simulate directly from the exponential and average the sample to the fourth power… In this specific situation, the (dreaded) harmonic mean estimator has a finite variance! To wit;

> y=rgamma(shape=5,n=10^5)
> pgamma(2,5,low=FALSE)*gamma(5)
[1] 22.73633
> integrate(f=function(x){x^4*exp(-x)},2,Inf)
22.73633 with absolute error < 0.0017
> pgamma(2,1,low=FALSE)/mean(y[y>2]^{-4})
[1] 22.92461
> z=rgamma(shape=1,n=10^5)
> mean((z>2)*z^4)
[1] 23.92876

So the harmonic means does better than the regular Monte Carlo estimate in this case!

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Typos sorted, at last!

Posted in Books, R, Statistics, University life with tags Bayesian Core, Introducing Monte Carlo Methods with R, Le Choix bayésien, Méthodes de Monte-Carlo avec R, Monte Carlo Statistical Methods, print on demand, Springer-Verlag, The Bayesian Choice, typos on March 24, 2011 by xi'an

After posting so many entries about typos in my books (making you wonder how there could be any text left!) and postponing their classification for so long, I decided on Saturday afternoon to collect those entries into a comprehensive pdf document that should be more useful for readers. I incidentally noticed that my book web-page had not been updated for the publication of Méthodes de Monte-Carlo avec R two months ago and did update it… I am obviously more than grateful to all those who contacted me, either thru the Og or by email to point out those numerous typos. As the Wheel had started rolling (or weaving), I took the opportunity to clean the original LaTeX files as well, including Introducing Monte Carlo methods with R. Since this is a print-on-demand edition, the changes should be immediate.

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