## best unbiased estimator of θ² for a Poisson model

Posted in Books, Kids, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , on May 23, 2018 by xi'an

A mostly traditional question on X validated about the “best” [minimum variance] unbiased estimator of θ² from a Poisson P(θ) sample leads to the Rao-Blackwell solution

$\mathbb{E}[X_1X_2|\underbrace{\sum_{i=1}^n X_i}_S=s] = -\frac{s}{n^2}+\frac{s^2}{n^2}=\frac{s(s-1)}{n^2}$

and a similar estimator could be constructed for θ³, θ⁴, … With the interesting limitation that this procedure stops at the power equal to the number of observations (minus one?). But,  since the expectation of a power of the sufficient statistics S [with distribution P(nθ)] is a polynomial in θ, there is de facto no limitation. More interestingly, there is no unbiased estimator of negative powers of θ in this context, while this neat comparison on Wikipedia (borrowed from the great book of counter-examples by Romano and Siegel, 1986, selling for a mere \$180 on amazon!) shows why looking for an unbiased estimator of exp(-θ) is particularly foolish: the only solution is -1 to the power S. (There is however a way to circumvent the difficulty if having access to an arbitrary number of generations from the Poisson, since the Forsythe – von Neuman algorithm allows for an unbiased estimation of exp(-F(x)).)

Posted in pictures, University life with tags , , , on May 22, 2018 by xi'an

Today, two statisticians (and good friends of mine) from Australia, Noel Cressie and Kerrie Mengersen, got elected at the Australian Academy of Sciences. Congratulations to them!

## Warwick locks [jatp]

Posted in pictures, Running, Travel with tags , , , , , , , , , , , on May 20, 2018 by xi'an

## miei primi cannoli

Posted in Kids, pictures, Travel, Wines with tags , , , , , , on May 19, 2018 by xi'an

Cannoli are among my favourite pastries and one of the first things I try to eat while in Italy (if never in Sicily)… (Or in Boston!) Especially since they cannot travel. Hence, when my wife brought me cannoli baking tubs from Venezia, I took the hint and did not wait many days to make a first attempt at baking my own cannoli. Preparing the dough takes a little while, although it does not much differ from a rich shortcrust pastry (with a wee bit of Sicilian sweet wine). But baking the tubes is not that complicated, thanks to the tubes, and they remain crusty for several days, while mastering the ricotta cheese filling will require many attempts before reaching the right consistency… The more attempts the better!!!

## a Swiss summer school on data assimilation

Posted in Books, Kids, Mountains, pictures, Statistics, Travel, University life with tags , , , , , on May 14, 2018 by xi'an

My friend Antonietta Mira sent me the announcement of a combined summer school and workshop on “Data Assimilation” that will take place from September 11th to 15th in Lugano, Switzerland. With Tamara Broderick, Philippe Moireau, and Andrew Stuart as teachers. (Registration, incl. lunches, is 120 CHF for the whole week.)