## 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(-2θ) is particularly foolish: the only solution is (-1) to the power S [for a single observation]. (There is however a first 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)). And, as a second way, as remarked by Juho Kokkala below, a sample of at least two Poisson observations leads to a more coherent best unbiased estimator.)

## nurse shark

Posted in Kids, pictures, Travel with tags , , on September 11, 2011 by xi'an

Some moderately successful attempts at photographing a nurse shark we swam with a few weeks ago… The first two pictures are taken with an underwater camera of the lowest quality! The buoy signals a shipwreck that housed many fish. Continue reading

## Sailing the Exumas

Posted in Kids, Mountains, pictures, Travel with tags , , , , , , , on September 4, 2011 by xi'an

A few weeks ago, taking advantage from being in Florida, the four of us sailed the Exumas islands in the Bahamas with Sylvia Richardson’s family (Sylvia, Jessie, and Alex) on a charter catamaran, Stray Cat, piloted by Captain Mark Pomerenke. Although “sea, sun, and sail” is not in my top ten list of outdoor activities, I enjoyed it very much, along with all members of the trip! Here at last is the shiplog of our week in the Exumas, mostly written by Sylvia (with a few additions of mine for the ‘Og). Continue reading