## 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

Posted in Books, Kids, Travel with tags , , , , , , , on September 3, 2011 by xi'an

Staying on the boat for several days meant some of us exhausted the few books we had brought and started looking for others’ discarded books. My daughter took David Gemmel’s John Shannow (in French) from my son, my son started The lies of Locke Lamora, by Scott Lynch,  as soon I finished it and he completed Abercrombie’s Best served cold, and no one touched my George Martin’s A Game of Thrones as Alex had already brought a copy! Anyway, several of us were eager for Sylvia to complete her Remarkable Creatures, by Tracy Chevalier, whose French translation I tried to impose on my daughter before we left. Captain Mark got the higher hand and managed to finish it on the last morning on the boat, then my wife and I shared it during the final days of our summer vacation…

This book is very fast to read, but very very enjoyable and not only as a summer book. It describes the beginning of fossil hunting at Lyme Regis on the Southern Coast of England and the consequences on science, (pre-)Victorian society [the story starts in 1810], religion, and the life of two remarkable women. (The title is thus ambiguous, describing both the remarkable specimens of extinct reptiles found there and the two women, given the lower status (pre-)Victorians accorded to women [despite being ruled by one of the greatest English monarchs!], as illustrated by the book. Some of the situations showing second-rate intelligences getting the better of those women simply for being male are indeed remarkable!) The book should appeal to scientific minds, as it describes how “discoverers”—Mary Anning who made the major fossil discoveries on Lyme Regis beach and managed to see the structure in them—get the better of theoreticians—like Buckland from Oxford and Cuvier from le Muséum d’Histoire Naturelle in Paris. Mary Anning and her friend Elizabeth Philpot drew inference from their observations, on their own, as for instance about coprolites (the name was coined by Buckland). Scientists like Cuvier and more clearly Buckland were hindered by the very classifications they had designed and by the religious constraints of a literal reading of the Bible. (This is oversimplifying as Cuvier was arguing from the 1790’s, i.e. before Mary Anning’s discovery, that fossils were the results of past extinctions, leading to a theory he called catastrophism.) The book is accompanied by a website that gives entries into those topics. My only reservation is that the dialogues often appear too modern, but this does not get annoying, quite the opposite. (The book got positive reviews from all passengers on the boat!) I had read a few years ago the Girl with the pearl earing and found the same pleasure in reading about the novelisation of historical events and this Dutch master of light…

## Keys in and out

Posted in Kids, pictures, Travel with tags , , , , , , on August 30, 2011 by xi'an

Spending a few days in the Keys was both relaxing, replacing the rudimentary comfort on the catamaran with the amenities of an American house, and frustrating, because of the sudden decrease in the intensity of our activities during those days. Indeed, finding snorkeling areas in the area was much less obvious than in the Bahamas, with less variety in the fish population. The best snorkeling spot happened to be a few meters away from Fort Zachary Taylor beacht in Key West, with even a reported sighting of a manatee a few meters away. The few accessible beaches in the remainder of the Keys were rather disappointing, with a band of dried sea grass that put the kids away, and long flat plateaus that prevented swimming and favoured bacterias. The best memory of the Keys is presumably the quality of its sunsets, which were uniformly magic. (A related disappointment was a fruitless search for a rising full moon, due to the lack of accessible beach anywhere close to our rental place…) We also enjoyed visiting the turtle hospital in Marathon, a private charity that provides medical care for turtles entangled in flotsam, jetsam, and fishing nets, damaged by boat [and idiotic jetski] propellers, suffering from plastic ingestion (impactation) or from fibropapilloma… This was in sharp contrast with the dolphin and shark “research centres” we briefly considered, which are nothing more than expensive petting zoos!

## Captain Mark

Posted in Kids, Mountains, pictures, Travel with tags , , , , , on August 21, 2011 by xi'an

The captain of the catamaran that took us sailing in the Bahamas was quite a character! Captain Mark Pomerenke reminded me of some mountain guides I have met by the quiet, methodical, and non-sensical way he handled things and people. A former USAF jet engineer, he was shot down by a missile during the Vietnam war and survived both the crash and the military hospitals he went thru (although he had to escape one under threat of amputation!). He took advantage of his years of physiotherapy to earn an MBA degree in hospital management he put to profit once he had retired from the USAF. After he quitted his medical practice management consulting firm, instead of settling down in Phoenix, he started taking chartered guests on his catamarans, mostly in the Caribbean. In addition, he loves telling stories about his time in the USAF and of his adventures at sea. Once started on the issue of being boarded by various law enforcement authorities, over and over, he is unstoppable! His story of facing Katrina because of an unlikely weather prediction is also amazing… As are, at another level, those stories about the whims and follies of some of his wealthy clients! The most incredible of his many stories is nonetheless the one about being caught by the high tide amplitude (5.5 meters today) in Folkerstone, England, and finding his boat hanging by the mooring at low tide!