## Tofino from the air [jatp]

Posted in Statistics with tags , , , , , , , , , , on August 21, 2018 by xi'an

## a funny mistake

Posted in Statistics with tags , , , , , , , , , , , on August 20, 2018 by xi'an

While watching the early morning activity in Tofino inlet from my rental desk, I was looking at a recent fivethirthyeight Riddle, which consisted in finding the probability of stopping a coin game which rule was to wait for the n consecutive heads if (n-1) consecutive heads had failed to happen when requested, which is

p+(1-p)p²+(1-p)(1-p²)p³+…

or

$q=\sum_{k=1}^\infty p^k \prod_{j=1}^{k-1}(1-p^j)$

While the above can write as

$q=\sum_{k=1}^\infty \{1-(1-p^k)\} \prod_{j=1}^{k-1}(1-p^j)$

or

$\sum_{k=1}^\infty \prod_{j=1}^{k-1}(1-p^j)-\prod_{j=1}^{k}(1-p^j)$

hence suggesting

$q=\sum_{k=1}^\infty \prod_{j=1}^{k-1}(1-p^j) - \sum_{k=2}^\infty \prod_{j=1}^{k-1}(1-p^j) =1$

the answer is (obviously) false and the mistake in separating the series into a difference of series is that both terms are infinite. The correct answer is actually

$q=1-\prod_{j=1}^{\infty}(1-p^j)$

which is Euler’s function. Maybe nonstandard analysis can apply to go directly from the difference of the infinite series to the answer!

## optimal approximations for importance sampling

Posted in Mountains, pictures, Statistics, Travel with tags , , , , , , , , , , , on August 17, 2018 by xi'an

“…building such a zero variance estimator is most of the times not practical…”

As I was checking [while looking at Tofino inlet from my rental window] on optimal importance functions following a question on X validated, I came across this arXived note by Pantaleoni and Heitz, where they suggest using weighted sums of step functions to reach minimum variance. However, the difficulty with probability densities that are step functions is that they necessarily have a compact support, which thus make them unsuitable for targeted integrands with non-compact support. And making the purpose of the note and the derivation of the optimal weights moot. It points out its connection with the reference paper of Veach and Guibas (1995) as well as He and Owen (2014), a follow-up to the other reference paper by Owen and Zhou (2000).