Jul 16th, 2008| 01:00 pm | Posted by vlk

The Χ^{2} distribution plays an incredibly important role in astronomical data analysis, but it is pretty much a black box to most astronomers. How many people know, for instance, that its form is exactly the same as the γ distribution? A Χ^{2} distribution with ν degrees of freedom is

**p(z|ν) = (1/Γ(ν/2)) (1/2)**^{ν/2} z^{ν/2-1} e^{-z/2} ≡ γ(z;ν/2,1/2) , where **z=Χ**^{2}.

Continue reading ‘chi-square distribution [Eqn]’ »

May 6th, 2008| 06:12 pm | Posted by vlk

The *gamma* function [not the Gamma -- note upper-case G -- which is related to the factorial] is one of those insanely useful functions that after one finds out about it, one wonders “why haven’t we been using this all the time?” It is defined only on the ~~positive~~ non-negative real line, is a highly flexible function that can emulate almost any kind of skewness in a distribution, and is a perfect complement to the Poisson likelihood. In fact, it is the conjugate prior to the Poisson likelihood, and is therefore a natural choice for a prior in all cases that start off with counts. Continue reading ‘*gamma* function (Equation of the Week)’ »

Aug 16th, 2007| 04:36 pm | Posted by hlee

I’ve been heard so much, without knowing fundamental reasons (most likely physics), about coverage problems from astrostat/phystat groups. This paper might be an interest for those: Interval Estimation in Exponential Families by Brown, Cai,and DasGupta ; Statistica Sinica (2003), **13**, pp. 19-49

*Abstract summary:*

The authors investigated issues in interval estimation of the mean in the exponential family, such as binomial, Poisson, negative binomial, normal, gamma, and a sixth distribution. The poor performance of the Wald interval has been known not only for discrete cases but for nonnormal continuous cases with significant negative bias. Their computation suggested that the equal tailed Jeffreys interval and the likelihood ratio interval are the best alternatives to the Wald interval. Continue reading ‘Coverage issues in exponential families’ »

Tags:

bias,

binomial,

coverage,

Edgeworth expansion,

exponential family,

gamma,

Gehrels,

interval,

Jeffreys,

likelihood ratio,

negative binomial,

normal,

Poisson,

Rao score,

Wald Category:

arXiv,

Stat,

Uncertainty |

Comment