Archive for Annals of Mathematical Statistics

mixtures as exponential families

Posted in Kids, Statistics with tags , , , , , , , on December 8, 2015 by xi'an

Something I had not realised earlier and that came to me when answering a question on X validated about the scale parameter of a Gamma distribution. Following an earlier characterisation by Dennis Lindley, Ferguson has written a famous paper characterising location, scale and location-scale families within exponential families. For instance, a one-parameter location exponential family is necessarily the logarithm of a power of a Gamma distribution. What I found surprising is the equivalent for one-parameter scale exponential families: they are necessarily mixtures of positive and negative powers of Gamma distributions. This is surprising because a mixture does not seem to fit within the exponential family representation… I first thought Ferguson was using a different type of mixtures. Or of exponential family. But, after checking the details, it appears that the mixture involves a component on ℜ⁺ and another component on ℜ⁻ with a potential third component as a Dirac mass at zero. Hence, it only nominally writes as a mixture and does not offer the same challenges as a regular mixture. Not label switching. No latent variable. Having mutually exclusive supports solves all those problems and even allows for an indicator function to permeate through the exponential function… (Recall that the special mixture processed in Rubio and Steel also enjoys this feature.)

reading classics (#11)

Posted in Books, Statistics, University life with tags , , , , , , , on March 21, 2013 by xi'an

Today was my last Reading Seminar class and the concluding paper chosen by the student was Tukey’s “The future of data analysis“, a 1962 Annals of Math. Stat. paper. Unfortunately, reading this paper required much more maturity and background than the student could afford, which is the reason why this last presentation is not posted on this page… Given the global and a-theoretical perspective of the paper, it was quite difficult to interpret without further delving into Tukey’s work and without a proper knowledge of what was Data Analysis in the 1960’s. (The love affair of French statisticians with data analysis was then at its apex, but it has very much receded since then!) Being myself unfamiliar with this paper, and judging mostly from the sentences pasted by the student in his slides, I cannot tell how much of the paper is truly visionary and how much is cheap talk: focussing on trimmed and winsorized means does not sound like offering a very wide scope for data analysis… I liked the quote “It’s easier to carry a slide rule than a desk computer, to say nothing of a large computer”! (As well as the quote from Azimov “The sound of panting“…. (Still, I am unsure I will keep the paper within the list next year!)

Overall, despite a rather disappointing lower tail of the distribution of the talks, I am very happy with the way the seminar proceeded this year and the efforts produced by the students to assimilate the papers, the necessary presentation skills including building a background in LaTeX and Beamer for most students. I thus think almost all students will pass this course and do hope those skills will be profitable for their future studies…