Archive for JASA

causal inference makes it to Stockholm

Posted in Statistics with tags , , , , , , , on October 12, 2021 by xi'an

Yesterday, Joshua Angrist and Guido Imbens, whose most cited paper is this JASA 1996 article with Don Rubin, were awarded the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel for 2021. It is one of these not-so-rare instances when econometricians get this prize, with causality the motive for their award. I presume this will not see the number of Biometrika submissions involving causal inference go down! (Imbens wrote a book on causal inference with Don Rubin, and is currently editor of Econometrica. And Angrist wrote Mostly Harmless Econometrics, with J.S. Pischke, which I have not read.)

ratio of Gaussians

Posted in Books, Statistics, University life with tags , , , , , , , , on April 12, 2021 by xi'an

Following (as usual) an X validated question, I came across two papers of George Marsaglia on the ratio of two arbitrary (i.e. unnormalised and possibly correlated) Normal variates. One was a 1965 JASA paper,

where the density of the ratio X/Y is exhibited, based on the fact that this random variable can always be represented as (a+ε)/(b+ξ) where ε,ξ are iid N(0,1) and a,b are constant. Surprisingly (?), this representation was challenged in a 1969 paper by David Hinkley (corrected in 1970).

And less surprisingly the ratio distribution behaves almost like a Cauchy, since its density is

meaning it is a two-component mixture of a Cauchy distribution, with weight exp(-a²/2-b²/2), and of an altogether more complex distribution ƒ². This is remarked by Marsaglia in the second 2006 paper, although the description of the second component remains vague, besides a possible bimodality. (It could have a mean, actually.) The density ƒ² however resembles (at least graphically) the generalised Normal inverse density I played with, eons ago.

Bayesian sufficiency

Posted in Books, Kids, Statistics with tags , , , , , , , , , on February 12, 2021 by xi'an

“During the past seven decades, an astonishingly large amount of effort and ingenuity has gone into the search fpr resonable answers to this question.” D. Basu

Induced by a vaguely related question on X validated, I re-read Basu’s 1977 great JASA paper on the elimination of nuisance parameters. Besides the limitations of competing definitions of conditional, partial, marginal sufficiency for the parameter of interest,  Basu discusses various notions of Bayesian (partial) sufficiency.

“After a long journey through a forest of confusing ideas and examples, we seem to have lost our way.” D. Basu

Starting with Kolmogorov’s idea (published during WW II) to impose to all marginal posteriors on the parameter of interest θ to only depend on a statistic S(x). But having to hold for all priors cancels the notion as the statistic need be sufficient jointly for θ and σ, as shown by Hájek in the early 1960’s. Following this attempt, Raiffa and Schlaifer then introduced a more restricted class of priors, namely where nuisance and interest are a priori independent. In which case a conditional factorisation theorem is a sufficient (!) condition for this Q-sufficiency.  But not necessary as shown by the N(θ·σ, 1) counter-example (when σ=±1 and θ>0). [When the prior on σ is uniform, the absolute average is Q-sufficient but is this a positive feature?] This choice of prior separation is somewhat perplexing in that it does not hold under reparameterisation.

Basu ends up with three challenges, including the multinomial M(θ·σ,½(1-θ)·(1+σ),½(1+θ)·(1-σ)), with (n¹,n²,n³) as a minimal sufficient statistic. And the joint observation of an Exponential Exp(θ) translated by σ and of an Exponential Exp(σ) translated by -θ, where the prior on σ gets eliminated in the marginal on θ.

an elegant book [review]

Posted in Books, Statistics, University life with tags , , , , , , , on December 28, 2020 by xi'an

“Handbook of Mixture Analysis is an elegant book on the mixture models. It covers not only statistical foundations but also extensions and applications of mixture models. The book consists of 19 chapters (each chapter is an independent paper), and collectively, these chapters weave into an elegant web of mixture models” Yen-Chi Chen (U. Washington)

inverse Gaussian trick [or treat?]

Posted in Books, Kids, R, Statistics, University life with tags , , , , , , , , , , , , , , on October 29, 2020 by xi'an

When preparing my mid-term exam for my undergrad mathematical statistics course, I wanted to use the inverse Gaussian distribution IG(μ,λ) as an example of exponential family and include a random generator question. As shown above by a Fortran computer code from Michael, Schucany and Haas, a simple version can be based on simulating a χ²(1) variate and solving in x the following second degree polynomial equation

\dfrac{\lambda(x-\mu)^2}{\mu^2 x} = v

since the left-hand side transform is distributed as a χ²(1) random variable. The smallest root x¹, less than μ, is then chosen with probability μ/(μ+x¹) and the largest one, x²=μ²/x¹ with probability x¹/(μ+x¹). A relatively easy question then, except when one considers asking for the proof of the χ²(1) result, which proved itself to be a harder cookie than expected! The paper usually referred to for the result, Schuster (1968), is quite cryptic on the matter, essentially stating that the above can be expressed as the (bijective) transform of Y=min(X,μ²/X) and that V~χ²(1) follows immediately. I eventually worked out a proof by the “law of the unconscious statistician” [a name I do not find particularly amusing!], but did not include the question in the exam. But I found it fairly interesting that the inverse Gaussian can be generating by “inverting” the above equation, i.e. going from a (squared) Gaussian variate V to the inverse Gaussian variate X. (Even though the name stems from the two cumulant generating functions being inverses of one another.)

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