**A**lthough it may sound like an excessive notion of optimality, one can hope at obtaining an estimator δ of a unidimensional parameter θ that is always closer to θ that any other parameter. In distribution if not almost surely, meaning the cdf of (δ-θ) is steeper than for other estimators enjoying the same cdf at zero (for instance ½ to make them all median-unbiased). When I saw this question on X validated, I thought of the Cauchy location example, where there is no uniformly optimal estimator, albeit a large collection of unbiased ones. But a simulation experiment shows that the MLE does better than the competition. At least than ~~three (above)~~ four of them (since I tried the Pitman estimator via Christian Henning’s smoothmest R package). The differences to the MLE empirical cd make it clearer below (with tomato for a score correction, gold for the Pitman estimator, sienna for the 38% trimmed mean, and blue for the median):I wonder at a general theory along these lines. There is a vague similarity with Pitman nearness or closeness but without the paradoxes induced by this criterion. More in the spirit of stochastic dominance, which may be achievable for location invariant and mean unbiased estimators…

## Archive for Cauchy distribution

## more concentration, everywhere

Posted in R, Statistics with tags best equivariant estimator, Cauchy distribution, cross validated, ecdf, Pitman best equivariant estimator, Pitman closeness, Pitman nearness, R, smoothmest R package, stochastic dominance, uniform optimality on January 25, 2019 by xi'an## complex Cauchys

Posted in Books, pictures, Statistics, Travel, University life with tags Augustin Cauchy, Cauchy distribution, complex numbers, confidence distribution, conjecture, Don Fraser, Nancy Reid, Peter McCullagh, Sceaux, seminar, Université Paris Dauphine, William Feller on February 8, 2018 by xi'an**D**uring a visit of Don Fraser and Nancy Reid to Paris-Dauphine where Nancy gave a nice introduction to confidence distributions, Don pointed out to me a 1992 paper by Peter McCullagh on the Cauchy distribution. Following my recent foray into the estimation of the Cauchy location parameter. Among several most interesting aspects of the Cauchy, Peter re-expressed the density of a Cauchy C(θ¹,θ²) as

f(x;θ¹,θ²) = |θ²| / |x-θ|²

when θ=θ¹+ιθ² [a complex number on the half-plane]. Denoting the Cauchy C(θ¹,θ²) as Cauchy C(θ), the property that the ratio aX+b/cX+d follows a Cauchy for all real numbers a,b,c,d,

C(aθ+b/cθ+d)

[when X is C(θ)] follows rather readily. But then comes the remark that

“those properties follow immediately from the definition of the Cauchy as the ratio of two correlated normals with zero mean.”

which seems to relate to the conjecture solved by Natesh Pillai and Xiao-Li Meng a few years ago. But the fact that a ratio of two correlated centred Normals is Cauchy is actually known at least from the1930’s, as shown by Feller (1930, Biometrika) and Geary (1930, JRSS B).

## at CIRM

Posted in Books, Mountains, Running, Statistics, Travel, University life, Wines with tags Cauchy distribution, CIRM, Kelker, Luminy, Marseiile, Paris, Rao-Blackwellisation, Sankhya, slice sampling, spherically symmetric distributions on March 1, 2016 by xi'anThanks to a very early start from Paris, and despite horrendous traffic jams in Marseilles, I managed to reach CIRM with ten minutes to spare before my course. After my one-hour class, I was suddenly made aware of the (simplistic) idea that the slice sampling uniforms are simply auxiliary, meaning they can be used in many different ways.

I noticed Natesh Pillai just arXived an extension of his earlier Cauchy paper with XL. He proves that the result on the Cauchy distribution of any convex combination of normal ratios still holds when the pair of vectors is distributed from a product of elliptically symmetric functions. Some of Natesh’s remarks reminded me of the 1970 Sankhyã paper by Kelker on spherically symmetric variables. Especially because of Kelker’s characterisation of elliptically symmetric functions as scale mixtures of normals, which makes perfect sense since the scale cancels.

As I skimmed through my slides yesterday, fearing everyone knew about the MCMC basics, I decided to present today the Rao-Blackwellisation slides I gave in Warwick a few months ago.

## Cauchy Distribution: Evil or Angel?

Posted in Books, pictures, Running, Statistics, Travel, University life, Wines with tags Boston, Box-Muller algorithm, Cauchy distribution, champagne, correlation, Harvard University, JSM 2014, Mathias Drton, Monte Carlo Statistical Methods, Mystic river, Natesh Pillai, Sommerville, Xiao-Li Meng on May 19, 2015 by xi'an**N**atesh Pillai and Xiao-Li Meng just arXived a short paper that solves the Cauchy conjecture of Drton and Xiao [I mentioned last year at JSM], namely that, when considering two normal vectors with generic variance matrix S, a weighted average of the ratios X/Y remains Cauchy(0,1), just as in the iid S=I case. Even when the weights are random. The fascinating side of this now resolved (!) conjecture is that the correlation between the terms does not seem to matter. Pushing the correlation to one [assuming it is meaningful, which is a suspension of belief!, since there is no standard correlation for Cauchy variates] leads to a paradox: all terms are equal and yet… it works: we recover a single term, which again is Cauchy(0,1). All that remains thus to prove is that it stays Cauchy(0,1) between those two extremes, a weird kind of intermediary values theorem!

Actually, Natesh and XL further prove an inverse χ² theorem: the inverse of the normal vector, renormalised into a quadratic form is an inverse χ² no matter what its covariance matrix. The proof of this amazing theorem relies on a spherical representation of the bivariate Gaussian (also underlying the Box-Müller algorithm). The angles are then jointly distributed as

and from there follows the argument that conditional on the differences between the θ’s, all ratios are Cauchy distributed. Hence the conclusion!

A question that stems from reading this version of the paper is whether this property extends to other formats of non-independent Cauchy variates. Somewhat connected to my recent post about generating correlated variates from arbitrary distributions: using the inverse cdf transform of a Gaussian copula shows this is possibly the case: the following code is meaningless in that the empirical correlation has no connection with a “true” correlation, but nonetheless the experiment seems of interest…

> ro=.999999;x=matrix(rnorm(2e4),ncol=2);y=ro*x+sqrt(1-ro^2)*matrix(rnorm(2e4),ncol=2) > cor(x[,1]/x[,2],y[,1]/y[,2]) [1] -0.1351967 > ro=.99999999;x=matrix(rnorm(2e4),ncol=2);y=ro*x+sqrt(1-ro^2)*matrix(rnorm(2e4),ncol=2) > cor(x[,1]/x[,2],y[,1]/y[,2]) [1] 0.8622714 > ro=1-1e-5;x=matrix(rnorm(2e4),ncol=2);y=ro*x+sqrt(1-ro^2)*matrix(rnorm(2e4),ncol=2) > z=qcauchy(pnorm(as.vector(x)));w=qcauchy(pnorm(as.vector(y))) > cor(x=z,y=w) [1] 0.9999732 > ks.test((z+w)/2,"pcauchy") One-sample Kolmogorov-Smirnov test data: (z + w)/2 D = 0.0068, p-value = 0.3203 alternative hypothesis: two-sided > ro=1-1e-3;x=matrix(rnorm(2e4),ncol=2);y=ro*x+sqrt(1-ro^2)*matrix(rnorm(2e4),ncol=2) > z=qcauchy(pnorm(as.vector(x)));w=qcauchy(pnorm(as.vector(y))) > cor(x=z,y=w) [1] 0.9920858 > ks.test((z+w)/2,"pcauchy") One-sample Kolmogorov-Smirnov test data: (z + w)/2 D = 0.0036, p-value = 0.9574 alternative hypothesis: two-sided

## More typos in Chapter 5

Posted in Books, R, Statistics with tags Cauchy distribution, integrate, Introducing Monte Carlo Methods with R, Monte Carlo Statistical Methods, simulation, typos on December 29, 2010 by xi'an**F**ollowing Ashley’s latest comments on Chapter 5 of ** Introducing Monte Carlo Methods with R**, I realised Example 5.5 was totally off-the-mark! Not only the representation of the likelihood should have used prod instead of mean, not only the constant should call the val argument of integrate, not only integrate uses lower and upper rather than from and to, but also the approximation is missing a scale factor of 10, squared root of the sample size… The corrected

**R**code is thus

> cau=rcauchy(10^2) > mcau=median(cau) > rcau=diff(quantile(cau,c(.25,.75)))/sqrt(10^2) > f=function(x){ + z=dcauchy(outer(x,cau,FUN="-")) + apply(z,1,prod)} > fcst=integrate(f,lower=-20,upper=20)$val > ft=function(x){f(x)/fcst} > g=function(x){dt((x-mcau)/rcau,df=49)/rcau} > curve(ft,from=-1,to=1,xlab="",ylab="",lwd=2) > curve(g,add=T,lty=2,col="steelblue",lwd=2)

and the corrected Figure 5.5 is therefore as follows. Note that the fit by the t distribution is not as perfect as before. A normal approximation would do better.

This mistake is most embarrassing and I cannot fathom how I came with this unoperating program! (The more embarrassing as Cauchy‘s house is about 1k away from mine…) I am thus quite grateful to Ashley for her detailed study of this example.