## on completeness

Posted in Books, Kids, Statistics with tags , , , , , , on November 19, 2020 by xi'an

Another X validated question that proved a bit of a challenge, enough for my returning to its resolution on consecutive days. The question was about the completeness of the natural sufficient statistic associated with a sample from the shifted exponential distribution

$f(x;\theta) = \frac{1}{\theta^2}\exp\{-\theta^{-2}(x-\theta)\}\mathbb{I}_{x>\theta}$

[weirdly called negative exponential in the question] meaning the (minimal) sufficient statistic is made of the first order statistic and of the sample sum (or average), or equivalently

$T=(X_{(1)},\sum_{i=2}^n \{X_{(i)}-X_{(1)}\})$

Finding the joint distribution of T is rather straightforward as the first component is a drifted Exponential again and the second a Gamma variate with n-2 degrees of freedom and the scale θ². (Devroye’s Bible can be invoked since the Gamma distribution follows from his section on Exponential spacings, p.211.) While the derivation of a function with constant expectation is straightforward for the alternate exponential distribution

$f(x;\theta) = \frac{1}{\theta}\exp\{-\theta^{-1}(x-\theta)\}\mathbb{I}_{x>\theta}$

since the ratio of the components of T has a fixed distribution, it proved harder for the current case as I was seeking a parameter free transform. When attempting to explain the difficulty on my office board, I realised I was seeking the wrong property since an expectation was enough. Removing the dependence on θ was simpler and led to

$\mathbb E_\theta\left[\frac{X_{(1)}}{Y}-\frac{\Gamma(n-2)}{\Gamma(n-3/2)}Y^\frac{-1}{2}\right]=\frac{\Gamma(n-2)}{n\Gamma(n-1)}$

but one version of a transform with fixed expectation. This also led me to wonder at the range of possible functions of θ one could use as scale and still retrieve incompleteness of T. Any power of θ should work but what about exp(θ²) or sin²(θ³), i.e. functions for which there exists no unbiased estimator..?

## arbitrary non-constant function [nonsensical]

Posted in Statistics with tags , , , , , , , , , , , on November 6, 2020 by xi'an

## artificial EM

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

When addressing an X validated question on the use of the EM algorithm when estimating a Normal mean, my first comment was that it was inappropriate since there is no missing data structure to anchor by (right preposition?). However I then reflected upon the infinite number of ways to demarginalise the normal density into a joint density

$$∫ f(x,z;μ)dz = φ(x–μ)$$

from the (slice sampler) call to an indicator function for $$f(x,z;μ)$$ to a joint Normal distribution with an arbitrary correlation. While the joint Normal representation produces a sequence converging to the MLE, the slice representation utterly fails as the indicator functions make any starting value of $$μ$$ a fixed point for EM.

Incidentally, when quoting from Wikipedia on the purpose of the EM algorithm, the following passage

Finding a maximum likelihood solution typically requires taking the derivatives of the likelihood function with respect to all the unknown values, the parameters and the latent variables, and simultaneously solving the resulting equations.

struck me as confusing and possibly wrong since it seems to suggest to seek a maximum in both the parameter and the latent variables. Which does not produce the same value as the observed likelihood maximisation.

## errno EFBIG

Posted in Books, Kids, Linux, Statistics, University life with tags , , , , , , on October 12, 2020 by xi'an

## linearity, reversed

Posted in Books, Kids with tags , , , , , on September 19, 2020 by xi'an

While answering a question on X validated on the posterior mean being a weighted sum of the prior mean and of the maximum likelihood estimator, when the weights do not depend on the data, which is true in conjugate natural exponential family settings, I re-read this wonderful 1979 paper of Diaconis & Ylvisaker establishing the converse, namely that when the linear combination holds, the prior need be conjugate! This holds within exponential families, but I cannot think of a reasonable case outside exponential families where the linearity holds (again with constant weights, as otherwise it always holds in dimension one, albeit with weights possibly outside [0,1]).