Archive for best unbiased estimator

efficiency and the Fréchet-Darmois-Cramèr-Rao bound

Posted in Books, Kids, Statistics with tags , , , , , , , , , , , on February 4, 2019 by xi'an

 

Following some entries on X validated, and after grading a mathematical statistics exam involving Cramèr-Rao, or Fréchet-Darmois-Cramèr-Rao to include both French contributors pictured above, I wonder as usual at the relevance of a concept of efficiency outside [and even inside] the restricted case of unbiased estimators. The general (frequentist) version is that the variance of an estimator δ of [any transform of] θ with bias b(θ) is

I(θ)⁻¹ (1+b'(θ))²

while a Bayesian version is the van Trees inequality on the integrated squared error loss

(E(I(θ))+I(π))⁻¹

where I(θ) and I(π) are the Fisher information and the prior entropy, respectively. But this opens a whole can of worms, in my opinion since

  • establishing that a given estimator is efficient requires computing both the bias and the variance of that estimator, not an easy task when considering a Bayes estimator or even the James-Stein estimator. I actually do not know if any of the estimators dominating the standard Normal mean estimator has been shown to be efficient (although there exist results for closed form expressions of the James-Stein estimator quadratic risk, including one of mine the Canadian Journal of Statistics published verbatim in 1988). Or is there a result that a Bayes estimator associated with the quadratic loss is by default efficient in either the first or second sense?
  • while the initial Fréchet-Darmois-Cramèr-Rao bound is restricted to unbiased estimators (i.e., b(θ)≡0) and unable to produce efficient estimators in all settings but for the natural parameter in the setting of exponential families, moving to the general case means there exists one efficiency notion for every bias function b(θ), which makes the notion quite weak, while not necessarily producing efficient estimators anyway, the major impediment to taking this notion seriously;
  • moving from the variance to the squared error loss is not more “natural” than using any [other] convex combination of variance and squared bias, creating a whole new class of optimalities (a grocery of cans of worms!);
  • I never got into the van Trees inequality so cannot say much, except that the comparison between various priors is delicate since the integrated risks are against different parameter measures.

absurdly unbiased estimators

Posted in Books, Kids, Statistics with tags , , , , , , , on November 8, 2018 by xi'an

“…there are important classes of problems for which the mathematics forces the existence of such estimators.”

Recently I came through a short paper written by Erich Lehmann for The American Statistician, Estimation with Inadequate Information. He analyses the apparent absurdity of using unbiased estimators or even best unbiased estimators in settings like the Poisson P(λ) observation X producing the (unique) unbiased estimator of exp(-bλ) equal to

(1-b)^x

which is indeed absurd when b>1. My first reaction to this example is that the question of what is “best” for a single observation is not very meaningful and that adding n independent Poisson observations replaces b with b/n, which gets eventually less than one. But Lehmann argues that the paradox stems from a case of missing information, as for instance in the Poisson example where the above quantity is the probability P(T=0) that T=0, when T=X+Y, Y being another unobserved Poisson with parameter (b-1)λ. In a lot of such cases, there is no unbiased estimator at all. When there is any, it must take values outside the (0,1) range, thanks to a lemma shown by Lehmann that the conditional expectation of this estimator given T is either zero or one.

I find the short paper quite interesting in exposing some reasons why the estimators cannot find enough information within the data (often a single point) to achieve an efficient estimation of the targeted function of the parameter, even though the setting may appear rather artificial.

best unbiased estimator of θ² for a Poisson model

Posted in Books, Kids, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , on May 23, 2018 by xi'an

A mostly traditional question on X validated about the “best” [minimum variance] unbiased estimator of θ² from a Poisson P(θ) sample leads to the Rao-Blackwell solution

\mathbb{E}[X_1X_2|\underbrace{\sum_{i=1}^n X_i}_S=s] = -\frac{s}{n^2}+\frac{s^2}{n^2}=\frac{s(s-1)}{n^2}

and a similar estimator could be constructed for θ³, θ⁴, … With the interesting limitation that this procedure stops at the power equal to the number of observations (minus one?). But,  since the expectation of a power of the sufficient statistics S [with distribution P(nθ)] is a polynomial in θ, there is de facto no limitation. More interestingly, there is no unbiased estimator of negative powers of θ in this context, while this neat comparison on Wikipedia (borrowed from the great book of counter-examples by Romano and Siegel, 1986, selling for a mere $180 on amazon!) shows why looking for an unbiased estimator of exp(-2θ) is particularly foolish: the only solution is (-1) to the power S [for a single observation]. (There is however a first way to circumvent the difficulty if having access to an arbitrary number of generations from the Poisson, since the Forsythe – von Neuman algorithm allows for an unbiased estimation of exp(-F(x)). And, as a second way, as remarked by Juho Kokkala below, a sample of at least two Poisson observations leads to a more coherent best unbiased estimator.)

an improvable Rao–Blackwell improvement, inefficient maximum likelihood estimator, and unbiased generalized Bayes estimator

Posted in Books, Statistics, University life with tags , , , , , , , , on February 2, 2018 by xi'an

In my quest (!) for examples of location problems with no UMVU estimator, I came across a neat paper by Tal Galili [of R Bloggers fame!] and Isaac Meilijson presenting somewhat paradoxical properties of classical estimators in the case of a Uniform U((1-k)θ,(1+k)θ) distribution when 0<k<1 is known. For this model, the minimal sufficient statistic is the pair made of the smallest and of the largest observations, L and U. Since this pair is not complete, the Rao-Blackwell theorem does not produce a single and hence optimal estimator. The best linear unbiased combination [in terms of its variance] of L and U is derived in this paper, although this does not produce the uniformly minimum variance unbiased estimator, which does not exist in this case. (And I do not understand the remark that

“Any unbiased estimator that is a function of the minimal sufficient statistic is its own Rao–Blackwell improvement.”

as this hints at an infinite sequence of improvement.) While the MLE is inefficient in this setting, the Pitman [best equivariant] estimator is both Bayes [against the scale Haar measure] and unbiased. While experimentally dominating the above linear combination. The authors also argue that, since “generalized Bayes rules need not be admissible”, there is no guarantee that the Pitman estimator is admissible (under squared error loss). But given that this is a uni-dimensional scale estimation problem I doubt very much there is a Stein effect occurring in this case.

best unbiased estimators

Posted in Books, Kids, pictures, Statistics, University life with tags , , , , , , , , , , , , on January 18, 2018 by xi'an

A question that came out on X validated today kept me busy for most of the day! It relates to an earlier question on the best unbiased nature of a maximum likelihood estimator, to which I pointed out the simple case of the Normal variance when the estimate is not unbiased (but improves the mean square error). Here, the question is whether or not the maximum likelihood estimator of a location parameter, when corrected from its bias, is the best unbiased estimator (in the sense of the minimal variance). The question is quite interesting in that it links to the mathematical statistics of the 1950’s, of Charles Stein, Erich Lehmann, Henry Scheffé, and Debabrata Basu. For instance, if there exists a complete sufficient statistic for the problem, then there exists a best unbiased estimator of the location parameter, by virtue of the Lehmann-Scheffé theorem (it is also a consequence of Basu’s theorem). And the existence is pretty limited in that outside the two exponential families with location parameter, there is no other distribution meeting this condition, I believe. However, even if there is no complete sufficient statistic, there may still exist best unbiased estimators, as shown by Bondesson. But Lehmann and Scheffé in their magisterial 1950 Sankhya paper exhibit a counter-example, namely the U(θ-1,θ-1) distribution:

since no non-constant function of θ allows for a best unbiased estimator.

Looking in particular at the location parameter of a Cauchy distribution, I realised that the Pitman best equivariant estimator is unbiased as well [for all location problems] and hence dominates the (equivariant) maximum likelihood estimator which is unbiased in this symmetric case. However, as detailed in a nice paper of Gabriela Freue on this problem, I further discovered that there is no uniformly minimal variance estimator and no uniformly minimal variance unbiased estimator! (And that the Pitman estimator enjoys a closed form expression, as opposed to the maximum likelihood estimator.) This sounds a bit paradoxical but simply means that there exists different unbiased estimators which variance functions are not ordered and hence not comparable. Between them and with the variance of the Pitman estimator.