**W**hen reading an entry on The Chemical Statistician that a sample median could often be a choice for a sufficient statistic, it attracted my attention as I had never thought a median could be sufficient. After thinking a wee bit more about it, and even posting a question on cross validated, but getting no immediate answer, I came to the conclusion that medians (and other quantiles) cannot be sufficient statistics for arbitrary (large enough) sample sizes (a condition that excludes the obvious cases of one & two observations where the sample median equals the sample mean).

In the case when the support of the distribution does not depend on the unknown parameter θ, we can invoke the Darmois-Pitman-Koopman theorem, namely that the density of the observations is necessarily of the exponential family form,

to conclude that, if the natural sufficient statistic

is minimal sufficient, then the median is a function of S, which is impossible since modifying an extreme in the *n>2* observations modifies S but not the median.

In the other case when the support does depend on the unknown parameter θ, we can consider the case when

where the set indexed by θ is the support of f. In that case, the factorisation theorem implies that

is a 0-1 function of the sample median. Adding a further observation y⁰ which does not modify the median then leads to a contradiction since it may be in or outside the support set.

Incidentally, if an aside, when looking for examples, I played with the distribution

which has θ as its theoretical median if not mean. In this example, not only the sample median is not sufficient (the only sufficient statistic is the order statistic and rightly so since the support is fixed and the distributions not in an exponential family), but the MLE is also different from the sample median. Here is an example with n=30 observations, the sienna bar being the sample median:

## Valen in Le Monde

Posted in Books, Statistics, University life with tags blogging, comments, False positive, Le Monde, Monsanto, p-values, Passeur de Sciences, statistical significance, UMPB test, uniformly most powerful tests, Valen Johnson on November 21, 2013 by xi'anValen Johnson made the headline inLe Monde, last week. (More precisely, to the scientific blogPasseur de Sciences. Thanks, Julien, for the pointer!) With the alarming title of “Une étude ébranle un pan de la méthode scientifique”(A study questions one major tool of the scientific approach). The reason for this French fame is Valen’s recent paper in PNAS,Revised standards for statistical evidence, where he puts forward his uniformly most powerful Bayesian tests (recently discussed on the ‘Og) to argue against the standard 0.05 significance level and in favour of “the 0.005 or 0.001 level of significance.”While I do plan to discuss the PNAS paper later (and possibly write a comment letter to PNAS with Andrew), I find interesting the way it made the headlines within days of its (early edition) publication: the argument suggesting to replace .05 with .001 to increase the proportion of reproducible studies is both simple and convincing for a scientific journalist. If only the issue with p-values and statistical testing could be that simple… For instance, the above quote from Valen is reproduced as “an [alternative] hypothesis that stands right below the significance level has in truth only 3 to 5 chances to 1 to be true”, the “truth” popping out of nowhere. (If you read French, the 300+ comments on the blog are also worth their weight in jellybeans…)## Share:

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