Archive for inequalities

about French inequalities [graphs]

Posted in Books, Kids, pictures with tags , , , , , , , on December 16, 2018 by xi'an

A first graph in Le Monde about the impact of the recent tax changes on French households as a percentage of net income with negative values at both ends, except for a small spike at about 10% and another one for the upper 1%, presumably linked with the end of the fortune tax (ISF).

A second one showing incompressible expenses by income category, with poorest households facing a large constraint on lodging, missing the fraction due to taxes. Unless the percentage is computed after tax.

A last and amazing one detailing the median monthly income per socio-professional category, not because of the obvious typo on the blue collar median 1994!, but more fundamentally because retirees have a median income in the upper part of the range. (This may be true in most developed countries, I was just unaware of this imbalance.)

life and death along the RER B, minus approximations

Posted in Statistics, Travel with tags , , , , , , , , , , , , , , on June 30, 2015 by xi'an

viemortrerbWhile cooking for a late Sunday lunch today [sweet-potatoes röstis], I was listening as usual to the French Public Radio (France Inter) and at some point heard the short [10mn] Périphéries that gives every weekend an insight on the suburbs [on the “other side’ of the Parisian Périphérique boulevard]. The idea proposed by a geographer from Montpellier, Emmanuel Vigneron, was to point out the health inequalities between the wealthy 5th arrondissement of Paris and the not-so-far-away suburbs, by following the RER B train line from Luxembourg to La Plaine-Stade de France…

The disparities between the heart of Paris and some suburbs are numerous and massive, actually the more one gets away from the lifeline represented by the RER A and RER B train lines, so far from me the idea of negating this opposition, but the presentation made during those 10 minutes of Périphéries was quite approximative in statistical terms. For instance, the mortality rate in La Plaine is 30% higher than the mortality rate in Luxembourg and this was translated into the chances for a given individual from La Plaine to die in the coming year are 30% higher than if he [or she] lives in Luxembourg. Then a few minutes later the chances for a given individual from Luxembourg to die are 30% lower than he [or she] lives in La Plaine…. Reading from the above map, it appears that the reference is the mortality rate for the Greater Paris. (Those are 2010 figures.) This opposition that Vigneron attributes to a different access to health facilities, like the number of medical general practitioners per inhabitant, does not account for the huge socio-demographic differences between both places, for instance the much younger and maybe larger population in suburbs like La Plaine. And for other confounding factors: see, e.g., the equally large difference between the neighbouring stations of Luxembourg and Saint-Michel. There is no socio-demographic difference and the accessibility of health services is about the same. Or the similar opposition between the southern suburban stops of Bagneux and [my local] Bourg-la-Reine, with the same access to health services… Or yet again the massive decrease in the Yvette valley near Orsay. The analysis is thus statistically poor and somewhat ideologically biased in that I am unsure the data discussed during this radio show tells us much more than the sad fact that suburbs with less favoured populations show a higher mortality rate.

a probabilistic proof to a quasi-Monte Carlo lemma

Posted in Books, Statistics, Travel, University life with tags , , , , , on November 17, 2014 by xi'an

As I was reading in the Paris métro a new textbook on Quasi-Monte Carlo methods, Introduction to Quasi-Monte Carlo Integration and Applications, written by Gunther Leobacher and Friedrich Pillichshammer, I came upon the lemma that, given two sequences on (0,1) such that, for all i’s,

|u_i-v_i|\le\delta\quad\text{then}\quad\left|\prod_{i=1}^s u_i-\prod_{i=1}^s v_i\right|\le 1-(1-\delta)^s

and the geometric bound made me wonder if there was an easy probabilistic proof to this inequality. Rather than the algebraic proof contained in the book. Unsurprisingly, there is one based on associating with each pair (u,v) a pair of independent events (A,B) such that, for all i’s,

A_i\subset B_i\,,\ u_i=\mathbb{P}(A_i)\,,\ v_i=\mathbb{P}(B_i)

and representing

\left|\prod_{i=1}^s u_i-\prod_{i=1}^s v_i\right| = \mathbb{P}(\cap_{i=1}^s A_i) - \mathbb{P}(\cap_{i=1}^s B_i)\,.

Obviously, there is no visible consequence to this remark, but it was a good way to switch off the métro hassle for a while! (The book is under review and the review will hopefully be posted on the ‘Og as soon as it is completed.)