Saturday 8 November 2014

Counterexamples to a Central Limit Theorem and a Weak Law of Large Numbers for capacities

P. Terán (2015). Statistics and Probability Letters 96, 185-189.


This note shows that the main results in a former SPL paper by Chareka are wrong. I approached Chareka with a counterexample to his central limit theorem and he didn't take it very well. He said the example was `pathological' and spoke to me quite dismissively saying that he and I were `not even in the same page'. He sent me an eminently reasonable (negative) referee report from SPL as an example of the wrongness I was incurring in. He somehow was able to overturn the report, I can't see how, and have his paper published. He said that I should accept his results since they had been checked by some of the world's leading experts in measure theory (veridical!!!), at which point I felt so deeply offended by the argument of authority that I just cut all communication.

Some time later I found a counterexample to the other main result in his paper, and said: Well, the world needs to know.

The nice thing about this short note is that is shows that ordinary intuitions about probabilities are misleading when studying the more general framework of capacities.


Up the line: 
·Laws of large numbers without additivity (2014).
·Non-additive probabilities and the laws of large numbers (plenary lecture 2011).

Down the line:

Some related papers have been submitted or are under preparation.
 

To download, click on the title or here.

Monday 24 February 2014

Strong consistency and rates of convergence for a random estimator a fuzzy set

P. Terán, M. López-Díaz (2014). Computational Statistics and Data Analysis 77, 130-145.


This paper is actually a continuation of work in my doctoral dissertation, which is why it is joint work with my Ph.D. advisor. The paper was in preparation when I moved to Zaragoza and, with the distance, it was never completed. Fast forward some years, we went back to working on it, appending to the theoretical results some interesting simulations and, eventually, an example with real data.

It is about approximating an unknown fuzzy set from the information in random samples taken from (again randomly sampled) alpha-cuts of the fuzzy set. Through the connection between fuzzy sets and nested random sets, that can also be recast as a problem of estimating a set conditionally on the value of another variable, when the set depends monotonically on the value of that variable.

We give rates of convergence for the approximants as a function of both sample sizes, in several metrics between fuzzy sets. Simulations suggest that sample sizes of 20-30 may be enough for the rate to be reliable. We present an example with breast cancer data, studying the range of the variable `cell size' as a function of `shape compactness', a measure of cell irregularity.


Up the line:
·P. Terán, M. López-Díaz (2004). A random approximation of set valued càdlàg functions. J. Math. Anal. Appl. 298, 352-362. (You can download it for free at the journal's site.)

Down the line:
Nothing so far. One referee wanted us to study more metrics, another different approximation schemes.



To download the paper, click on the title or here