2

I am regularly doing econometrics on various distributions. But I wonder if one should theoretically think of the regularity of this distribution.

To be more, clear, should not one "check" that a distribution is $\mathcal{L}^2$ before doing anything ? And how to do so ? On a particular sample, how to be sure that the standard error of the salaries or the taxes or whatever sample it is makes sense ? What if the sample was in reality derived from a Cauchy distribution ?

Is there any test whether a sample distribution refers to a $\mathcal{L}^2$ distribution ? Is my question even making sense ?

kjetil b halvorsen
  • 77,844
Anthony Martin
  • 1,197
  • 3
  • 11
  • 27
  • What is $\mathcal{L}^2$? – Sycorax Oct 22 '15 at 16:31
  • 1
    A distribution $X$ which density $f_X$ is Square-integrable : $\int_{-\infty}^{\infty} |f_X(x)|^2dx < \infty$ – Anthony Martin Oct 22 '15 at 16:38
  • Thanks! I just learned something new! (I've only see $\mathcal{L}^2$ in reference to Lebesgue measurable sets.) – Sycorax Oct 22 '15 at 16:40
  • 3
    $L^2$ is not any measure of "regularity." Why would you care whether a density function is square integrable? What statistical property in your problem would that be relevant to? Are you perhaps asking how to check whether the variance is finite? (Even the density of a Cauchy distribution is square integrable. In fact, all bounded densities are square integrable, as an easy inequality will show.) – whuber Oct 22 '15 at 18:19
  • If I have a random sample drawn from a Cauchy distribution, I will be able to calculate a mean of my sample, but it will not have any real meaning. If I have a real life sample of the distribution of something, how can I know whether the mean I calculate can be interpreted as the mean of my distribution

    And I believe the expectation of a Cauchy distribution is not defined, and its density is not integrable. $1/x$ for $x \in \mathbb{N}$ is bounded yet the series of $1/x$ does not exists.

     – Anthony Martin Oct 22 '15 at 21:20
  • And more generally, all econometric results rely on the law of large numbers or equivalent results, that assume some regularities of the random variables we face (I would have said that assume they are $\mathcal{L}2$ but maybe my formulation is improper. Have we a way to check that ? – Anthony Martin Oct 22 '15 at 21:45

1 Answers1

0

Your question is not entirely clear, but the real preoccupation seems to be that the distribution might not have a mean (as the case of Cauchy distribution). But there is no hypothesis test of the null hypothesis that the expectations is finite!

That was asked and answered for the variance, not the mean, but the sam e answer works for the mean, see Test for finite variance?

So, I think the real answer is the following: If you have any reason to doubt the existence of the mean, then you should not use statistical methods based on the sample mean, period. See Central limit theorem and the Pareto distribution and links therein.

kjetil b halvorsen
  • 77,844