Comparing to zero

Eric Maris maris at NICI.RU.NL
Fri Jun 24 11:08:11 CEST 2005

Hi Fieldtrippers,

Some comments to Robert's contribution:

> The test that Vladimir refers to does not test the hypothesis of the
> expected value of the mean being zero, but it tests the null  hypothesis
> of exchangeability of the sign of the data. If it is very  improbable (say
> p<0.05)  that the sign can be exchanged, you would  accept the alternative
> hypothesis which states that the random  distribution of your data is not
> symmetric with respect to zero. That  is already very close to what
> Vladimir is interested in.

Sounds good. What you propose is a randomization test for symmetry around
zero. This null hypothesis implies the hypothesis of interest (expectation
equal to 0) but unfortunately not the other way around, as Robert also
notes. However, by proper choice of your test statistic one can try to make
it insensitive to other deviations from the null hypothesis (symmetry around
0) than a nonzero expectation.

A terminological remark: exchangeability refers to an array and not to a
single element, such as the sign of the data. What Robert means is usually
denoted as "the sign has a Bernoulli distribution with probability 0.5".
This ("the sign etc.") is a property of random variable that is symmetric
around zero.

> What you could do, after finding with the randomization test that the
> distribution is not symemtric around zero, is
> 1) assume that that assymetry is caused by a shift in the mean and  not
> test it (which seems fair enough to me in quite some cases, e.g.  when you
> know that the random distribution is close to a normal  distribution), or
> 2) perform a second test in which you test whether the distribution  of
> the random variable is symmetric around its estimated mean (i.e.  repeat
> the same test as done before, after subtracting the mean of  the random
> variable).

This second test has much intuitive appeal, but it still has to be shown
that it controls the false alarm rate. The null hypothesis of interest is
the following: the distribution is symmetric around an unknown expectation
X. Because X is unknown, one may consider replacing it by its sample mean,
as Robert proposes, but trick may spoil your false alarm rate control.

Eric Maris

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