Re: Lilliefors Test : 40 years

In article <3036213.1209467099549.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
Luis A. Afonso <licas_@xxxxxxxxxxx> wrote:
Why one should prefer the Lilliforss test


An advice (if you allow me) Herman Rubin
Do not make definitive statements before to think closely what the problem is, and do not forget the context, namely that all values are imprecise at Statistical Grounds.
You said:
The Kolmogorov-Smirnov test is a good test if the correct distribution is used. If the population distribution is completely specified, there is no problem. If it is not, the distribution depends on the way parameters are estimated, translated appropriately.
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I add:
The Lillieforss Test has critical values that are 2/3 (approximately) shorter than he K-S ones. In consequence using the latter ones the confidence intervals are INCORRECTLY LARGER than they should be, fail to reject H0 (when we REALY should do so) is MUCH MORE PROBABLE than what the chosen K-S ALPHA VALUE seems to indicate.

The Lilliefors Test tests whether a distribution is
normal, and is in no way intended to obtained confidence
limits for the parameters if normal. Similarly, the
Kolmogorov-Smirnov tests tests for a particular normal;
if one wants confidence intervals for the parameters,
or some of them, use the more classical tests.

A test generally should not be used to test what it is
not intended to test. K-S tests for a particular
distribution; Lilliefors adjusts the critical values
so that the test operates correctly if the parameters
allowed to vary in testing a form are estimated.

So there is a separate set of critical values if
one is testing for a normal with known variance
and unknown mean, known mean and unknown variance,
or both unknown. Similarly for other distributions;
the asymptotics for a double-exponential with known
scale parameter can be explicitly calculated, as
the restriction on the Brownian bridge consists
in setting it to 0 at 1/2, and for the uniform it
only reduces n by the number of endpoints estimated,
but otherwise, it is hard to obtain except by
simulation.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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