Re: consistency

On 13 Nov, 02:00, Randy Yates <ya...@xxxxxxxx> wrote:

At this point, all I can do is say "OK, if you say so." I'm
not sure how to show this to myself rigorously (analytically),
so I guess I'll just shutup...

Don't. Don't shut up.

It doesn't matter what the analysis says. The accepted use[*]
of the term 'consistent estimator' is exactly what I
already said: That the estimator is unbiased *and* the
variance vanishes when the number of samples become large.

[[*] I don't use the term 'definition' since I haven't found
anything that qualifies as a definition in the textbooks;
that's probably because the term is considered to be so
trivial everyone are expected to know. ]

So when the OP asks "what does it mean that an estimator
is unbiased but not consistent" the answer is that "the
estimator's variance doesn't vanish as the number of data
increases."

You just can't analyze your way out of that. The estimate
for the mean is either biased or unbiased. The estimate
for the variance either vanishes or doesn't when the number
of samples increases to infinity.

The mean doesn't govern the asymptotic behaviour of the
variance, so you can tell nothing about the asymptotic
behaviour of the variance by analyzing the mean.
Which is the very reason why one uses *both* mean *and*
variance to characterize statistics. (If they were that
closely interconnected, one would not need to estimate
both.) This is statistics 101.

One example of a non-consistent estimator the periodogram,
where the mean is

E[periodogram(f,N)] = Power Spectrum Density(f)

where periodogram(f,N) means "periodogram coefficient at
freqency f based on N data points". I am sure you have
seen somewhere (or know where to look up) that

Var(periodogram(f,N)) = PSD(f)^2

That is, the variance doesn't depend on the number of
samples so it will not improve (that is, reduce) as
the number of samples increases.

So the periodogram is not a consistent estimator for
the PSD. Which is the direct reason why there are so
many roundabout, awkward ways to estimate the PSD.

Randy, I know you have taken a few classes and put
in a lot of work over the past few years, so I am
confident that you know where to look to check these
things out for yourself.

As you do, and grow more confident that all that work
you invested in the classes actually paied off, one
unfortunate side effect is that you might start
evaluating what you hear from colleagues and peers,
and you will come to see that not everything they say
make sense. That's OK, no one can be right every
time. The difficult part is how people relate to this
simple fact of life.

Some people are just too stupid or too stubborn to
realize that they need to learn, and refuse to go back
to the basics more or less as a matter of principle.
There's nothing you can do about that, except map out
who those people might be.

Such people will inhibit you in your struggles to get
a pay-off for you hard-earned education. So you need
to know what to look for, in order to identify (and
maybe handle) them.

Some common traits:

- They have a carreer in a different field than
where they have their education or base training
- They very often have a high academic degree
(PhD) in a more 'aristocratic' field than where
they have their carreer
- They may have decades of seniority (but in
administrative positions)
- They hold hold positions of percieved authority
- They have never been challenged or had to defend
a position in professional matters
- They may have been over-promoted and try to compensate
for professional insecurity

Rune
.

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