Re: consistency

On 13 Nov., 02:00, Randy Yates <ya...@xxxxxxxx> wrote:
Andor <andor.bari...@xxxxxxxxx> writes:
Randy Yates wrote:
Rune Allnor <all...@xxxxxxxxxxxx> writes:
On 10 Nov, 14:32, Randy Yates <ya...@xxxxxxxx> wrote:
HardySpicer <gyansor...@xxxxxxxxx> writes:
What is consistency as regards random variables. I know that to be
unbiased we have E[x] = the actual mean but does consistency mean that
an estimator could be unbiased yet not consistent? What in words does
this mean?

Hi Hardy,

[mcdonough] defines an estimator x_n(_y_), where _y_ is a
length-n vector, as "consistent" when it has the property
that, for an small number e,

  lim_{n --> infty} P( | x_n(_y_) - x | > e) = 0.

This is also called "convergence in probability" (see, e.g.,
[viniotis]).

Maybe this is the same as you said, but my understanding
is that a consistent estimator is an estimator which
has the two properties that

1) the estimate is unbiased

I think that's true.

But it's not true (see below).

2) the variance vanishes as the number of samples goes to
   infinity.

I think so, Rune.

This second point is necessary for consistency.

If you assume that the estimator x_n results from estimating some
parameter from n i.i.d. observations and let n->oo, then a consistent
estimator may be biased for all n but the bias has to disappear as n-
oo ("asymptotically unbiased") and the variance has to disappear,
too.

Hi Andor,

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...

It would be very interesting to me, however, if you could explain why
you had to condition that statement with the assumption that the
estimator inputs are iid. Aren't there potentially consistent-but-biased
estimators for other classes of random processes?

Hi Randy

It's more a matter of convenince. You need a sequence of random
variables x_n in the first place (consistency is a convergence result,
so you need a sequence that converges to somewhere). Estimators on
growing sets of "data" (itself random varibles) are typical sequences
that can converge. Assuming the "data" to be i.i.d just makes things
simple.

Regards,
Andor
.

Relevant Pages

  • Re: consistency
    ... Andor writes: ... an estimator could be unbiased yet not consistent? ... as "consistent" when it has the property ... estimator may be biased for all n but the bias has to disappear as n- ...
    (comp.dsp)
  • Re: Find a consistent estimator?
    ... >It is possible estimator, but there may be no simple explicit solution ... >for the MLE to demonstrate consistency. ... >the MLE not to be consistent. ... Herman Rubin, Department of Statistics, Purdue University ...
    (sci.stat.math)
  • Re: consistency
    ... an estimator could be unbiased yet not consistent? ... as "consistent" when it has the property ... estimator may be biased for all n but the bias has to disappear as n- ... An example of an unbiased estimator where the variance does not ...
    (comp.dsp)
  • Re: consistency
    ... an estimator could be unbiased yet not consistent? ... as "consistent" when it has the property ... the variance vanishes as the number of samples goes to ... infinity. ...
    (comp.dsp)
  • Will you account for example the gang, if Elisa dramatically confuses the firm?
    ... Plenty of revivals cheerfully pin the ... Ghassan and Rashid drawed the tragic surfaces because of consistent ... Some warm orthodox oxygen convicts bowels in addition to ... sequence wants, Nelly assesses more than coming, comparative ...
    (rec.arts.drwho.moderated)