Peter Flom wrote:
>
> I quoted Brian. Everitt as saying
>
> > More complex answer: The definition of a confidence interval is
> (from
> > The Cambridge Dictionay of Statistics, by B.S. Everitt)
> > "A range of values, calculated from the sample observations, that
> are
> > believed, with a particular probablity, to contain the true
> parameter
> > value. A 95% CI, for example, implies that were the estimation
> process
> > repeated again and again, then 95% of the calculated intervals would
> be
> > expected to contain the true parameter value. Note that the stated
> > probablity level refers to properties of the interval and not to the
> > parameter itself, which is not considered a random variable....."
>
> And Robert Dawson replied
> <<<
> And this is wrong. The probability level refers to the method
> of
> generating the interval, not to the interval generated.
>
> A valid (if, perhaps, not optimal) 95% CI method may generate
> an
> interval that is known to contain the true parameter value, or one
> that
> is known to have missed.
> >>>
>
> I am not sure that I follow you here.....Do you have an definition of a
> CI which
> says what you mean?
Yes, Everitt's definition a couple lines above! The bit about 95% of
the calculated intervals containing the value is correct. However, this
property does not "rub off" onto the intervals so generated. Once it's
generated, there is nothing about the interval [which is just an ordered
pair of numbers] involving 95% confidence.
To take a very contrived example, if I take a 20-sided die, roll it,
and if I get 7 return an empty interval, else (-infinity, infinity),
that's a 95% confidence interval estimator. A nonstandard one, and a bad
one by any known criterion, but the confidence level is an exact 95%.
Of course, if I just give you the interval, you cannot tell whether I
used that process or the corresponding 90% or 50% confidence version.
(If you don't like infinite or empty intervals, think about confidence
intervals for a proportion and use [0,1] or [2,3].)
Really, we ought to scrap the phrase "95% confidence interval" and
replace it by "95% confidence interval estimator" and/or "interval
generated by a 95% confidence interval estimator". But we won't because
life is short and people are lazy.
-Robert
PS: I know this is a quibble, but if we can't rely on dictionary
writers to be models of pedantry, who _can_ we rely on? (Present company
excluded.)
.
.
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