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Topic: A "plausible range" for a random variable
Replies: 9   Last Post: Jun 11, 2013 7:42 PM

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 deltaquattro@gmail.com Posts: 77 Registered: 7/21/06
Re: A "plausible range" for a random variable
Posted: Jun 8, 2013 8:36 AM

> 1)
> Selecting Lower and Upper limits, L and U, as you notice, is
> usually done symmetrically (if not one-tailed). That is done,
> mainly, for lack of any other good reason to outweigh the
> equal emphasis on each end.
>
> Statistical estimation theory occasionally looks at the
> "narrowest" CI. That is *the* important characteristic
> of one-tailed tests, determining UMP (Uniformly Most
> Powerful). Because tails can be asymmetrical, no two-tailed
> test is UMP.
>
> Decision theory would suggest that you apply a loss-function
> to determine what degree of asymmetry might apply -- I
> was intrigued, long ago, by the suggestion that the "power" of
> standard research might be improved by splitting the conventional
> 5% into 4% at the "expected" end and 1% at the other end,
> for a gain in general power without losing the right to report
> stronger effects in the opposite direction. I read that at least
> 30 years ago, so you can see that the idea never caught on.
>
> 2)
> A parametric approach to L and U for Extreme Values is not
> going to be at all efficient. What is used for estimation is what
> your bootstrapping would converge to: The CI based for L
> (or U) based on rank-order in the original sample.
>
> Poisson consideration gives a good approximation for small
> proportions. This is applied for your N=2000, 2 1/2%, as follows.
>
> Rank 50 is the point estimate of L. The +/- 2SD range for Poisson
> can be estimated as ( Square(Sqrt(L) - 1), Square(Sqrt(L) + 1) )
>
> The square root of 50 is about 7; the square of 6 is 36, and the
> square of 8 is 64. That gives (approximately) the CI for L=50
> is (37, 65).
>
> --
> Rich Ulrich

Great! Thanks a lot, Rich, that's precisely what I needed. You mention that this Poisson approximation is valid for small proportions: what about U, i.e., the 97.5-th percentile? This is a "big" proportion, I mean, it's close to 1. Can I still claim that the +/- 2SD CI for U=1950 is given by [(sqrt(U)-1)^2, (sqrt(U)+1)^2]? Or do I need another formula?
Also, which is the formula for a general C.I., like 90% or 99% or 99.9%, etc.? When using the normal approximation, one substitutes the value 1.96 with the corresponding percentile of the normal distribution, z_{1/2+alpha/2} where alpha={.90, .95, .99,...} etc. I'm not familiar with this Poisson approximation, however, so I don't know how to proceed in this case.

Thanks again,

Best Regards

Sergio Rossi