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Geoffrey,

You wrote:

> My "proof", as you were kind enough to call it, has nothing to say
> regarding the size of the 'hat.


I did not suggest that it had. But your proof leads to the conclusion
that there is a 0.25 probability that the true position lies within the
cocked hat _regardless_ of the size of that 'hat. That is
counter-intuitive. Either there is a reason why what seems intuitive is
wrong or else there is some error in your proof that I cannot see. I was
  wondering whether you could dismiss the latter possibility by pointing
to the reason for disregarding my argument that the number cannot always
be 0.25.

> However, I think you are confusing yourself by then talking about
> "confidence circles" and Most Probable Positions in relation to 'hats.
> Getting from the statistics about the size and distribution of 'hats to the
> statistics of Most Probable Positions and circles of confidence is not
> trivial as they are quite different animals.
>
> I have some expertise with the statistics of groups and grouping where
> bullets are hitting a target. The statistics of MPP's is, I think, quite
> similar to finding the centre of a group.
>
> If you fire just one shot at a target and ask yourself where the centre of
> the group is, the best you can do is say it is in the centre of the bullet
> hole. You cannot do statistics on one datum point. But if you have two data
> points, you can do all the statistics in the world.


Maybe not all of them!

But with a cocked hat, we do have the equivalent of two shots. A bullet
hole approximates to a zero-dimensional point and you need two to do any
statistics. LOPs are one-dimensional and you need two to get a fix of
any kind but, once you have three LOPs you should be in the same
position as having two bullet holes to work with.


> Similarly, if you only have one 'hat, then placing the MPP in the centre of
> it is the best you can do.


That is indeed the MPP, if the 'hat is equilateral. But the size of the
'hat also allows estimation of the confidence intervals around the MPP.
(Indeed, the equations were posted early on in this thread.) Again,
three LOPs are enough to start playing with stats, even though
additional ones would, of course, allow greater precision.

Trevor Kenchington


--
Trevor J. Kenchington PhD                         Gadus@iStar.ca
Gadus Associates,                                 Office(902) 889-9250
R.R.#1, Musquodoboit Harbour,                     Fax   (902) 889-9251
Nova Scotia  B0J 2L0, CANADA                      Home  (902) 889-3555

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