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Steven Wepster wrote:

>Let me add one cent. I don't understand what _the_ _true_ form of a
>confidence region should mean.  IMO a confidence region is a region that
>contains the MPP with a specified probability, often taken as 95%. It can
>be circular, elliptical, square, heart-shaped or whatever you prefer, for
>whatever theoretical, practical, or other reasons are important to you.
>

Steven,

My use of statistical terminology is probably not what it should be
(comes of picking up stats as a biologist with little formal training!)
and I have no idea of whatever conventions may have developed in the
application of statistics to navigation. My lack of the right words
should not, however, hide the existence of the thing I was trying to
refer too.

Given an MPP and sufficient information on the associated uncertainties,
one could draw any number of boundaries around it, each defined so that
there was a 95% probability (or any other probability one might select)
that the true position lay within the boundary. Each boundary would
encompass some areas where there was a high probability that the true
position would lie and other areas where there would be a low
probability. But there would only be one such boundary that not only met
the criterion of a 95% probability of the true position lying within it
but also had, at all points along the boundary, an equal probability of
the true position lying at that point. That boundary (and only that
boundary) would encompass the smallest area that had a 95% probability
of containing the true position. It would also be the only boundary
which did not exclude some areas where there was a relatively high
probability of the true position lying -- all other boundaries excluding
small high-probability areas but taking in larger low-probability ones
to maintain the 95% encompassed.

It is the area surrounded by that one boundary (for any chosen
probability of enclosing the true position) which is also a "contour"
(or isopleth, if you prefer) of constant probability which I meant by
the "true" confidence region. I would contend that it, rather than the
area with some other boundary, is the confidence region that we would
like to have when using an MPP in practical navigation.

The problem, of course, is that we do not have complete information on
the shape of the boundary and thus must fall back on estimates from the
available data. Moreover, we do not (at least, I do not) go to sea with
the sort of computing power needed to make statistically-optimal
estimates from such data as we do have. Hence, in practical
applications, we fall back on assuming that our true position is
somewhere inside a cocked hat, a circle around the MPP, an ellipse or
some other simple figure that we can plot. My only reason in seeking to
go beyond that, in a discussion of theory, is to explore the reliability
or otherwise of such simple approximations to a complex topic. If, for
example, George was right that the true position falls within the cocked
hat only 25% of times (and I have yet to see his defence of that
conclusion), navigators should know not to rely on that old rule of
thumb. The same would be true of computed ellipses if they fail us too
often.


Trevor Kenchington

   

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