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    Re: Bias and cocked hats
    From: Bill Noyce
    Date: 2007 Apr 19, 08:47 -0400

    Thanks for running these simulations.  The description is rather
    terse, so let me see if I understand it.  You're trying to answer the
    question, "What is the probability that the cocked hat surrounds the
    true position, when there's some systematic error in addition to
    random scatter in the observations?"  You've chosen a "best case" of
    three observations equally-spaced in azimuth, and the error of each
    observation consists of a fixed bias (common to all three) plus a
    normally-distributed random error.
    
    Summarizing the results, we see that when the bias is zero, the chance
    of the cocked hat surrounding the true position is 25% -- confirming
    what we already knew.  When the bias is  more than 3 sd, the cocked
    hat surrounds the true position in over 99% of the cases.  When the
    bias is 0.8 sd, the cocked hat surrounds the true position in 50% of
    the cases.  It doesn't matter whether the bias is positive or negative
    (I assume differences of 1% are the expected fluctuation from the
    Monte Carlo process).  And the actual magnitudes of the errors don't
    matter (for determining this percentage) -- only their relative sizes.
    
    It would be interesting to see similar results when the observations
    are spaced only 60 degrees in azimuth, so the common error tends to
    push the cocked hat away from the true position.
        -- Bill N.
    
    On 4/18/07, d walden  wrote:
    >
    > For three equally spaced observations with constant bias from -1 min to 1 
    min and independent normally distributed random errors with standard 
    deviation labeled sd results from the previously listed FORTRAN Monte-Carlo 
    simulation using 100,000 runs are as shown.
    >
    > Results can be nondimensionalized by sd.
    >
    >
    >
    >
    >    sd=.5     0.25    1
    >   -1  93%  100%  60%
    >   -0.9  90%  100%  55%
    >   -0.8  85%  100%  50%
    >   -0.7  78%  99%  45%
    >   -0.6  69%  98%  40%
    >   -0.5  60%  93%  36%
    >   -0.4  50%  85%  32%
    >   -0.3  40%  69%  29%
    >   -0.2  32%  50%  27%
    >   -0.1  27%  32%  26%
    >   0  25%  25%  25%
    >   0.1  27%  33%  26%
    >   0.2  33%  50%  27%
    >   0.3  41%  70%  29%
    >   0.4  50%  84%  33%
    >   0.5  60%  93%  36%
    >   0.6  70%  98%  41%
    >   0.7  78%  99%  45%
    >   0.8  84%  100%  50%
    >   0.9  90%  100%  55%
    >   1  93%  100%  60%
    >
    >
    >          ________________________________
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    >
    
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