🚀 go-pugleaf

wsimpson@uwpg02.uwinnipeg.ca writes:

>I am wondering about the statistical treatment of quantile plots.
>Suppose we have a quantile plot that looks linear.  This reveals
>a uniform distribution.

>My question is this: how can we fit a line to the points?
>Linear regression won't do since the observations in a quantile
>plot are not independent.  I am particularly interested in
>estimating parameters of the distribution from the best fitting line
>and finding confidence intervals.

The quantiles of a distribution completely determine the distribution
and vice versa.  Likewise, a complete set of sample quantiles is
essentially a list of the order statistics of the sample, and is
equivalent to the empirical distribution function.

So your question is essentially equivalent to "How can we fit a
probability model to a data sample?"  It is hard to think of a broader
question within the field of statistics!  You will have to be much
more specific about what are the objectives of your analysis and what
else you know about the problem before you can have any hope of
obtaining a reasonable, non-trivial answer to your question.

If you have a parametric probability model in mind (the uniform model,
for example) then the problem may be solvable (depending on how
complete a set of quantiles are available) using maximum likelihood
methods.  The specifics depend on the probability model assumed.

You say that you want to "fit a line" to the quantile points.  This is
only a reasonable procedure if the data are known to have a uniform
distribution.  The true quantile plot for all other distributions will
be nonlinear.  On the other hand, if you know that the distribution
really is uniform, then the maximum likelihood estimate of the true
quantile plot is the line connecting the minimum and maximum observed
quantile points.
--
T. Scott Thompson              email:  thompson@atlas.socsci.umn.edu
Department of Economics        phone:  (612) 625-0119
University of Minnesota        fax:    (612) 624-0209