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- From: thompson@atlas.socsci.umn.edu (T. Scott Thompson)
- Newsgroups: sci.math.stat
- Subject: Re: quantile plots
- Message-ID: <thompson.726338057@daphne.socsci.umn.edu>
- Date: 6 Jan 93 16:34:17 GMT
- References: <4JAN93.17433200@uwpg02.uwinnipeg.ca>
- Sender: news@news2.cis.umn.edu (Usenet News Administration)
- Reply-To: thompson@atlas.socsci.umn.edu
- Organization: Economics Department, University of Minnesota
- Lines: 40
- Nntp-Posting-Host: daphne.socsci.umn.edu
-
- 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
-