Thread View: sci.math.stat
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Started by wsimpson
Mon, 04 Jan 1993 23:43
quantile plots
Author: wsimpson
Date: Mon, 04 Jan 1993 23:43
Date: Mon, 04 Jan 1993 23:43
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1035 bytes
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. I am not especially interested in the particular example of uniform distribution. I am interested in the general problem of fitting curves to quantile plots. Thanks very much for any help. Bill Simpson
Re: quantile plots
Author: T. Scott Thompson
Date: Wed, 06 Jan 1993 16:34
Date: Wed, 06 Jan 1993 16:34
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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
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