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From |
"Verkuilen, Jay" <JVerkuilen@gc.cuny.edu> |

To |
<statalist@hsphsun2.harvard.edu> |

Subject |
RE: Re: st: Dependent continuous variable with bounded range |

Date |
Thu, 17 Apr 2008 17:09:51 -0400 |

Nick Cox wrote: >> >>I wrote: In point of fact, the variance function of the beta distribution is the same as the binomial, up to an additional free scale constant. Both are proportional to E(X)(1-E(X)). You would definitely want to free up the scale parameter for continuous data, though. << Good point. In fact a little thought shows that if a variable is bounded on [0,1] then as the mean goes to either 0 or 1 the variance must go to 0, because the mean can only approach 0 or 1 if all values approach 0 or 1. That is true regardless of whether the variable is discrete or continuous. (Same is true for any finite bounds.) << The classic leaf blotch data analyzed by Wedderburn used a variance function E(X)^2 (1 - E(X))^2. There are other distributions in the unit interval, e.g., Barndorff-Nielsen and Jorgensen's simplex distribution or the Johnson SB distribution, which have different variance functions. (The simplex distribution is generated from the inverse Gaussian in the same basic way the beta is from the gamma; the Johnson SB is similarly generated from the lognormal.) I am not aware of a distribution with Wedderburn's variance function, but I suppose one could always manufacture it. As for finitely bounded data, the two distinguishable cases are: (1) Known bounds. In this case, we are free to use these bounds to rescale to any convenient interval, provided our statistical model does not make use of the particular values of the boundaries (as nearly all do not). (2) Unknown bounds. In this case, we have a markedly more complex estimation problem. The article by Nguyen found in Handbook of Beta Distribution and Its Applications edited by A.K. Gupta and S. Nadarajah (2004, CRC Press) discusses this situation for the univariate case. It is not pretty. I'm betting it would get markedly worse as a regression model. Because everything I deal with is in category (1), I've not thought much about category (2). Jay * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

**References**:**RE: Re: st: Dependent continuous variable with bounded range***From:*"Nick Cox" <n.j.cox@durham.ac.uk>

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