From: mwande@qal.qal.berkeley.edu (Mike Anderson)
Newsgroups: sci.math.stat
Subject: RE: modelling distributions
Date: 12 Nov 1992 00:55:07 GMT
Organization: University of California, Berkeley
Distribution: world
NNTP-Posting-Host: qal.qal.berkeley.edu

mwande@graunt.qal.berkeley.edu (Mike Anderson) writes:

>>Suppose I have data on two variables - age and income of a person. I would like
>>to estimate income curves by age using spline regression, but my problem is
>>this: to protect the identity of the individuals, incomes were topcoded, so
>>that any person reporting incomes of > $100,000 gets coded at $100,000. I am
>>interested in getting very accurate estimates, and given the long right tail
>>of the income distribution, these truncated values may be throwing off my
>>estimates quite a bit.
>>       So I would like to replace the truncated values with my own "tail". My
>>question is, how do I go about modelling the income distribution and tacking
>>on my own tail? Off the top of my head, I would precede thusly: If there are

>>(stuff deleted)

>I assume that you apply this procedure separately for each age group.
>I also assume that you scale the log of income (rather than income
>itself) to have mean zero, since a log-normal variable cannot have
>mean zero.  I have no idea why you would want to do this.

Let me give a few more specifics about what I'm doing. Ignore the spline for
the moment - I'm using that to fit a curve to the average income at each age
across age. Suppose, for the purposes of the "tail-fitting", that I am just 
trying to model the income at  a particular age, to get an average. In scaling
the income to mean zero, I had in mind using any number of distributions,
not just a log-normal, to try against the distributions. I guess
I could just set the mean of the model distribution  equal to the population
mean without the censored points, point being that the scaling is arbitrary; 
I was originally think of a normal dist, which is why is was thinking mean 
zero, but then I realized a log normal might have a better tail for income.

thompson@atlas.socsci.umn.edu writes:
>Except for the bit about scaling the log of incomes to mean 0, the
>procedure that you describe is approximately equivalent to what you
>would do to fit a censored log-normal distribution to the data by
>maximum-likelihood.  An ML estimator would not insist that the
>distribution predict the observed number of censored observations
>exactly, nor would it require that the mean of the log of the
>uncensored income variables be zero, since generally this value will
>not give the best fit.

>The ML estimate is optimal if the true uncensored distribution is
>really log-normal, and may be arbitrarily bad otherwise, depending on
>what that unobserved tail really looks like.  On the other hand, if
>you know that the uncensored distribution is log-normal then you don't
>need to use splines to estimate this distribution.  Catch-22.

As above, I'm not using the spline on the distributions I'm modelling,
rather on the average at each age after adjusting the age-specific distribu-
tions for the censored observations. The distirbution  across age is nowhere
near a log-normal, or any typical distribution I am aware of.

>The fact of the matter is that there is no "right" way to do this.
>You simply have no information in your data about what the upper tail
>looks like.  Any attempt to "simulate" the upper tail is as arbitrary
>as any other in the absence of additional information.  Hopefully you
>don't really need it for your ultimate purposes.  For example, any
>estimates of mean income produced from this data will be unreliable,
>since mean income cannot be determined without knowing how much the
>Donald Trumps of the world earned.

Suppose I look at outside data sources to estimate the incomes of the
Donald Trumps. How could I incorporate this in the estimation of the various
distribution parameters?

-- 
Mike Anderson    Dept. of Demography   UC Berkeley   mwande@QAL.Berkeley.EDU
"And I would say to those out around the country. 'Take a hard look now.
Don't let that rabbit be pulled out of the hat by one hand and 25 other 
rabbits dumped on you in another.'" - George Bush, 1/24/90 
Newsgroups: sci.math.stat