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My problem arises from this situation: I have the mean and standard deviation of body-mass index (BMI) for a given population. Contemporary BMI distributions are not close to a normal distribution, i.e., there is a much 'fatter' tail to the right.

Unfortunately, this is where my statistics knowledge ends. What I would like to be able to do, are the following things:

  1. Find a distribution that fits my idea of a BMI distribution more appropriately
    1a.)This distribution would likely need to be bounded (very few people have a BMI <14 or >50)
  2. From this distribution, be able to find the area under the curve for a given value

EDIT: With reference to the area under the curve, I am looking for a CDF, so I can find the proportion of the population for a given value of BMI.

From some initial research, it seems like a beta-distribution is potentially what I need. But these seem to need parameters within bounds of 0 and 1?

This was all very straightforward with a normal distribution, however the normal distribution is not nearly an accurate representation of the distribution of BMIs.

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    $\begingroup$ Why do you need the area "under the curve"? (do you actually mean you need the area to the left of the specified value -- i.e. the cdf? What for?) $\endgroup$ – Glen_b Aug 9 '16 at 9:14
  • $\begingroup$ I would say that the criterion for using a bounded distribution should be based more on the fact that it is impossible to have a BMI <=0, not that very few people have a BMI <14 or >50 (many distributions would allow for very low frequencies at some values). $\endgroup$ – mkt - Reinstate Monica Aug 9 '16 at 9:31
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Body Mass Index data typically look asymmetric (right-skewed). In many cases the tails look normal, but in other cases the right tail is heavier. There are numerous families of parametric distributions that can capture this behaviour (skew-symmetric distributions and two-piece distributions). I recommend the following paper for a review on these:

On Families of Distributions with Shape Parameters, M.C. Jones (2014). International Statistical Review.

Nonparametric density estimators can also be easily computed for this kind of data sets as follows (which also shows how to fit a skew normal density):

library(sn)
# Simulated data
bmi = rsn(1000,20,3,5)

# histogram and kernel density estimator
hist(bmi,probability=T)
points(density(bmi),type="l",col="red",lwd=2)

# parametric approximation using skew normal

log.lik = function(par){
if(par[2]>0) return( -sum(dsn(bmi,par[1],par[2],par[3],log=T)))
else return(Inf)
}

# MLE 
MLE = optim(c(20,3,5),log.lik)$par

# Fitted skew normal density
fit.den = Vectorize(function(x) dsn(x,MLE[1],MLE[2],MLE[3]))

hist(bmi,probability=T)
curve(fit.den,18,30,add=T,col="blue",lwd=2)
  • Regarding your comment on "the proportion of the population for a given value of BMI", what you need is a quantile of the fitted distribution, rather than the ROC curve.
  • The Beta distribution is not appropriate for modelling BMI data since its support is (0,1), while BMI data is positive, and typically way beyond (0,1).
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