Fitting distributions and comparing tails Suppose I have some data points (0.6695, 0.5968, 0.7641, 0.7252, and 0.7779) and want to fit different distributions to this dataset, say, log-normal and log-t (with different df), and then compare the tails of these distributions using graphics. How this can be done in R?
 A: Fitting a log-$t$ distribution is not recommended for small samples. However, I am going to present a R code for estimating the parameters $(\mu,\sigma,\nu)$ using MLE.
# Your data
data = c(0.6695, 0.5968, 0.7641, 0.7252, 0.7779)
n=length(data)

# -log-likelihood of (mu,sigma,nu)
lt = function(par){
if(par[2]>0&par[3]>0) return( -n*(log(gamma(0.5*(par[3]+1)))    -log(gamma(0.5*par[3]))-0.5*log(par[3])) + 0.5*(par[3]+1)*sum(log(1+ (log(data)-par[1])^2/(par[2]^2*par[3]))) + n*log(par[2])  )
else return(Inf)
}

# MLE obtained numerically
optim(c(0,0.1,10),lt)

The estimators are $(\hat\mu,\hat\sigma,\hat\nu)=(-0.35162295 ,0.09719402,342.22742778)$. As you can see, $(\hat\mu,\hat\sigma)$ are very similar to those obtained for the lognormal  in your previous question. In addition, the estimator of the degrees of freedom is very large $(342.22742778)$, suggesting that the lognormal model is reasonable in this case.
The tails of the log-$t$ are always heavier than those of the lognormal, therefore there is not much point on comparing them graphically. But, again, the MLE suggest that the lognormal is reasonable.
In order to compare these models you can use the AIC (which again might be affected due to the small sample size), which in this case favours the lognormal model.
library(MASS)

# -loglikelihood for the lognormal model
lln = function(par){
if(par[2]>0) return( - sum(log(dlnorm(data,par[1],par[2]))) )
else return(Inf)
}

# AIC for the lognormal model   
optim(c(0,0.1),lln)$value + 2*2

# AIC for the log-t model
optim(c(0,0.1,10),lt)$value + 2*3

Best regards.
A: If you fit a distribution to a small sample, a good fit will only really measure how well the body of the distribution fits the data because you won't see much of the tails of a distribution unless you have a very large dataset.  Distributions can look very similar in the body and yet very different in the tails. Comparing the tails of the distributions, as Procrastinator pointed out, can pretty much be done by knowing the particular distributions being selected without knowing anything about the data and in a small sample what you know from the data won't tell much if anything about the tails.
So I don't see where comparing the tails of competing distributions makes any sense in data analysis.
