Dr.  Sir/Madam

I have been using the package fExtremes for the statistical analysis of
financial time series.  I find the package very useful. 

 

My query concerns the function hillPlot and, in particular, the
estimation of the Hill index. 

 

If I am not wrong the formula for the Hill index (in latex form) is
given by:

 

Gamma(k) = 1/k sum_{i=1}^{k}[ ln(x_{n-i+1})-ln(x_{n-k})]   

 

Where k+1 is the threshold and x are the ordered statistics. 

The corresponding formula in the R function is:

 

loggs = log(ordered)

avesumlog = cumsum(loggs)/(1:n) 

xihat = c(NA, (avesumlog - loggs)[2:n]) -

 

the second code line, as far as I can understand (probably I am wrong),
is equivalent to

 

Gamma(k) = 1/k sum_{i=1}^{k}[ ln(x_{n-i})-ln(x_{n-k})]   

 

That is the first term in the summation is ln(x_{n-i}) rather than
ln(x_{n-i+1}). 

 

Should it be   xihat = c(NA, (avesumlog[-1] - logs[-n]) ?

 

I apologize for the long email .

 

Best,

Efthymios

 


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