[R] Need help..

[yudi@ucd.ie]

> From:          Ajit K Jena <akj at its.hk-r.se>
> Subject:       [R] Need help..
> To:            r-help at stat.math.ethz.ch

> > pareto(1000, 1.06, 5000) -> k
> Pareto mean : theoritical 17666.67 generated 6125.007 
> 
What is guaranteed by the law of large numbers is that
(sum(x_i) - n*mu)= o(n), nothing more. This o(n) term
means o(n)/n goes to zero, but o(n) itself can be quite
big, even tending to infinity like n/log(n). 
Some probabilists have probably analysed the
Pareto case carefully.

We are on safe ground if variance exists, since in this case
the central limit theorem holds and the o(n) term above
is of order sqrt(n).

What is happening in the Pareto case is that the theoretical
large mean is determined by large deviations of tiny probabilities,
these events do not occur in simulation with n=5000. For
a = 1.06 you need to increase n to around 1000,000,000 (maybe), 
(I tried up to 10,000,000 and increase the observed mean
to around 12,000).

-Yudi-

------------------------------
Yudi Pawitan: yudi at stat.ucc.ie
Department of Statistics, UCC
Cork, Ireland
Ph : 353-21-902 906
Fax: 353-21-271 040
------------------------------

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