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Topic: Is " exp[ log-normal ] " a defined distribution ?
Replies: 1   Last Post: May 17, 2006 11:54 PM

 Reef Fish Posts: 60 Registered: 5/13/06
Re: Is " exp[ log-normal ] " a defined distribution ?
Posted: May 17, 2006 11:54 PM

Ray Koopman wrote:
> Are sure the model shouldn't be Y = a * X^b * error ?

That would be a better formulation, not assuming that a = 1, or
the intercept of the transformed model to be zero.

But in either case, the OP is concerned with the error distribution
in the transformed model is lognormal (which would be the case,
if the errors in the un-transformed model is normal).

BTW, the question of WHAT to do with the estimation of those
loglinear models with resulting lognormal error is a little can of
worms yet to be opened. :-)

> > Is " exp[ log-normal ] " a defined distribution ?

W is lognormally distributed iff log (W) is normally distributed.

If U is a lognormal, then log(U) is normal, but exp(U) is a horse
of different color from both U and W.

Since your r.v. V is exp [log-normal], then log V would be lognormal,
and log[logV] would be normal, so that one might say V has a
log-lognormal distribution, but I know nothing that one.

> > Shall I give up and travel to Monte-Carlo ?

Or read Chapter 14 (Lognormal and related distributions) of the
2nd edition of Johnson, Kotz, and Balakrishnan's Volume 1
(which starts with Chapter 12) on "Continuous Univariate
Distributions", enroute to Monte Carlo, or while you run out
of interesting things to do there.

-- Bob.

>
> tiagorubin@gmail.com wrote:

> > Hello group,
> >
> > A friend at the Bank of Canada challenged me to find a solution to the
> > following problem :
> >
> > Given a model
> > y=(X^b)*(error)
> >
> > log tranformed it becomes:
> >
> > log(y) = b*log(X) + log(error)
> >
> > The fit is impressively good, but it turns out that log(error) is
> > distributed as a log-normal distribution
> >
> > log(error)~exp[Normal]
> >
> > If I want to use this regression to do some prediction, how should I
> > account for the error term? How can I convert it back to the initial
> > setting?
> > Is " exp[ log-normal ] " a defined distribution ? Shall I give up and
> > travel to Monte-Carlo ?
> >
> > And now, what if it is serially correlated?
> >
> > If some of you know the solution (or a book that does), please let me
> > know where I should look. Any kind of information would be
> > appreciated. I would also welcome other transformation offers.
> >
> > Thanks a lot.
> >
> > Strabudje
> >
> > P.S: Please be gentle in your explanations. I am somewhat new to the
> > field.