MLE for weighted AND censored samples

From: Carsten Steinhoff (carsten.steinhoff_at_stud.uni-goettingen.de)
Date: 03/22/05 

Date: Tue, 22 Mar 2005 23:27:58 +0100

Hi,

I have to fit a distribution to a truncated Dataset. I just want to
verify with this mail that Im on the right way. If not I hope to get
many constructive reply!

The data I get is truncated from below. That means, all observed values
under a known threshold are not recorded in the Database.
These data won't have all the same weight. For example the Value X1000
shall be considered only half times compared with the other database
values.So far so good.

What I do now is transforming the density-function that I put into my
MLE algorithm (I use the R-Function "fitdistr" to do that)

For the truncation I take:

        density(x,parameters)/1-CDF(threshold,parameters)

For the weighting I introduce an exponent to this function like:

        (density(x,parameters)/1-CDF(threshold,parameters))^a

Where a will be a variable for the weighting of each observation. So I
get something like a * f(...,parameters) for the log-likeliehood
computation.

My questions:

(1) Is this proceeding right?

(2) Has there be a change in the weighting algorithm if I had an
ADDITIONALY truncation from above? Apart from replacing the denominator
by: CDF(upper_thresholc,parameters)-CDF(lower_threshold,parameters)

Thanks a lot for your help!

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