[R] random truncation

[Rolf Turner]

On 13/07/19 10:54 AM, Spencer Graves wrote:

> Hello:
> 
> 
>        What do you suggest I do about modeling random truncation?

Good question!  Probably the best answer is "Give up and go to the pub!" :-)

But seriously, there is a package DTDA on CRAN which purports to analyse 
randomly truncated data.

> 
>         I have data on a variable Y in strata S[0], S[1], ..., S[n], 
> where Y is always observed in S[0] but is less often observed in the 
> other strata.  I assume that the probability of observing Y is a 
> monotonically increasing function of Y and a monotonically decreasing 
> function of d[i] = the distance from S[0] to S[i].
> 
> 
>        There is a section on "random truncation" in the Wikipedia 
> article on "Truncated distribution".[1]  It would be nice if I had an R 
> package that would make it relatively easy to model the truncation as a 
> function of "d" and / or publication that described someone doing it in 
> R.  (I also have a couple of other variables that influence the 
> distribution of Y.)
> 
> 
>        Thanks,
>        Spencer Graves
> 
> 
> [1] https://en.wikipedia.org/wiki/Truncated_distribution#Random_truncation

I'd just like to add that many many years ago, probably back before you 
were born, I struggled mightily for a while with random truncation, 
modelling the truncation variable to have a density that was uniform on 
some interval [a,b].

I took the underlying ("untruncated") variable to have a Gaussian 
distribution N(mu,sigma^2).

The distribution of the randomly truncated variable has thus four 
parameters: a, b, mu and sigma.  I was able to write down the likelihood 
and attempted to maximise it, but this was a nightmare since the partial 
derivatives of the log likelihood are undefined for b=x_i where x_1, 
..., x_n are the i.i.d. observations from the randomly truncated 
distribution.  I fought with this for a long while, could not get 
estimates based on simulated data to agree at all well with the true 
values, and eventually chucked it all in.

This all happened back before there was R, or even S!!!

Let us hope that the authors of DTDA are far clever than I.  (Not at all 
unlikely!)

cheers,

Rolf

-- 
Honorary Research Fellow
Department of Statistics
University of Auckland
Phone: +64-9-373-7599 ext. 88276