[R] Regression with partial info about the dependent variable

[maneesh deshpande]

Hi,

I have the following problem which I would appreciate some help on.

A variable y  is to be modelled as a  function of  a set of variables 
Vector(x).
The twist is that there is another variable z in  the problem with the 
property that y(i) <= z(i).
So the data set is divided into three categories

I.    y(i) = z(i)
II.   Both y(i) and z(i) are known and y(i) < z(i)
III.  y(i) is not known but z(i) is known ( But y(i) is guaranteed to be < 
z(i) )

The data in categories I + II can be satisfactorily modelled via a OLS 
regression of the form:
y ~ Vec(x)
The question is how to incorporate the information contained in the category 
III data?
The category II data can be used to construct a model for y given z. Indeed 
log(z(i)-y(i))
is reasonably normal and so the following is a decent approximation:
y(i) = z(i) + A*exp( N(0,1) )
This model can be improved by including Vec(x).

After this I am not sure how to proceed :-( :-(

Thanks in advance,

Maneesh