Regression with partial info about the dependent variable
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
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 <
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
The category II data can be used to construct a model for y given z. Indeed
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).