Firstly, we don't really need all your working.
Just the problem you want solve.
However, I'm still having difficulty understanding this.
> I'm observing Y[i] = (X[i]'b+e) given Y[i]>(z[i]'c+f) where e and
> f are normally distributed with standard deviations s and t,
> respectively, i = 1:n. I want the total of all the Y's, including those
> truncated (and not observed).
> (x[i]'b+e)>(z[i]c+f)
> (x[i]'b-z[i]'c)>(f-e),
(1) What does the tick (or single quote) mean?
(2) What do you mean by "X[i]" and "observing Y[i]"?
(3) Are e and f random variables, or vectors of (observed) errors?
My intuitive understanding of (2), in the given context, would be that X
and Y are vectors of observations.
However, if e and f are random variables, wouldn't that would make Y[i] a
random variable too? In which case I'm not sure what you mean by
"observing".
Conversely, if e and f are vectors of (observed) errors, shouldn't they be
indexed (for consistency)? And in which case, there's no random component
in your expressions.
(4) Is lower case "x[i]" the same as upper case "X[i]", and what's "z[i]"?
(5) Is the mean of e and f, zero?
(6) So, you want to estimate b and c?
And wouldn't that make this a parameter estimation problem?
(In which case, you don't necessary need to model any distributions).
> Pr{Y[i] observed} = Phi((x[i]'b-z[i]'c)/sqrt(s^2+t^2))
> where Phi is the cdf of the standard normal.
This implies that "Y[i]" is a is binary (true or false) random variable.
Do you mean Pr (Y <= y) where Y is a random variable and y is a possible
value (from Y's sample space), that Y can be less than or equal to?
You can replace y with y[i] if you want, but the principle is the same.
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