Distribution of interval between normally distributed variables of known order X > Y > Z > 0


Hoping to find some ideas here to resolve a problem that I'm working on:

As part of a study, I have a set of age-dated samples (say: variables X, Y, and Z) of known order X > Y > Z > 0. The uncertainty of each age value is captured in a normal distribution with a given mean (μx, μy, and μz, respectively) and standard deviation (σx^2, σy^2, and σz^2, respectively). I am interested to know the distribution of difference (time) between these variables (i.e., X - Y, X - Z, and Y - Z).

From normal difference distribution theory, I know that μ(x-y) = μx - μy and σ(x-y)^2 = σx^2+ σy^2 for independent variables. It should be possible to narrow these distributions knowing that each variable > 0 and that the variables have a known order (i.e., X - Y > 0, X - Z > 0, and Y - Z > 0).

Is it statistically valid to perform simple truncations for this? Should I convert to log-normal distributions?

Any help would be greatly appreciated. Thanks!


Ambassador to the humans
Interesting question. Do you *have* to require that Z>0? For any normally distributed variable there is a non-zero probability that it will be negative. Or do you really want to work with truncated normals even before the differencing takes place?