Statistical Estimation
The preceding section outlined the statistical model for S which SEPATH attempts to fit to the sample data. If the model fits perfectly in the population, then Equation 41 holds. Such perfection is, of course, extremely unlikely to happen.
It is, in fact, virtually certain that Equation 41 does not hold exactly for your statistical population, and that in fact an additional error term Epop should be added to the right side of the equation. The size of the elements of this error matrix would reflect how badly a particular model fits in the population. You could find out what Epop was if you somehow knew S. (You would simply input S to SEPATH and fit your model to it.) If you did, you would be faced with a difficult problem of exactly how to quantify the information in Epop.
There is an additional complication. In practice, you do not know S. You only have S, an estimate of S from sample data. It is this estimate, usually the ordinary sample covariance matrix based on N independent observations, which one attempts to fit with SEPATH.
Consequently, in practice one attempts to fit S rather than S with the model of Equation 41, and one obtains, as a result of this model fitting procedure, a sample matrix of residuals Esamp. In general the object of the estimation process is to make the elements of Esamp as "small as possible" in some sense. This notion of "smallness" is quantified in a "discrepancy function."