A underlying premise of stochastic models is that exact outcomes cannot be predicted with certainty.

Though we are often tempted to express predictions made by a good model as a point estimate of a parameter value, on average, our predictions will be wrong by one standard error (SE).

Thus stochastic models require an expression of this uncertainty, usually as a SE of prediction or estimation.

Useful expressions of uncertainty are not necessarily those with the smallest SE, but those that you can state with some specified confidence that they include the 'true' value, e.g., 19 times out of 20 (alpha=0.05).

Undesirably small SEs can result from badly fit models whose increased estimation bias results in those SEs not encompassing the 'true' value of a parameter.

An important goal of goodness-of-fit testing is to detect such poor and potentially misleading models.

A model's SEs are generally calculated and expressed by a covariance matrix that assumes asymptotic conditions, i.e., a very large sample size.

As such, SEs are assumed to apply to a Gaussian (normal) distribution of uncertainty around a point estimate, in accordance with the central limit theorem, and have minimal bias.

If such conditions are not met, SEs can be biased and the distribution of uncertainty can be highly skewed (non-normal).

A covariance matrix can be calculated by inverting a Hessian matrix calculated using analytical second partial derivatives of the likelihood function with respect to the parameters.

Under asymptotic conditions the values of these second partial derivatives remain constant for all values of a particular parameter, especially near the maximum-likelihood point parameter estimates.

However, for an analytically calculated Hessian matrix there is no easy way to assess that the second partial derivatives are constant throughout that restricted parameter space, and thus conformance with the central limit theorem cannot be ascertained.

This limitation arises from the analytical derivatives generally being calculated only for the point values of the maximum-likelihood estimates.

However, when the covariance matrix is determined by inverting a Hessian matrix calculated using numerical second partial derivatives of the likelihood function with respect to the parameters, it is possible to assess the non-constancy of the second partial derivatives.

This is achieved by systematically taking small incremental steps away from the point parameter estimates then approximating the second partial derivatives using a finite difference calculation.

Differences in values for the second partial derivatives at different distances from the point parameter values are diagnostic of the degree of conformance with the asymptotic conditions of the central limit theorem.