Linearized Variance
Linearization is a method for approximating variance of a non-linear estimator.
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Description
If an estimator is of the form X = f(Y1, Y2, ... Yn) where the Yi are themselves estimators, and if f is a linear function, then the variance of X is straightforward to calculate using the basic properties of variance under linear transformations. If f is non-linear but smooth, then the function can be approximated with Taylor linearization. This allows for an approximation of the variance of X.
Given a smooth f, the first order Taylor approximation (by convention, calculated around the mean) is:
Taking the variance of both sides, noting that Var[a + b] = Var[a] + Var[b]:
The first term on the RHS is a constant, so has zero variance. The equation reduces to:
More properly, the estimator should be described as being of the form X = F(Y) = <f1(Y), f2(Y)>, and the linearization is in terms of a Jacobian matrix J. Therefore, X ≈ F(μ) + J(μ)(Y - μ). Note that linear transformations operate on a covariance matrix as Var[AX] = A Var[X] AT.
