Estimability

Estimability is a concept in statistics that describes whether a parameter or a function of parameters can be unbiasedly and uniquely determined from observed data under a given model. It is most commonly used in the context of linear models, where it determines which linear combinations of the regression coefficients can be consistently and uniquely estimated from the design matrix.

In the linear model y = Xβ + ε (X of size n×p), a linear function c'β is said to be estimable if there exists a vector a (of length n) such that a'X = c'. Equivalently, c' must lie in the row space of X (or c in the column space of X'). If the design matrix X has full column rank (rank p), all components of β are estimable; if rank is deficient, only certain linear combinations are estimable. For an estimable function, the Gauss–Markov theorem gives a unique best linear unbiased estimator (BLUE). Non-estimable combinations cannot be unbiasedly recovered from the data without additional assumptions or constraints.

Estimability is related but not identical to identifiability: identifiability refers to the theoretical uniqueness of parameters in the model, while estimability concerns practical recovery from observed samples and the design. In practice, lack of estimability is addressed by reparameterization, imposing constraints, using prior information, or applying regularization methods, each changing which quantities can be meaningfully estimated.