Fisher consistency

In statistics, Fisher consistency, named after Ronald Fisher, is a desirable property of an estimator asserting that if the estimator were calculated using the entire population rather than a sample, the true value of the estimated parameter would be obtained.

Suppose we have a statistical sample X₁, ..., X_n where each X_i follows a cumulative distribution F_θ which depends on an unknown parameter θ. If an estimator of θ based on the sample can be represented as a functional of the empirical distribution function F̂_n:

the estimator is said to be Fisher consistent if:

As long as the X_i are exchangeable, an estimator T defined in terms of the X_i can be converted into an estimator T′ that can be defined in terms of F̂_n by averaging T over all permutations of the data. The resulting estimator will have the same expected value as T and its variance will be no larger than that of T.

If the strong law of large numbers can be applied, the empirical distribution functions F̂_n converge pointwise to F_θ, allowing us to express Fisher consistency as a limit — the estimator is Fisher consistent if

Suppose our sample is obtained from a finite population Z₁, ..., Z_m. We can represent our sample of size n in terms of the proportion of the sample n_i / n taking on each value in the population. Writing our estimator of θ as T(n₁ / n, ..., n_m / n), the population analogue of the estimator is T(p₁, ..., p_m), where p_i = P(X = Z_i). Thus we have Fisher consistency if T(p₁, ..., p_m) = θ.

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