Fisher consistency

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:

${\displaystyle {\hat {\theta }}=T({\hat {F}}_{n})\,,}$ 

the estimator is said to be Fisher consistent if:

${\displaystyle T(F_{\theta })=\theta \,.}$ ^[2]

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 consistentif

${\displaystyle T\left(\lim _{n\rightarrow \infty }{\hat {F}}_{n}\right)=\theta .\,}$ 

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 consistencyifT(p₁, ..., p_m) = θ.

Suppose the parameter of interest is the expected value μ and the estimator is the sample mean, which can be written

${\displaystyle n^{-1}\sum _{i=1}^{n}\sum _{j=1}^{m}I(X_{i}=Z_{j})Z_{j},}$ 

where I is the indicator function. The population analogue of this expression is

${\displaystyle n^{-1}\sum _{i=1}^{n}\sum _{j=1}^{m}p_{j}Z_{j}=n^{-1}\sum _{i=1}^{n}\mu =\mu ,}$ 

so we have Fisher consistency.

Maximising the likelihood function L gives an estimate that is Fisher consistent for a parameter bif

${\displaystyle E\left[{\frac {d\ln L}{db}}\right]=0{\text{ at }}b=b_{0},\,}$ 

where b₀ represents the true value of b.^[3][4]

The term consistency in statistics usually refers to an estimator that is asymptotically consistent. Fisher consistency and asymptotic consistency are distinct concepts, although both aim to define a desirable property of an estimator. While many estimators are consistent in both senses, neither definition encompasses the other. For example, suppose we take an estimator T_n that is both Fisher consistent and asymptotically consistent, and then form T_n + E_n, where E_n is a deterministic sequence of nonzero numbers converging to zero. This estimator is asymptotically consistent, but not Fisher consistent for any n.

The sample mean is a Fisher consistent and unbiased estimate of the population mean, but not all Fisher consistent estimates are unbiased. Suppose we observe a sample from a uniform distribution on (0,θ) and we wish to estimate θ. The sample maximum is Fisher consistent, but downwardly biased. Conversely, the sample variance is an unbiased estimate of the population variance, but is not Fisher consistent.

A loss function is Fisher consistent if the population minimizer of the risk leads to the Bayes optimal decision rule.^[5]