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Consistent Estimator

Also Consistency · Consistent · Konsistenter Schätzer · Konsistenz

An estimator θ̂(y₁, …, y_N) is consistent if, for every ε > 0, the probability that it lies in [θ_true − ε, θ_true + ε] tends to one as N → ∞.

Properties#

Sufficient condition: an estimator is consistent if
1. it is asymptotically unbiased, and
2. its covariance tends to zero as $N \to \infty$ .
Stronger and more desirable than asymptotic unbiasedness: it concerns the estimate itself in a single experiment, not only its average over many experiments.

MSI script p. 19 (§2.4, Definition 3), MSI Slides 02 > 2.4 Statistical Estimators (not published)

Connections

Resistance Estimation ExampleMSI
The script’s computations in §2.5 take the limit of the estimate itself ( $\lim_{N\to\infty}\hat R^{[N]}$ , via the law of large numbers) and read the result as the limit of its expectation, as in eq. (2.10). For these estimators both limits… §
Asymptotically Unbiased Estimator
Towards consistency: asymptotically unbiased and covariance $\to 0$ as $N \to \infty$ ⇒ consistent. §
Estimator
convergence in probability to the true value: Consistent Estimator. §
Law of Large Numbers
Relation to consistency: it says that the Sample Mean is a consistent estimator of the mean. §
Sample Mean
Consistent: unbiased with variance $\to 0$ (Consistent Estimator). For $N \to \infty$ , $M(y) \to \mu_y$ (Law of Large Numbers). §
Point Estimation
An estimator turns random measurements into an estimate of a fixed but unknown parameter, so the estimate is a random variable. Two questions judge it: is it right on average (unbiased, or at least asymptotically unbiased), and how much do… §
Consistent Estimator — the estimate itself concentrates at the true value (Definition 3) §