Estimator's performance
In this section, we examine an estimator's performance by exploring its bias and consistency properties. We illustrate how bias can impact estimates and how the consistency of the estimator can impact the accuracy of the results. This analysis will provide a better understanding of the performance of the estimator in various statistical applications.
Estimator performance criteria seek to minimize the error average, which represents the bias of the estimator (see equation (1)), as well as the variance of the assessor (see Equation (2)), a measure of the variability of estimates around their average. The latter, sometimes also called the range of fluctuations, reflects the estimator's ability to produce consistent and unvariable results in different situations.
Definition 1. The estimator is said to be biassed or not biassed if the estimate bias is null.
\(\hat{\theta}\) unbiased \(\implies\) b\(\left(\hat{\theta}\right)=E\left[\hat{\theta}\right]-\theta=\mathrm{0}\) \(\iff\) \(E\left[\hat{\theta}\right]=\theta\)
Definition 2. The \(\hat{\theta}\) estimator is said to be asymptotically non-biassed if the bias exists but tends to zero when the duration of observation increases.
\(\hat{\theta}\) asymptotically unbiased \(\implies\) \(\lim\limits_{N \to \infty} b(\hat{\theta}) = 0\)
Definition 3. When the duration of the observation increases, the estimator tends in probability towards the exact value of the parameter.
\(\hat{\theta}\) consistent \(\iff\) \(p{\lim\limits_{N \to \infty} b\hat{\theta}_(x)} = \theta\)
In practice, the variance is a positive quantity.
Definition 4. If the duration of the observation increases and overturns the estimate variance, the estimator will remain consistent.
\(\hat{\theta}\) consistent \(\iff\) \(\lim\limits_{N \to \infty}\) \(\sigma_E^2 = 0\)