Minimum-variance unbiased estimator

Minimum-variance unbiased estimator (MVUE) refers to a statistical estimator that, among all the unbiased estimators of a given parameter, has the lowest variance. It is a key concept in the field of statistics and estimation theory, playing a crucial role in the selection of optimal estimators under certain conditions. The importance of MVUE lies in its ability to provide the most precise estimate of a parameter without bias, making it a valuable tool in statistical analysis and data science.

Definition

An estimator \(\hat{\theta}\) of a parameter \(\theta\) is said to be unbiased if its expected value is equal to the true value of the parameter for all values of \(\theta\). Mathematically, this is expressed as \(E[\hat{\theta}] = \theta\). An unbiased estimator is considered minimum-variance if, among all unbiased estimators of \(\theta\), it has the smallest variance. The formal definition is given by:

\[ \text{Var}(\hat{\theta}_{MVUE}) \leq \text{Var}(\hat{\theta}), \forall \hat{\theta} \text{ unbiased estimators of } \theta. \]

Properties

The MVUE is desirable in statistical estimation for several reasons:

• Efficiency: By definition, the MVUE is the most efficient estimator among all unbiased estimators because it has the smallest variance, leading to more precise estimates.
• Consistency: Under certain conditions, the MVUE is consistent, meaning that as the sample size increases, the estimator converges in probability to the true parameter value.
• Sufficiency: The concept of sufficiency is closely related to the MVUE. An estimator is sufficient if it captures all the information about the parameter that is present in the sample. The Lehmann-Scheffé theorem states that if there exists a sufficient statistic for a parameter, then any function of that statistic that is an unbiased estimator of the parameter is the MVUE.

Finding the MVUE

The process of finding the MVUE involves several steps and principles from statistical theory:

• Identification of unbiased estimators for the parameter of interest.
• Calculation of the variance of these estimators.
• Application of the Cramér-Rao lower bound (CRLB), which provides a theoretical lower bound for the variance of an unbiased estimator.
• Use of the Rao-Blackwell theorem to improve an unbiased estimator by conditioning it on a sufficient statistic.

Examples

A classic example of an MVUE is the sample mean \(\bar{X}\) as an estimator for the population mean \(\mu\) in a normal distribution with known variance. The sample mean is unbiased, and no other unbiased estimator of \(\mu\) has a smaller variance.

Limitations

While the MVUE has many desirable properties, it is not always possible to find such an estimator for every parameter in every statistical model. Additionally, the requirement for unbiasedness can sometimes lead to estimators that are not robust to outliers or are inefficient compared to biased estimators in certain contexts.

Applications

The concept of MVUE is widely applied in various fields such as econometrics, biostatistics, engineering, and more. It is particularly useful in the design of experiments and in the analysis of data where the goal is to make precise and accurate inferences about population parameters.