Estimator

Summary

• Given a sample \(x_1,x_2,...,x_n\) , an set of assumption s about the joint distribution of these random variables is called a statistical model .
• A statistical model may include one or several unknown parameters \(θ_1,…,θ_k\) .

Example

• \(x_1,x_2,...,x_n\) is an IID random sample where each \(x_i \sim N\left( μ,σ^2 \right)\) . This is a statistical model with two unknown parameters,
• When we use a specific statistic to infer the value of an unknown parameter \(θ\) , such a statistic is called an estimator .

Example (continued)

• The sample mean \(\bar{x}\) can be used as an estimator of the expected value \(μ\) and the sample variance \(s^2\) can be used as an estimator of the (true) variance \(σ^2\) .
• An estimator   \(\hat{θ}=g(x_1,…,x_n)\) is said to be a linear estimator if \(g\) is a linear function.