Estimators

An estimator is a rule for calculating some estimate of a desired quantity using data. The rule is the estimator, the thing to be estimated is the estimand and the result of applying the rule to data is the estimate.

Properties

Let data \(x \sim p(x)\), \(\theta\) be the estimand and \(\hat{\theta}(\dot)\) be an estimator based on \(x\). We define the following properties for the estimator:

Bias

\[B(\hat{\theta}) = \mathbb{E}_{p(x)}[\hat{\theta}(x)] - \theta\]

Variance

\[\mathbb{V}_{p(x)} = \mathbb{E}_{p(x)}[(\hat{\theta}(x) - \mathbb{E}_{p(x)}[\hat{\theta}(x)])^2]\]

Mean Squared Error

\[MSE(\hat{\theta}(x)) = \mathbb{E}_{p(x)}[(\hat{\theta}(x) - \theta)^2 ]\]

Bias-Variance Tradeoff

One commonly referenced topic in introductory ML courses is a so-called “bias-variance” tradeoff, which is the fact that the MSE is exactly the sum of the variance plus the bias squared; consequently, for a given MSE, attempting to minimize the variance of an estimator necessary introduces bias and vice versa. To show why this is, we drop \(x\) and \(p(x)\) for brevity:

\[\begin{align} MSE(\hat{\theta}) &= \mathbb{E}_{p(x)}[(\hat{\theta} - \theta)^2 ]\\ &= \mathbb{E}[(\hat{\theta} + \mathbb{E}[\hat{\theta}] - \mathbb{E}[\hat{\theta}] - \theta)^2 ]\\ &= \mathbb{E}[(\hat{\theta} + \mathbb{E}[\hat{\theta}] )^2] + 2 \mathbb{E}[(\hat{\theta} + \mathbb{E}[\hat{\theta}] )(\mathbb{E}[\hat{\theta}] - \theta)] - \mathbb{E}[\hat{\theta}] - \theta)^2 ]\\ &= \mathbb{V}_{p(x)} + B(\hat{\theta})^2 \end{align}\]

Estimator Desiderata

Consistent

An estimator is consistent if

\[\]

Unbiased

An estimator is said to be unbiased if the bias \(B(\hat{\theta}) = 0\). An unbiased estimator means that on average, the estimate \(\hat{\theta}(x)\) equals the estimand \(\theta\).

Efficient

A sequence of

Minimal Variance