Estimators

Objective. Evaluate bias, consistency and mean squared error.

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Learn: the key idea

An estimator is a rule or formula, such as the sample mean X̄, used to infer an unknown population parameter from sample data, and its quality is judged by several properties. An estimator is unbiased if E(θ̂) = θ, meaning it is correct on average across repeated sampling; the sample mean is an unbiased estimator of the population mean. Consistency means the estimator converges in probability to the true parameter as sample size n increases, even if it is biased for small n. Mean squared error, MSE(θ̂) = Var(θ̂) + [Bias(θ̂)]², combines both variance and bias into a single measure, and a slightly biased estimator with much smaller variance can outperform an unbiased one by this criterion.

Worked example

An estimator θ̂ has bias 2 and variance 9. Find the mean squared error of this estimator.

1 $Recall the formula MSE(θ̂) = Var(θ̂) + [Bias(θ̂)]².$
2 $Substitute the variance: Var(θ̂) = 9.$
3 $Square the bias: [Bias(θ̂)]² = 2² = 4.$
4 $Add the two values together: 9 + 4.$

Answer: MSE(θ̂) = 13

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Common mistake. Students assume unbiasedness alone makes an estimator good, ignoring variance entirely; an unbiased estimator with enormous variance can have far worse mean squared error than a slightly biased but low-variance alternative.

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