ECMT1020 Chap.7 Bias, Efficiency and Mean Squared Error
Bias, Efficiency and Mean Squared Error
Estimator quality has several dimensions. Bias compares an estimator's expected value with the parameter, variance measures repeated-sample spread, and mean squared error combines variance with squared bias. This chapter calculates those quantities, explains why a precise estimator can still target the wrong centre, and distinguishes finite-sample comparisons from consistency as sample size grows.
The direction of bias remains substantively useful even though mean squared error squares it. Efficiency requires a stated comparison class, because lower variance is meaningful only among estimators aimed at the same target under comparable conditions. Consistency introduces a sequence of estimators indexed by sample size.
A finite-sample defect may shrink, persist or worsen as n grows, so large-sample claims must examine both displacement and concentration instead of relying on the slogan that more data solve every problem.
What this chapter covers
- 01
Estimator bias
- 02
Sampling variance
- 03
Efficiency comparisons
- 04
Mean squared error
- 05
Bias-variance trade-offs
- 06
Consistency
Worked example · free
Choose by mean squared error
- +1MSE equals variance plus squared bias.
- +1MSE(A)=9+0²=9.
- +1The squared bias of B is one.
- +1MSE(B)=4+1=5.
- +1B has lower MSE despite being biased.
Key terms
- Bias
- Expected estimator minus the parameter it targets.
- Efficiency
- Relative variance performance among estimators of the same parameter.
- Mean squared error
- Variance plus squared bias around the target parameter.
- Consistency
- Concentration of an estimator on the parameter as sample size grows.
- Precision
- Tightness of an estimator's repeated-sample distribution.
- Squared bias
- The square of an estimator's expected displacement from its target. Squaring removes direction and places bias on the same squared-distance scale as variance when the two are combined in mean squared error.
- Comparison class
- The stated collection of estimators aimed at the same parameter under comparable conditions. An efficiency claim is incomplete without this set because low variance has meaning only relative to the alternatives being considered.
- Asymptotic property
- A feature describing an estimator as sample size increases without bound. It can differ from finite-sample behaviour and therefore requires an explicit limiting argument.
Bias, Efficiency and Mean Squared Error FAQ
Can a biased estimator be preferable?
Yes, under mean squared error. A modest squared bias can be more than offset by a large reduction in variance, producing smaller expected squared distance from the target.
What does efficiency compare?
It compares sampling variance among estimators aimed at the same parameter under a stated setup. If estimators have different biases, the requested loss criterion must also be considered.
Does one accurate estimate prove unbiasedness?
No. Unbiasedness concerns an expectation over possible samples. One estimate can land near the truth by chance even when the estimator is systematically displaced across repeated applications.
Exam move
Build a three-column table for every estimator: bias, variance and MSE. Calculate each column before choosing, and write the criterion in the concluding sentence. When formulas depend on sample size, inspect both centre and spread as n grows rather than assuming that a large sample automatically removes every problem. Begin comparisons with a target diagram.
Put the parameter at the centre, then draw the expected estimator location and a band representing spread. Translate that picture into signed bias and variance before using numbers. This prevents a tight but displaced distribution from being called accurate without qualification. Keep the bias sign in its own column and create a separate squared-bias column for MSE.
Never compare variance directly with unsquared bias because they are on different scales. For two unbiased estimators, lower variance determines the more efficient one within the stated class. When biases differ, follow the criterion requested by the question and say explicitly whether your conclusion concerns unbiasedness, variance or MSE.
Practise threshold comparisons by solving for how large a variance reduction must be to offset a given bias. That algebra makes the trade-off concrete. For consistency, write bias and variance as functions of n and evaluate their limits separately. If the expected estimator approaches the parameter and the dispersion contracts appropriately, explain why concentration follows.
If variance shrinks around a centre that remains displaced, do not call the estimator consistent. Create counterexamples for revision: an unbiased but noisy estimator, a biased estimator with lower MSE, and a sequence with vanishing variance but persistent bias. Use each to answer a different property question. After calculations, interpret direction.
Positive bias means systematic overestimation on average; negative bias means systematic underestimation. MSE discards direction because it measures squared distance, but a real application may care differently about the two errors. Conclude with the exact criterion and comparison set, avoiding an unqualified statement that one estimator is simply better.
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