Actuarial Statistics · Estimators and their properties

Mean Squared Error and Efficiency of Estimators

Q: What is the formula for the mean squared error of an estimator?

MSE(θ̂) = E[(θ̂ − θ)²] = Var(θ̂) + [bias(θ̂)]². Here bias(θ̂) = E[θ̂] − θ. For an unbiased estimator the MSE is just the variance.

Q: How do I compare two estimators using efficiency?

Compute the MSE of each. The efficiency of the first relative to the second is MSE(second) ÷ MSE(first). If it is above 1, the first estimator is better. For two unbiased estimators you can use variances instead.

Q: Can a biased estimator be better than an unbiased one?

Yes. If a small bias brings a larger fall in variance, the MSE is lower. The divisor-n variance estimator for a normal sample is a standard example.

Q: Does a smaller MSE mean the estimator is consistent?

Not by itself. Consistency is about what happens as n grows. If MSE tends to 0 as n tends to infinity, the estimator is consistent. A small MSE at one fixed n tells you nothing about that.

Updated 11 October 2026 · Fact-checked

The mean squared error of an estimator θ̂ is MSE(θ̂) = E[(θ̂ − θ)²] = Var(θ̂) + bias(θ̂)². To compare two estimators, find each MSE. The one with the smaller MSE is better. Relative efficiency is the ratio of their MSEs, and for unbiased estimators it is the ratio of their variances.

Understand Mean Squared Error and Efficiency

An estimator is a rule that turns sample data into a guess for a parameter θ. Different rules give different guesses, so you need a fair way to say which rule is better. The mean squared error (MSE) does this. It is the average squared distance between the estimator and the true parameter.

MSE has two parts. Bias measures systematic error: bias(θ̂) = E[θ̂] − θ. Variance measures how much the estimator moves from sample to sample. The result is MSE = variance + bias². An estimator can be wrong on average (bias), wrong by scatter (variance), or both. Squaring the bias stops positive and negative bias from cancelling and matches the squared-error idea.

If an estimator is unbiased, bias is zero and MSE equals variance. This is why, among unbiased estimators, you simply pick the one with the smaller variance. But unbiased is not always best. A slightly biased estimator with much lower variance can have a smaller MSE. This is the bias-variance trade-off.

To compare two estimators, use relative efficiency. The efficiency of θ̂1 relative to θ̂2 is MSE(θ̂2) ÷ MSE(θ̂1). A value above 1 means θ̂1 is better. A value below 1 means θ̂2 is better. Note that MSE can depend on the unknown θ, so one estimator may be better for some values of θ and worse for others.

MSE also links to consistency. If MSE(θ̂) tends to 0 as the sample size n grows, then θ̂ is consistent. This happens when both bias and variance tend to 0.

Key rules to remember

Bias: bias(θ̂) = E[θ̂] − θ; Zero for an unbiased estimator. It can be positive or negative. Always subtract θ from the expected value.
Mean squared error: MSE(θ̂) = E[(θ̂ − θ)²]; The definition. It is the average squared error around the true value θ.
MSE decomposition: MSE(θ̂) = Var(θ̂) + [bias(θ̂)]²; The main working formula. Prove it by writing θ̂ − θ = (θ̂ − E[θ̂]) + (E[θ̂] − θ) and expanding. The cross term has zero expectation.
Unbiased case: If E[θ̂] = θ, then MSE(θ̂) = Var(θ̂); Compare unbiased estimators by variance.
Relative efficiency: eff(θ̂1 relative to θ̂2) = MSE(θ̂2) ÷ MSE(θ̂1); Above 1 means θ̂1 is better. For two unbiased estimators this is Var(θ̂2) ÷ Var(θ̂1).
Variance of sample mean: Var(X̄) = σ² ÷ n; For independent observations with common variance σ². X̄ is unbiased for μ.
Variance estimators for a normal sample: S² (divisor n−1): bias 0, MSE = 2σ⁴ ÷ (n−1). Divisor n: bias = −σ² ÷ n, MSE = (2n−1)σ⁴ ÷ n²; Valid for a random sample from a normal distribution. The divisor n−1 is unbiased but the divisor n has smaller MSE.
MSE and consistency: MSE(θ̂) → 0 as n → ∞ implies θ̂ is consistent; This is a sufficient condition. It is not necessary.

How to solve Mean Squared Error and Efficiency questions

Use this method for any question that asks you to compute, compare or justify estimators using MSE or efficiency.

1Write down the estimator θ̂ clearly in terms of the sample, and note what you know about the sample (independence, distribution, n).
2Find E[θ̂] using linearity of expectation. Then compute bias = E[θ̂] − θ.
3Find Var(θ̂) using the variance rules. Check independence before adding variances. For a constant c, Var(cX) = c²Var(X).
4Compute MSE = Var(θ̂) + bias². Keep the answer as an expression in θ, σ² and n if the parameter is unknown.
5To compare, compute the MSE of each estimator. Then form the ratio MSE(θ̂2) ÷ MSE(θ̂1) for relative efficiency.
6If the MSEs depend on θ, state for which values of θ each estimator is better. Solve the inequality MSE(θ̂1) < MSE(θ̂2) if asked.
7Finish with a clear conclusion in words. Mention the bias-variance trade-off if one estimator is biased but has lower variance.

Quickest way: Bias and variance table

When to use it: Use it when you must compare two or three estimators quickly, especially in the multiple-choice section.

Make a small table with one row per estimator and columns for E[θ̂], bias, variance and MSE.
Fill in bias first. If it is zero, the MSE is just the variance, so skip squaring.
For scaled estimators cX̄ use the shortcut: bias = (c − 1)μ and variance = c²σ² ÷ n.
Compare MSE values as numbers. Do not compare variance alone unless both estimators are unbiased.
Check that relative efficiency is written as the worse MSE divided by the better MSE if the question asks how much better the best one is.

Common mistakes in Mean Squared Error and Efficiency

Writing MSE = variance + bias instead of bias squared.
Students remember the two ingredients but forget the square. The units then do not match.
Fix: Always write MSE = Var + (bias)². Check units: variance is in squared units, so bias must be squared too.
Comparing biased estimators by variance alone.
Variance is easy to compute and is the right comparison for unbiased estimators, so students apply it everywhere.
Fix: Compute full MSE for every estimator unless both are unbiased.
Inverting relative efficiency.
The ratio can be written either way. Students do not state which estimator is the reference.
Fix: Write which estimator is relative to which. Use MSE of the other estimator divided by MSE of yours. A value above 1 means yours is better.
Getting the sign of bias wrong.
Students compute θ − E[θ̂] instead of E[θ̂] − θ.
Fix: Bias is expected value minus truth. The sign does not change MSE, but it matters when you are asked to state the bias.
Adding variances of dependent terms or forgetting c² when scaling.
Students apply Var(X + Y) = Var(X) + Var(Y) without checking independence, or write Var(cX) = cVar(X).
Fix: Check independence first. Use Var(cX) = c²Var(X) every time.
Concluding that an unbiased estimator is always the best.
Unbiasedness is taught first and feels like the ideal property.
Fix: Remember that a small bias can be worth accepting if the variance falls by more. Judge by MSE.

Worked examples

Example 1

X1, ..., X10 is a random sample from a distribution with mean μ and variance 25. Estimator A is X̄. Estimator B is 0.8X̄. (a) Find the MSE of each estimator. (b) For μ = 10 and for μ = 2, say which is better and give the relative efficiency of the better estimator.

Show the solution

Estimator A: E[X̄] = μ, so bias = 0. Var(X̄) = 25 ÷ 10 = 2.5. So MSE(A) = 2.5.
Estimator B: E[0.8X̄] = 0.8μ, so bias = 0.8μ − μ = −0.2μ.
Var(B) = 0.8² × 2.5 = 0.64 × 2.5 = 1.6.
MSE(B) = 1.6 + (0.2μ)² = 1.6 + 0.04μ².
For μ = 10: MSE(B) = 1.6 + 0.04 × 100 = 5.6. This is more than 2.5, so A is better. Efficiency of A relative to B = 5.6 ÷ 2.5 = 2.24.
For μ = 2: MSE(B) = 1.6 + 0.04 × 4 = 1.76. This is less than 2.5, so B is better. Efficiency of B relative to A = 2.5 ÷ 1.76 = 1.42 (to two decimal places).

Answer: MSE(A) = 2.5 and MSE(B) = 1.6 + 0.04μ². For μ = 10, A is better (efficiency 2.24). For μ = 2, B is better (efficiency about 1.42). B beats A when 1.6 + 0.04μ² < 2.5, that is when μ² < 22.5.

Example 2

A random sample of size n = 5 is taken from a normal distribution with variance σ². Compare two estimators of σ²: S² (divisor n − 1) and T = (1 ÷ n) Σ(Xi − X̄)² (divisor n). Find the MSE of each and the efficiency of T relative to S².

Show the solution

For a normal sample, (n − 1)S² ÷ σ² follows a chi-square distribution with n − 1 degrees of freedom. Its variance is 2(n − 1).
So Var(S²) = σ⁴ × 2(n − 1) ÷ (n − 1)² = 2σ⁴ ÷ (n − 1). S² is unbiased, so MSE(S²) = 2σ⁴ ÷ 4 = 0.5σ⁴.
T = ((n − 1) ÷ n) S² = 0.8S². So E[T] = 0.8σ² and bias = −0.2σ² = −σ² ÷ n.
Var(T) = 0.8² × 0.5σ⁴ = 0.32σ⁴.
MSE(T) = 0.32σ⁴ + (0.2σ²)² = 0.32σ⁴ + 0.04σ⁴ = 0.36σ⁴. This agrees with (2n − 1)σ⁴ ÷ n² = 9σ⁴ ÷ 25.
Efficiency of T relative to S² = MSE(S²) ÷ MSE(T) = 0.5 ÷ 0.36 = 25 ÷ 18 = 1.39 (to two decimal places).

Answer: MSE(S²) = 0.5σ⁴ and MSE(T) = 0.36σ⁴. T is biased but has the smaller MSE, with relative efficiency about 1.39. This is the bias-variance trade-off.

Exam tips

Always show the decomposition MSE = Var + bias² as a line of working. Examiners award marks for the method even if the arithmetic slips.
In written questions you may be asked to prove the decomposition. Learn the short proof: add and subtract E[θ̂] inside the square and show the cross term vanishes.
State assumptions such as independence and normality. The MSE formulas for variance estimators hold for normal samples only.
When a question says 'compare', end with a sentence that names the better estimator and says why, including the dependence on θ if there is one.
In the computer-based paper, you can check your algebra by simulating many samples and averaging squared errors. The simulated MSE should be close to your formula.

Practice questions from Estimators and their properties

Mean Squared Error and Efficiency: frequently asked questions

What is the formula for the mean squared error of an estimator?

MSE(θ̂) = E[(θ̂ − θ)²] = Var(θ̂) + [bias(θ̂)]². Here bias(θ̂) = E[θ̂] − θ. For an unbiased estimator the MSE is just the variance.

How do I compare two estimators using efficiency?

Compute the MSE of each. The efficiency of the first relative to the second is MSE(second) ÷ MSE(first). If it is above 1, the first estimator is better. For two unbiased estimators you can use variances instead.

Can a biased estimator be better than an unbiased one?

Yes. If a small bias brings a larger fall in variance, the MSE is lower. The divisor-n variance estimator for a normal sample is a standard example.

Does a smaller MSE mean the estimator is consistent?

Not by itself. Consistency is about what happens as n grows. If MSE tends to 0 as n tends to infinity, the estimator is consistent. A small MSE at one fixed n tells you nothing about that.