Skip to content

IAI Actuarial Core Principles · Actuarial Statistics

Estimators and Their Properties: Chapter Guide for IAI CS1

An estimator is a rule that turns sample data into a guess for an unknown parameter. You judge it by bias, mean squared error, consistency, sufficiency and the Cramér-Rao lower bound. To solve questions, write the estimator, find its mean and variance, then test each property in turn.

What this chapter covers

This chapter asks one question: given a sample, how good is a formula for estimating a parameter? You meet the main ideas of point estimation. An estimator is a random variable built from the sample. An estimate is the number you get from one observed sample. Then you learn the tests used to compare estimators: bias, mean squared error (MSE), efficiency, consistency and sufficiency. The chapter ends with the Cramér-Rao lower bound (CRLB), which sets a floor on the variance of unbiased estimators.

The chapter sits in the Statistical inference part of CS1, which the 2026 syllabus weights at 25%. It builds on random variables and distributions, because you need expectations, variances and standard results such as E(X̄) = μ and Var(X̄) = σ² ÷ n. It also feeds straight into the topics after it. Method of moments and maximum likelihood give you estimators, and this chapter gives you the tools to judge them. Confidence intervals, hypothesis tests and regression all rely on the same ideas.

The work is mostly algebra with expectations and variances, plus a few definitions you must state precisely. You will see it in both the multiple-choice section and the written questions of Paper A. Paper B can also test it, for example by simulating an estimator in R to see its bias and variance. Learn the definitions first, then practise the calculations until they feel routine.

Almost every later inference topic asks you to choose or justify an estimator, so the properties here are used again and again. Written questions often award marks for a clean definition, a correct expectation calculation and a clear conclusion, and these are marks you can secure with practice. Multiple-choice questions on bias and MSE are quick to answer once the method is automatic. Since statistical inference carries a large share of the CS1 syllabus, time spent here pays back across the whole paper.

Estimators and their properties: topics in the order to study them

  1. 1Point Estimation and EstimatorsStart here because you need the vocabulary of estimator, estimate and sampling distribution before you can judge anything.
  2. 2Unbiasedness and Bias of EstimatorsBias is the simplest property to test and needs only an expectation, so it builds your core calculation habit.
  3. 3Mean Squared Error and EfficiencyMSE extends bias by adding variance, so you must know both first; efficiency then compares estimators using it.
  4. 4Consistency of EstimatorsConsistency is about behaviour as the sample size grows, and it usually reuses your bias and variance results.
  5. 5Sufficiency and Cramér-Rao Lower BoundThis is the most abstract part, so leave it until the basic properties are secure; the CRLB then gives a benchmark for variance.

How to prepare Estimators and their properties

Treat this chapter as a short toolkit. Learn each definition exactly, then practise applying it to the same few standard estimators until you can do it without notes.

  1. Write the definitions on one page in your own words: estimator, estimate, bias, MSE, consistency, sufficiency. Check each against the course notes.
  2. Memorise the standard results: E(X̄) = μ, Var(X̄) = σ² ÷ n, and E(S²) = σ² when S² uses the divisor n − 1. Know why the divisor n gives a biased estimate of σ².
  3. Practise the bias routine: write the estimator, take its expectation, subtract the true parameter. Then do the same with MSE using MSE = Var + bias².
  4. Test consistency with a quick check: if bias → 0 and variance → 0 as n → ∞, the estimator is consistent. Practise this on at least three different estimators.
  5. Work through the CRLB with the formula, then check whether a given unbiased estimator reaches it. Practise finding the Fisher information for common distributions.
  6. For sufficiency, practise the factorisation criterion on a few distributions until you can spot the split of the likelihood quickly.
  7. Finish with timed past-style questions, and in R simulate an estimator many times to see its bias and variance for Paper B practice.

Common mistakes in Estimators and their properties

  • Confusing the estimator with the estimate.

    Fix: Say that the estimator is a random variable with a distribution, and the estimate is one number. Take expectations only of the estimator.

  • Using the divisor n for sample variance and calling it unbiased.

    Fix: Remember that dividing by n − 1 gives an unbiased estimator of σ². Show the expectation if asked to prove it.

  • Forgetting the bias term in MSE.

    Fix: Always write MSE = Var + bias² first, then check whether the bias is zero.

  • Applying the CRLB to biased estimators or ignoring regularity conditions.

    Fix: State that the simple bound applies to unbiased estimators, and note that the usual regularity conditions must hold, for instance that the range of the variable does not depend on θ.

  • Claiming consistency from unbiasedness alone.

    Fix: Unbiasedness is about the average at a fixed n. Consistency needs behaviour as n grows, so check the variance as well.

  • Choosing an estimator on bias alone.

    Fix: Compare MSE. A slightly biased estimator with much smaller variance can have the lower MSE, and you should say so in your conclusion.

Last-day revision: Estimators and their properties

  • An estimator is a random variable; an estimate is the number from one observed sample.
  • Bias of θ̂ = E(θ̂) − θ. The estimator is unbiased if this is 0 for every value of θ.
  • MSE(θ̂) = E[(θ̂ − θ)²] = Var(θ̂) + [bias(θ̂)]².
  • For unbiased estimators, MSE equals the variance.
  • A lower MSE means a better estimator; a biased estimator can beat an unbiased one on MSE.
  • Sample variance with divisor n − 1 is unbiased for σ²; the divisor n version is biased.
  • Consistency: the estimator converges to the true value as n → ∞. Bias → 0 and variance → 0 together are enough.
  • A statistic is sufficient if it carries all the information in the sample about the parameter.
  • Factorisation criterion: the likelihood splits as g(t, θ) × h(x), where t is the statistic.
  • CRLB: for an unbiased estimator, Var(θ̂) ≥ 1 ÷ I(θ), where I(θ) is the Fisher information of the whole sample, under the usual regularity conditions.
  • An unbiased estimator that reaches the CRLB is efficient, but many good estimators do not reach it.
  • State the conditions behind any rule you quote, such as unbiasedness for the CRLB.

Estimators and their properties practice questions

Estimators and their properties: frequently asked questions

Which topics in this chapter come up most in the exam?

Bias, MSE and consistency are the most calculation-friendly, so they suit both multiple-choice and written questions. The CRLB and sufficiency appear too, usually as a short derivation or a check of a given estimator. Prepare all five, but make the first three automatic.

How do I prove an estimator is consistent?

Find its expectation and variance. If the bias tends to 0 and the variance tends to 0 as n → ∞, the estimator is consistent. State both limits clearly in your answer.

Is an unbiased estimator always better than a biased one?

No. Compare them by MSE, which combines variance and squared bias. A biased estimator can have a smaller MSE and so be preferred.

Do I need R for this chapter?

The theory is tested mainly in Paper A. In Paper B you may be asked to simulate samples and compare the mean and variance of different estimators. Practise a short simulation loop so you can do it quickly.