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Explain what the bias of an estimator is and what it indicates about the estimator.

FRM® Part I · Quantitative Analysis · Sample Moments · FRM1.QA.5.d

The idea

The bias of an estimator is its expected value minus the true parameter: Bias = E(estimator) − θ. It measures the systematic tendency to overshoot or undershoot when the estimator is applied to many samples, not the error in a single sample. The sample mean is unbiased, and so is the sample variance with divisor n − 1. The divisor-n variance has a bias of minus σ² ÷ n. Zero bias does not mean precise, because an unbiased estimator can have a large variance. Mean squared error equals variance plus bias squared, so a slightly biased estimator can be better overall. Remember also that even the n − 1 sample standard deviation is biased low, because the square root is concave.

Formulas

Bias of an estimator

Bias = E[θ̂] − θ

θ̂
the estimator
θ
the true parameter

An expectation across repeated samples, not the error in one sample. The divisor-n variance has bias −σ² ÷ n.

Mean squared error of an estimator

MSE = Var(θ̂) + Bias²

Var(θ̂)
the variance of the estimator
Bias
its expected value minus the true parameter

Lets a slightly biased but less variable estimator beat an unbiased one. Zero bias alone does not make an estimator precise.

A practice question

From the question bank, with the reason each wrong option is wrong.

An estimator of a parameter is found to have zero bias. What does this tell us?

  1. Averaged over repeated samples, the estimator equals the true parameter value. - correct
  2. Every estimate it produces equals the true parameter value.Why not: Unbiased estimators still produce estimates that differ from the parameter in any given sample.
  3. It has the smallest variance among all possible estimators.Why not: Unbiasedness says nothing about variance; a low-variance biased estimator can be preferable.
  4. It is guaranteed to be consistent.Why not: An unbiased estimator need not converge as n grows; for example, using only the first observation to estimate a mean.

Zero bias is a statement about the center of the sampling distribution only. Precision (variance) and large-sample behaviour (consistency) are separate properties.

A harder one

Returns are independent with variance 9. An analyst uses the sample variance with divisor n on a sample of n = 25, for which E[estimator] = (n - 1)/n x sigma². What is its bias, closest to?

  1. +0.36Why not: Gets the size but the wrong sign: the divisor-n estimator is biased downward.
  2. -0.36 - correct
  3. -0.375Why not: Divides by n - 1 instead of n when computing sigma²/n.
  4. 8.64Why not: Reports the expected value of the estimator rather than its bias (expected value minus sigma²).

Bias = E[estimator] - sigma² = 9 x 24/25 - 9 = -0.36. It equals -sigma²/n and shrinks as n grows.

The hardest tier

Estimator A is unbiased with variance 4.0. Estimator B has a bias of 0.5 and variance 3.0. Which is preferable under mean squared error (MSE = variance + bias²), and why?

  1. A, because zero bias always means the lowest MSE.Why not: Ignores that MSE also includes variance; a small bias can be a worthwhile trade for lower variance.
  2. B, with an MSE of 3.00, because bias does not enter MSE.Why not: Drops the bias term from MSE; it enters squared.
  3. B, with an MSE of 3.50, because MSE adds the bias to the variance.Why not: Adds the bias unsquared (3.0 + 0.5) instead of squaring it (3.0 + 0.25).
  4. B, with an MSE of 3.25 against 4.00 for A. - correct

MSE = variance + bias². For A this is 4.0 + 0 = 4.0; for B it is 3.0 + 0.25 = 3.25. B is closer to the truth on average despite being biased.

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