2 marks, −0.667StatisticsPoint and interval estimationProperties of estimatorsHardabout 3 min

GATE 2021 Question 22: Point and interval estimation

If $S^2=\dfrac{1}{n-1}\sum_{i=1}^n(x_i-\bar x)^2$ is an unbiased and consistent estimator of the population variance, then one can conclude that $S=\sqrt{\dfrac{1}{n-1}\sum_{i=1}^n(x_i-\bar x)^2}$ is a(an) ______ estimator of the population standard deviation.

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Answer and worked solution

Answer: (B)

$S^2$ is given as unbiased for $\sigma^2$ : $E[S^2]=\sigma^2$ .

Bias of $S=\sqrt{S^2}$ : the square root function is strictly concave. By Jensen's inequality, for a strictly concave function $g$ , $E[g(X)] < g(E[X])$ (when $X$ is non-degenerate). Applying this with $g(\cdot)=\sqrt{\cdot}$ and $X=S^2$ :

$E[S] = E[\sqrt{S^2}] < \sqrt{E[S^2]} = \sqrt{\sigma^2} = \sigma$

So $E[S] < \sigma$ strictly — $S$ is a biased (downward) estimator of $\sigma$ , even though the underlying $S^2$ is unbiased for $\sigma^2$ . Unbiasedness does NOT carry over through a nonlinear transformation in general.

Consistency of $S$ : consistency, unlike unbiasedness, DOES carry over through continuous transformations (by the continuous mapping theorem) — since $S^2 \xrightarrow{p} \sigma^2$ (consistency of $S^2$ ) and the square root function is continuous, $S=\sqrt{S^2} \xrightarrow{p} \sqrt{\sigma^2}=\sigma$ . So $S$ remains a consistent estimator of $\sigma$ .

Therefore the answer is (B): biased and consistent.

The trap: (A) unbiased and consistent

Students often assume that if $S^2$ is 'unbiased and consistent,' then simply taking its square root preserves both properties automatically — but unbiasedness is a property about EXPECTATIONS, which generally does NOT survive nonlinear transformations (per Jensen's inequality), while consistency (a property about convergence in probability) DOES survive continuous transformations. Conflating these two very different mathematical properties is the core trap here.

Plain-text version

If S²=1/(n-1)Σᵢ₌₁ⁿ(xᵢ-x̄)² is an unbiased and consistent estimator of the population variance, then one can conclude that S=√(1/(n-1)Σᵢ₌₁ⁿ(xᵢ-x̄)²) is a(an) estimator of the population standard deviation. (A) unbiased and consistent (B) biased and consistent (C) unbiased and inconsistent (D) biased and inconsistent

Also written as: If S^2=1/(n-1) sum_(i=1)^n(x_i-x_bar)^2 is an unbiased and consistent estimator of the population variance, then one can conclude that S=sqrt(1/(n-1) sum_(i=1)^n(x_i-x_bar)^2) is a(an) estimator of the population standard deviation.

Point and interval estimation has come up 28 times in 6 exams since 2008.

What this question tests

This question is on Properties of estimators, part of Point and interval estimation in Statistics.

What makes an estimator good: unbiased, consistent and efficient.

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