An estimator θ̂ satisfies E[θ̂] = θ + 5/n for all sample sizes n. Which of the following is correct?
Aθ̂ is biased at every finite n, but consistent — the bias 5/n shrinks to zero as n → ∞
Bθ̂ is both biased and inconsistent — it never equals θ exactly
Cθ̂ is unbiased — the formula shows it equals θ plus a correction term
Dθ̂ is unbiased and consistent because the correction term is small for large n
Bias = E[θ̂] − θ = 5/n ≠ 0 for any finite n, so θ̂ is biased. But as n → ∞, the bias 5/n → 0, and if the variance also shrinks appropriately, θ̂ converges in probability to θ — making it consistent. This demonstrates that unbiasedness and consistency are distinct properties: an estimator can be biased at every finite sample size yet still be consistent.
Question 2 Multiple Choice
You are estimating average income in a city that includes a few billionaires. Which statement best captures the robustness-efficiency tradeoff between the sample mean and median?
AThe sample median is robust to the billionaires' extreme values but less efficient than the sample mean if income were normally distributed
BThe sample mean is robust because averaging spreads the effect of extreme values across all observations
CThe sample median is strictly better than the sample mean — more efficient and more robust in every real-world setting
DNeither estimator is affected by extreme values once the sample is large enough
The sample mean is efficient under normality (minimum variance among unbiased estimators) but is sensitive to outliers — a single billionaire can drastically pull the mean away from the typical income. The sample median, depending only on the middle rank, is unaffected by how extreme the outliers are. The tradeoff: efficiency under the assumed model (mean) vs. safety when the model is violated (median). In income data, where skewness is severe, robustness often matters more.
Question 3 True / False
An estimator can be consistent without being unbiased at any finite sample size.
TTrue
FFalse
Answer: True
True. Consistency is an asymptotic property: the estimator converges in probability to the true parameter as n → ∞. This is compatible with having nonzero bias at every finite n, as long as the bias shrinks to zero sufficiently fast (and variance also shrinks). For example, an estimator with E[θ̂] = θ + 1/n and Var[θ̂] → 0 is biased for every n yet consistent. Unbiasedness (E[θ̂] = θ for all n) and consistency are independent properties.
Question 4 True / False
An unbiased estimator is generally consistent.
TTrue
FFalse
Answer: False
False. Unbiasedness means E[θ̂] = θ at every sample size — the estimator aims correctly on average. But it says nothing about whether estimates cluster more tightly around θ as n grows. An unbiased estimator could have constant or even growing variance, meaning larger samples provide no additional accuracy. Consistency requires both that the estimator aims at θ and that its variance shrinks to zero as n → ∞. Unbiasedness guarantees the aim; it does not guarantee convergence.
Question 5 Short Answer
Explain in your own words why an unbiased estimator is not necessarily consistent.
Think about your answer, then reveal below.
Model answer: Unbiasedness means the estimator is correct on average — its expected value equals the true parameter — but it does not constrain how spread out individual estimates are or whether that spread decreases with more data. An estimator could be perfectly centered (no bias) yet have high variance that stays constant as n grows, so collecting more data still leaves you with highly variable estimates. Consistency requires that as n increases, the estimates concentrate around the true value (variance → 0). Unbiasedness addresses the center; consistency addresses the spread.
The clearest way to see this is to construct a counterexample: define θ̂ = X₁ (just the first observation, ignoring the rest). E[θ̂] = μ (unbiased), but Var[θ̂] = σ² regardless of n — it never shrinks. This estimator is unbiased but inconsistent. Real estimators rarely behave this pathologically, but the conceptual distinction matters for understanding what large samples can and cannot guarantee.