An estimator θ̂ is defined as 'always return the first observation X₁, regardless of how many observations are collected.' Which properties does this estimator have?
AIt is both unbiased and consistent
BIt is unbiased but not consistent
CIt is consistent but not unbiased
DIt is neither unbiased nor consistent
E[X₁] = μ for any sample size n, so the estimator is unbiased — it hits the true mean on average. But its distribution never concentrates around μ as n grows; it always has variance σ². Consistency requires P(|θ̂ − θ| > ε) → 0 as n → ∞, which fails here because no data beyond X₁ is ever used. This is the clearest demonstration that unbiasedness does not imply consistency.
Question 2 Multiple Choice
The MLE for a normal distribution's variance divides by n rather than n−1. How would you characterize this estimator?
AUnbiased and consistent — MLE guarantees both properties in large samples
BBiased and inconsistent — the n denominator creates error that never disappears
CBiased but consistent — the bias is −σ²/n which shrinks to zero as n grows
DUnbiased but inconsistent — the large-sample properties of MLE correct the bias
The MLE variance estimator has E[θ̂] = (n−1)σ²/n ≠ σ², so it is biased by −σ²/n at every finite n. But as n → ∞, this bias shrinks to 0 and the variance of the estimator also shrinks — so it converges in probability to σ². This is the canonical example of a consistent but biased estimator, showing directly that consistency does not require unbiasedness.
Question 3 True / False
An unbiased estimator is generally more accurate than a biased estimator for the same parameter.
TTrue
FFalse
Answer: False
Unbiasedness means E[θ̂] = θ — on average, you're right. But an unbiased estimator can have very high variance, meaning individual estimates scatter widely around the true value. A slightly biased estimator with much lower variance can produce estimates consistently closer to the truth in practice. Accuracy in the sense of mean squared error depends on both bias and variance (MSE = Bias² + Variance), and reducing variance sometimes justifies accepting some bias.
Question 4 True / False
An estimator can be unbiased at every fixed sample size n while still failing to converge to the true parameter as n → ∞.
TTrue
FFalse
Answer: True
The 'always return X₁' estimator demonstrates this. It is unbiased for every n (E[X₁] = μ regardless of sample size), but its distribution never concentrates around μ — its variance is always σ². Consistency requires the distribution to collapse around the true value as n grows; unbiasedness only requires the mean of that distribution to be correct at each fixed n. These are completely different requirements.
Question 5 Short Answer
Why are unbiasedness and consistency described as 'independent' properties? Give an example showing that one does not imply the other.
Think about your answer, then reveal below.
Model answer: They are independent because each concerns a different aspect of an estimator's behavior at different scales. Unbiasedness is a finite-sample property: E[θ̂] = θ at each fixed n. Consistency is asymptotic: the estimator converges in probability to θ as n → ∞. Example of unbiased but inconsistent: θ̂ = X₁ (always use the first observation) — E[X₁] = μ for all n, but its variance never shrinks. Example of consistent but biased: MLE for variance (divides by n) has bias −σ²/n → 0, so it converges even though it's biased at every finite n.
The independence of these properties is practically important: in small-sample settings, unbiasedness may matter more (you can't wait for asymptotics); in large-sample settings, consistency and efficiency (asymptotic variance) often matter more than finite-sample unbiasedness.