ISS 2016 Statistics Paper-2 Solution: Question 36 (Consistency vs Unbiasedness of Estimators)
Continuing our question-by-question walk through the ISS Statistics Paper-2 archive, today we take up Question 36 from the 2016 paper. If you have been following this series from Question 1 onward, you will notice a pattern: the exam loves to test whether you truly understand the definitions behind estimator properties, not just their names.
Quick Summary
Topic: Properties of estimators — consistency versus unbiasedness
Question Reference: ISS 2016, Statistics Paper-2, Question 36
Correct Answer: Option (b) — Statements 1 and 3 only
The Question
Consider the following statements regarding the properties of an estimator:
Every consistent estimator is unbiased.
If an estimator tn based on n observations is a consistent estimator of θ, then for any ε > 0, Prob {|tn − θ| < ε} tends to one as n → ∞.
Since consistency is a limiting property, no consistent estimator actually exists as sample size is finite.
Which of the above statements are not correct?
(a) 2 and 3 only
(b) 1 and 3 only
(c) 1 and 2 only
(d) 1, 2 and 3
Step 1: Get the two definitions straight before touching the statements
Before judging any of the three statements, write down the two definitions being tested, side by side, because the whole question is really just checking whether you keep them separate.
Unbiasedness is an exact, finite-sample property. An estimator tn of a parameter θ is unbiased if
E(tn) = θ, for every sample size n.
This has to hold exactly, even for n = 1 or n = 5. There is no "as n grows" clause anywhere in this definition.
Consistency, on the other hand, is an asymptotic (large-sample, limiting) property. An estimator tn is said to be a consistent estimator of θ if, for every ε > 0,
Prob {|tn − θ| < ε} → 1 as n → ∞.
In words: as you collect more and more data, the probability that the estimator lands within any small neighbourhood of the true θ gets closer and closer to 1. This is exactly convergence in probability of tn to θ, written tn →P θ.
Step 2: Examine Statement 1 — "Every consistent estimator is unbiased"
This statement tries to force a link between the two definitions above, but no such link exists in general. Consistency is about what happens in the limit as n → ∞; unbiasedness is about the expectation being exactly θ at every finite n. An estimator can satisfy one without satisfying the other.
A standard textbook counterexample makes this concrete. Take a random sample X1, X2, ..., Xn from a population with mean μ and finite variance σ2, and consider the (biased) estimator of σ2 given by
Sn2 = (1/n) Σ (Xi − X̄)2.
Its expectation works out to E(Sn2) = ((n−1)/n) σ2, which is strictly less than σ2 for every finite n — so Sn2 is biased for every n. Yet as n → ∞, the factor (n−1)/n → 1, and by the weak law of large numbers Sn2 converges in probability to σ2. So Sn2 is consistent but never unbiased. This single example is enough to break Statement 1: it is not correct.
Step 3: Examine Statement 2 — the probability limit condition
Statement 2 says: if tn is a consistent estimator of θ, then for any ε > 0, Prob{|tn − θ| < ε} tends to one as n → ∞.
Compare this word-for-word with the definition of consistency written out in Step 1. They are identical. Statement 2 is not an extra claim that needs proving — it is the formal definition of consistency, restated. So Statement 2 is correct.
Step 4: Examine Statement 3 — the "does consistency even exist" claim
Statement 3 argues that because consistency is defined only "in the limit," no estimator can actually be consistent for a finite sample, so consistent estimators effectively "do not exist." This confuses the mathematical language used to define a property with the existence of that property.
A limit statement like n → ∞ describes the behaviour of a whole sequence of estimators {tn} as n increases without bound — it does not claim that any single tn must literally be evaluated "at infinity" to count as consistent. Plenty of everyday estimators satisfy this limiting condition perfectly well. The sample mean X̄ is a textbook example: it is consistent for the population mean μ by the weak law of large numbers, and it is a perfectly ordinary, computable statistic at every finite n. So consistent estimators very much exist; Statement 3 is not correct.
Pro Tip: Whenever a statement pairs an asymptotic property (consistency, asymptotic efficiency, asymptotic normality) with a claim about "existence" or with a finite-sample property like unbiasedness, slow down and ask whether the statement is confusing the *definition* of a limit with a claim about what happens *before* the limit. That single habit — separating "how a property is defined" from "what it implies about finite samples" — is the way this distinction is usually drilled into shape, through repeated practice on exactly this kind of statement-based MCQ, until spotting the trap becomes automatic.
Step 5: Combine the results and match to the options
Putting the three findings together:
Statement 1:not correct
Statement 2: correct
Statement 3:not correct
The question asks which statements are not correct, so the answer set is {1, 3}. Scanning the options, this is exactly option (b) — "1 and 3 only."
This matches the official answer key for ISS 2016 Statistics Paper-2, Question 36, which also lists option (b)as the correct response. Our independent derivation and the official key agree, so we can state the final answer with confidence:Option (b) — Statements 1 and 3 only.
Why This Question Matters
Estimator properties — unbiasedness, consistency, efficiency, sufficiency — are among the most heavily tested topics in ISS Statistics Paper-2, and examiners frequently probe them through exactly this kind of "which statements are true/false" format rather than direct computation. Getting comfortable with the precise definitions, and with standard counterexamples like the biased-but-consistent variance estimator used above, pays off across dozens of similar questions in inference, not just this one.
Frequently Asked Questions
Is every unbiased estimator automatically consistent?
No. Unbiasedness only guarantees the expectation equals θ at each finite n; it says nothing about the variance shrinking as n grows. An estimator can be unbiased for every n yet fail to converge in probability to θ if its variance does not vanish as n → ∞, so unbiasedness alone does not imply consistency.
What is the easiest way to remember the difference between unbiasedness and consistency?
Unbiasedness is a "here and now" property: it must hold exactly at whatever sample size you have. Consistency is a "what happens eventually" property: it only describes behaviour as the sample size grows without bound. Keeping this exact-versus-eventual distinction in mind resolves most statement-based questions on this topic.
Can you give another common example of an estimator that is consistent but biased?
Yes — the maximum likelihood estimator of the variance of a normal population, (1/n) Σ (Xi − X̄)2, is the standard example, and it is exactly the Sn2 used in this solution. It is biased downward for every finite n but converges in probability to the true variance as n → ∞.
Does the weak law of large numbers relate directly to consistency?
Yes, very directly. The weak law of large numbers is itself a statement that the sample mean converges in probability to the population mean, which is precisely the definition of consistency applied to X̄ as an estimator of μ. Many consistency proofs in this syllabus reduce to invoking this law.
Why is Statement 3 considered a logical trap rather than a statistical one?
Statement 3 does not misuse any formula; it misuses logic. It takes the true fact that consistency is defined via a limit and draws the false conclusion that no finite-sample estimator can possess the property, when in fact the property describes the long-run behaviour of the sequence of finite-sample estimators itself. Recognising this kind of reasoning error is as important as recognising a wrong formula.
Are consistency and sufficiency the same thing?
No, they are unrelated properties. Sufficiency is about whether a statistic captures all the information in the sample about θ (it can be checked at any fixed n via the factorisation theorem), while consistency is about convergence of the estimator's value to θ as n → ∞. An estimator can be sufficient without being consistent, and vice versa.
If any step above felt unclear, or if you solved this one differently, drop a comment below — working through the disagreement is often where the real learning happens. And if this solution helped, consider sharing it with a fellow ISS aspirant who is also grinding through the estimator-properties chapter.
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