ISS 2016 Statistics Paper-2 Solution: Question 29 (Likelihood Ratio Test Basics)
This post continues our question-by-question walk through the ISS Statistics Paper-2 previous year papers, where we take one MCQ at a time and work it out from first principles rather than just naming the correct option. Today we are on 2016, Question 29, which tests a topic that trips up a lot of aspirants: the exact definition of a likelihood ratio test and how its critical region is built.
Quick Summary
Topic:Likelihood Ratio Tests (LRT) — a theory/statement-based question
Question Reference:ISS 2016, Statistics Paper-2, Question 29
Correct Answer:Option (d) — Statements 1, 2 and 3 are all correct
The Question, As Asked
Consider the following statements pertaining to likelihood ratio tests:
The likelihood ratio criterion offers a method of constructing tests of hypotheses which do not fall within the purview of Neyman–Pearson Lemma.
The likelihood ratio test statistic λ is the ratio of the supremum of the likelihood under H0 to the supremum of the likelihood in the unrestricted parametric space.
The critical region of the likelihood ratio test is the left hand tail of the distribution of λ.
Which of the above statements are correct?
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
Step 1: What Exactly Is a Likelihood Ratio Test?
Before checking each statement, let's build the idea from scratch, the way a beginner should see it for the first time.
Suppose you have data with a likelihood function L(θ), where θ is an unknown parameter living in some parameter space Θ. You want to test:
H0: θ ∈ Θ0 against H1: θ ∈ Θ − Θ0
where Θ0 is a subset of the full parameter space Θ (it could correspond to a single value, like θ = θ0, or to a whole restricted region).
The idea behind a likelihood ratio test (LRT) is simple: find the "best possible" likelihood value you could get if H0 is true (that is, restricting θ to lie only in Θ0), and compare it with the "best possible" likelihood value you could get if there were no restriction at all (letting θ range over the whole of Θ). If restricting to H0 barely hurts the likelihood, H0 looks plausible. If restricting to H0 hurts the likelihood a lot, H0 looks implausible.
"Best possible likelihood" here means the supremum (the least upper bound — for the well-behaved distributions you see in this exam, you can simply read this as the maximum) of the likelihood function over the relevant set of parameter values. That single idea is what lets us define the test statistic in Statement 2.
Step 2: Checking Statement 1 — Does LRT Go Beyond the Neyman–Pearson Lemma?
The Neyman–Pearson (NP) Lemma is the tool most students meet first when learning hypothesis testing. It tells you exactly how to build the most powerful test, but only for a very specific situation: testing one simple hypothesis against another simple hypothesis, i.e.
H0: θ = θ0 against H1: θ = θ1
Both the null and the alternative are single, fully-specified points. In real problems, however, you very often need to test something like H0: θ = θ0 against H1: θ ≠ θ0 (a composite alternative), or compare two nested models with several unknown parameters (as in ANOVA or regression). The NP Lemma, in its basic form, simply does not cover these cases — it was built for the simple-versus-simple setting.
This is exactly where the likelihood ratio test earns its importance as a general-purpose construction method: it works whether H0 and H1 are simple or composite, single-parameter or multi-parameter. So Statement 1 is correct: LRT is a method for constructing tests precisely in situations that fall outside what the Neyman–Pearson Lemma by itself can handle.
Step 3: Checking Statement 2 — The Formal Definition of λ
Now we write down the LRT statistic precisely. Using the notation from Step 1:
λ = [ supθ ∈ Θ0 L(θ) ] ÷ [ supθ ∈ Θ L(θ) ]
In words: the numerator is the largest value the likelihood can achieve when θ is forced to satisfy H0 (restricted to Θ0). The denominator is the largest value the likelihood can achieve when θ is allowed to roam over the entire parameter space Θ (unrestricted, sometimes written as the "full" or "unconstrained" maximum). This is precisely what Statement 2 describes, so Statement 2 is correct.
A useful thing to notice immediately: since Θ0 is a subset of Θ, the maximum over the smaller set can never exceed the maximum over the bigger set. In other words, the numerator can never be larger than the denominator, so:
0 ≤ λ ≤ 1
Keep this fact in your pocket — it is exactly what you need for Statement 3.
Step 4: Checking Statement 3 — Why Is the Critical Region the Left Tail?
Think about what a small value of λ actually means. λ is small when the numerator (best likelihood under H0) is much smaller than the denominator (best likelihood overall). That happens precisely when forcing θ into Θ0 costs you a lot of likelihood — in other words, when the data fit much better under the unrestricted model than under H0. That is strong evidence against H0.
On the other hand, if λ is close to 1, restricting θ to Θ0 barely reduced the likelihood at all, meaning the data are just as well explained by H0 as by the unrestricted model. That is evidence in favour of H0.
So the decision rule for an LRT is: reject H0 when λ is small, i.e. when λ ≤ k for some suitably chosen constant k (0 < k < 1), where k is fixed using the desired level of significance. Since we already established λ ∈ [0, 1], "small values of λ" is exactly the left-hand tail of λ's distribution. That confirms Statement 3 is correct too.
Pro Tip: The single easiest way to get this type of question wrong under exam pressure is to reflexively think "reject for large values of the test statistic," because that is the muscle memory built from chi-square and F tests. With λ, it is the opposite direction, precisely because λ is a ratio of a restricted maximum to an unrestricted maximum, capped at 1. Whenever a question about LRT throws a "which tail" statement at you, don't try to recall it as a fact — rederive it in ten seconds from the definition of λ, the way it's usually taught: numerator restricted, denominator unrestricted, ratio at most 1, small ratio means bad fit under H0, so reject on the left. Students who drill this three-line derivation until it's automatic (a habit that gets built early on at coaching setups like Sunrise Classes) stop losing marks to exactly this kind of directional trap.
Step 5: Final Answer
All three statements check out on independent verification:
Statement 1 is correct — LRT is a general construction method that extends testing beyond the simple-versus-simple world of the Neyman–Pearson Lemma.
Statement 2 is correct — λ is exactly the ratio of the restricted supremum of the likelihood to the unrestricted supremum.
Statement 3 is correct — because λ ∈ [0, 1] and small λ signals poor fit under H0, the critical region is the left tail of λ's distribution.
So the correct option is (d) 1, 2 and 3. This matches the official answer key exactly, which lists "D" for Question 29 — so there is no discrepancy to flag here.
Why This Question Matters
Likelihood ratio tests are one of the most heavily examined theory topics across ISS Paper-2, because they sit at the intersection of estimation theory and testing theory — the same maximization idea you use for MLEs resurfaces here as the backbone of a testing procedure. Questions on this topic are rarely about heavy calculation; they are about whether you truly understand the definition well enough to reason about direction (left tail vs right tail), scope (simple vs composite hypotheses), and the restricted-versus-unrestricted comparison, rather than having memorized a one-line summary. Once this definition is solid, a whole family of related questions — on the asymptotic chi-square distribution of −2 log λ, on nested model comparisons, and on the relationship between LRT and other optimal tests — becomes far easier to answer correctly and quickly.
Frequently Asked Questions
Why is the LRT statistic λ always between 0 and 1?
Because the numerator maximizes the likelihood over Θ0, a subset of the full parameter space Θ, while the denominator maximizes over all of Θ. A maximum taken over a smaller set can never exceed a maximum taken over a larger set that contains it, so the numerator is always less than or equal to the denominator, keeping the ratio between 0 and 1.
Is the Neyman–Pearson Lemma ever used together with the likelihood ratio test?
Yes. In the simple-versus-simple case, the most powerful test given by the Neyman–Pearson Lemma actually turns out to be a monotone function of the likelihood ratio itself, so the two approaches agree there. The distinction in this question is about scope: NP Lemma is proven only for simple-versus-simple hypotheses, while the LRT construction extends naturally to composite hypotheses where NP does not directly apply.
Why do we reject H0 for small λ instead of large λ?
Small λ means the likelihood achievable under H0 is much worse than the likelihood achievable without any restriction, which is evidence that the data do not fit H0 well. Large λ (close to 1) means restricting to H0 barely changed the best achievable likelihood, which supports H0. So the rejection region is naturally the region of small λ values, i.e. the left tail.
What is the practical distribution used for λ in exams and applications?
Exact small-sample distributions of λ are often complicated, so in practice we commonly use the large-sample (asymptotic) result that −2 log λ is approximately distributed as a chi-square random variable, with degrees of freedom equal to the difference in the number of free parameters between the unrestricted and restricted models. That large-sample chi-square test is what you will see used most often in applied LRT problems.
How is this different from the F-test used in ANOVA or regression?
The F-test you use in ANOVA or regression is actually a special case that arises from applying the likelihood ratio idea (or an equivalent geometric argument) under normal-theory linear models; in that setting the general LRT reduces algebraically to a ratio of sums of squares that follows an exact F distribution. So the F-test is not a separate concept — it is what the likelihood ratio machinery gives you once you add the assumption of normal errors in a linear model.
Do I need to memorize the formula for λ, or is it better to derive it each time?
It is far more reliable to understand the construction — restricted maximum over unrestricted maximum — than to memorize a symbol string, because exam questions like this one test whether you can reason about direction, scope, and range (0 to 1) rather than just recognizing a formula. Once the construction is internalized, the "which tail," "what range," and "when does it reduce to NP" questions all answer themselves.
If any part of this derivation felt unclear, drop your doubt in the comments and we'll walk through it together — and if you know a fellow ISS aspirant working through the same Paper-2 syllabus, consider sharing this post with them; going through these statement-based theory questions one at a time is one of the most reliable ways to build the kind of solid conceptual footing this exam rewards.
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