ISS 2016 Statistics Paper-2 Solution: Question 26 (SPRT Stopping Rule Statements)
Continuing our question-by-question walk through the ISS Statistics Paper-2 archive, we now arrive at Question 26 of the 2016 paper. This one sits in the sequential analysis portion of the syllabus, and it is a favourite trap-setter because it tests whether you actually remember which direction the likelihood ratio points, not just whether you remember the names "SPRT" and "Wald boundaries".
Quick Summary
Topic: Sequential Probability Ratio Test (SPRT) — stopping/continuation rule
Question reference: ISS 2016, Statistics Paper-2, Question 26
Correct answer: Option (d) — Statement 3 only
The Question as Asked
For testing the hypothesis H0 : θ = θ0 against the alternative H1 : θ = θ1, λ is the likelihood ratio; A and B (B < A) are the constants. Under the SPRT criteria, consider the following statements:
If λ ≥ A, we terminate the process with the acceptance of H0
If λ ≤ B, we terminate the process with the rejection of H0
If B < λ < A, we continue sampling by taking an additional observation
Which of the above statements is/are correct?
(a) 1, 2 and 3
(b) 1 and 3 only
(c) 2 and 3 only
(d) 3 only
Step 1: Set Up the Likelihood Ratio Correctly
Before touching the three statements, fix what λ actually means, because the whole question hinges on this. In Wald's Sequential Probability Ratio Test, after n observations x1, x2, ..., xn have been collected, we compute:
λn = [ L1n ] / [ L0n ] = [ ∏ f(xi ; θ1) ] / [ ∏ f(xi ; θ0) ], for i = 1 to n
So λn is the likelihood of the data under H1 divided by the likelihood of the data under H0. This ordering — H1 on top, H0 on the bottom — is the standard convention used in Wald's original formulation and in the SPRT chapters of the textbooks this syllabus draws from. Everything else in the question follows from reading this ratio correctly.
Step 2: Recall the Actual Wald Decision Rule
Given a desired size α and power constraint related to Type II error β, we fix two constants B < A (commonly approximated as A ≈ (1 − β)/α and B ≈ β/(1 − α)). After each new observation, we recompute λn and apply:
If λn ≥ A:stop and reject H0 (i.e., accept H1).
If λn ≤ B:stop and accept H0.
If B < λn < A:continue sampling — take one more observation and recompute.
The intuition is simple once you hold on to Step 1: a large λn means the data are much more consistent with H1 than with H0, so a large λn should push you towards rejecting H0, not accepting it. Symmetrically, a small λn (data far more consistent with H0) should push you towards accepting H0.
Pro Tip: The single most common way students lose marks on this exact question type is by mixing up "large λ favours H1" with "large λ favours H0". A quick sanity anchor that is usually drilled until it becomes automatic: λ is built as (likelihood under the alternative) ÷ (likelihood under the null), so a
Step 3: Test Each Statement Against the Real Rule
Statement 1:"If λ ≥ A, we terminate the process with the acceptance of H0." The correct rule says λ ≥ A leads to rejecting H0 (accepting H1), not accepting it. Statement 1 has the conclusion backwards —false.
Statement 2:"If λ ≤ B, we terminate the process with the rejection of H0." The correct rule says λ ≤ B leads to accepting H0, not rejecting it. This is also backwards —false.
Statement 3:"If B < λ < A, we continue sampling by taking an additional observation." This matches the Wald rule exactly — the region strictly between B and A is the inconclusive zone where sampling continues —true.
Step 4: Match to the Options
Only Statement 3 is correct, and Statements 1 and 2 are both false (each one simply swaps "accept" and "reject"). Scanning the four options, this corresponds exactly to option (d): 3 only.
This matches the official answer key, which also lists D for Question 26. Independent derivation and the official key agree — no discrepancy here.
Why This Question Matters
SPRT questions recur across ISS papers because the test is genuinely important — it is the classical example of a test that, for a given pair of error probabilities, needs a smaller expected sample size than any fixed-sample-size test with the same error rates (the Wald–Wolfowitz optimality result). Examiners like to test it not through heavy computation but through exactly this kind of statement-matching question, because it separates candidates who memorised the shape of the rule from those who understand why the ratio moves the way it does. Once the "H1 on top" convention is internalised, all three boundary conditions become a one-line check rather than something to recall from memory.
Frequently Asked Questions
What exactly is the Sequential Probability Ratio Test?
SPRT is a hypothesis testing procedure, due to Abraham Wald, where the number of observations is not fixed in advance. After each new observation, the cumulative likelihood ratio is compared against two boundaries, and sampling stops as soon as the ratio crosses either boundary, or continues otherwise.
Why does a ratio greater than A mean we reject H0 instead of accepting it?
Because λ is defined as the likelihood under H1 divided by the likelihood under H0. A value of λ well above 1 (and above the threshold A) means the observed data are much more probable under H1 than under H0, which is evidence against H0, not in favour of it.
How are the constants A and B chosen in practice?
They are approximated from the desired significance level α and Type II error probability β using A ≈ (1 − β)/α and B ≈ β/(1 − α). These approximations, due to Wald, make the boundaries easy to compute without solving for the exact stopping distribution.
Can the SPRT go on forever without ever stopping?
In theory the continuation region could be re-entered indefinitely, but in practice SPRT terminates with probability 1 under either hypothesis being true — this is one of its proven properties, and it is part of why the test is considered reliable for real sampling schemes.
Is SPRT only for simple hypotheses like H0: θ = θ0?
The classical Wald SPRT, including the version examined here, is built for simple versus simple hypotheses. Extending it to composite hypotheses requires modifications and is a separate, more advanced topic.
Why is this type of question common in ISS Paper-2 rather than Paper-1?
Sequential analysis and SPRT sit within inference and testing theory, which is core Paper-2 territory in the ISS syllabus, alongside estimation, sufficiency, and testing of hypotheses — distinct from the more applied and methods-based content typically seen in Paper-1.
If any part of this SPRT stopping rule still feels shaky, drop a comment with where it gets confusing — and if you know a fellow ISS aspirant working through the same paper, sharing this one along might save them the same mix-up on exam day.
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