In a likelihood ratio test, the statistic Λ = L(θ̂₀)/L(θ̂) is computed and found to be 0.97. What does this indicate?
AStrong evidence against H₀, because the ratio is close to 1 and the null model explains 97% of the data
BLittle evidence against H₀, because the null model achieves nearly the same maximum likelihood as the unconstrained model
CThat the test is invalid, because a valid Λ must be below 0.5 to reject H₀
DThat θ̂₀ = θ̂, so the null and alternative hypotheses are indistinguishable
Λ close to 1 means the best fitting model under H₀ (the numerator) achieves almost as high a likelihood as the best fitting model overall (the denominator). The data is fit nearly equally well whether or not we impose the null constraint — so there is no strong reason to reject H₀. The test rejects when Λ is close to 0: the constrained model fits the data much worse than the unconstrained model, indicating the null constraints are inconsistent with the data. Confusing 'large Λ = reject' with 'small Λ = reject' is a very common error.
Question 2 Multiple Choice
You test H₀: μ = 0 in a Normal(μ, σ²) model with both μ and σ² unknown. The full model has 2 free parameters; under H₀, only σ² is free. By Wilks' theorem, −2 log Λ is asymptotically distributed as:
Aχ²₂, because the full model has 2 parameters
Bχ²₁, because H₀ imposes 1 constraint (fixing μ), reducing the parameter space by 1
CNormal(0, 1), because the test involves a single mean
Dt₁, because one mean is being tested from a normal distribution
Wilks' theorem states that −2 log Λ converges in distribution to χ²_r, where r is the number of constraints imposed by H₀ — equivalently, the difference in the number of free parameters between the full and null models. Here: full model has 2 free parameters (μ, σ²); null model has 1 free parameter (σ² only, since μ is fixed at 0). So r = 2 − 1 = 1, and −2 log Λ ~ χ²₁ asymptotically. The chi-squared distribution has 1 degree of freedom, not 2. In this case the LRT is equivalent to the squared t-statistic.
Question 3 True / False
The likelihood ratio statistic Λ always lies between 0 and 1, because the maximum likelihood under the full model is at least as large as the maximum likelihood under the restricted null model.
TTrue
FFalse
Answer: True
Since the null parameter space Θ₀ is a subset of the full parameter space Θ, any parameter value allowed under H₀ is also allowed in the unrestricted optimization. The unrestricted MLE therefore achieves a likelihood at least as high as the constrained MLE. This means sup_{θ∈Θ} L(θ|x) ≥ sup_{θ∈Θ₀} L(θ|x), so Λ = L(θ̂₀)/L(θ̂) ≤ 1. Since likelihoods are non-negative, Λ ≥ 0. The bound Λ ∈ [0,1] is not an assumption but a logical consequence of the nested structure of H₀ within H₁.
Question 4 True / False
The likelihood ratio test is primarily applicable when the null hypothesis specifies a single fixed value of the parameter (a simple null hypothesis).
TTrue
FFalse
Answer: False
The LRT was designed specifically to generalize beyond simple null hypotheses. The Neyman-Pearson lemma handles simple nulls (H₀: θ = θ₀ vs. H₁: θ = θ₁) by comparing two fixed likelihoods. The LRT extends this to composite hypotheses — where H₀ specifies a set of parameter values — by replacing fixed likelihoods with the best achievable likelihood under each model (the constrained and unconstrained MLEs). This is precisely the LRT's contribution: a universal framework for testing any constraint on a parametric model, simple or composite.
Question 5 Short Answer
Explain why the LRT uses maximum achievable likelihoods (MLEs under each hypothesis) rather than fixed parameter values, and what advantage this provides over the Neyman-Pearson likelihood ratio.
Think about your answer, then reveal below.
Model answer: The Neyman-Pearson likelihood ratio compares L(θ₁|x) / L(θ₀|x) where θ₀ and θ₁ are single specified values — it only works for simple hypotheses. For composite hypotheses (where H₀ and H₁ specify sets of values), there is no single 'the' likelihood under H₀. The LRT solves this by asking: what is the best the null hypothesis can possibly do? It uses the MLE under H₀ in the numerator and the unrestricted MLE in the denominator, so the ratio compares the null hypothesis at its best against the unrestricted model at its best. This makes the test applicable to any nested parametric hypothesis.
The key conceptual move is from 'compare two likelihoods at fixed points' to 'compare the best likelihoods achievable under each model.' This is a natural generalization: if the null model, given every opportunity to fit the data, still fits much worse than the unrestricted model, that is evidence against the null. Wilks' theorem then provides a universal reference distribution (chi-squared) for the test statistic, making the LRT a general-purpose framework rather than a collection of special-case tests for each model type.