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These multiple-choice questions help you verify your understanding of key concepts from the lectures.

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Please answer the following multiple-choice questions. They are designed to help you verify your understanding of key concepts from the lectures.

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Which of the following statements is true?

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Let $\\underline{X} := (X_1, \\dotsc, X_n) \\sim f(\\theta)$, where $f(\\theta)$ is a distribution parametrised by $\\theta \\in \\mathbb{R}$ with probability density function $f(\\, \\cdot \\,; \\theta)$. Let $L(\\theta; \\underline{x})$ be the likelihood for a given realisation $\\underline{x}$ of $\\underline{X}$. Which of the following statements best describes the likelihood?

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Which statement about a large-sample confidence interval for a parameter $\\theta$ based on the ML estimator is most correct?

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Let $\\underline{X} = (X_1, ..., X_n)$ be a random sample from $f_X(\\theta)$, where $f_X(\\theta)$ is a distribution -- parameterised by $\\theta \\in \\mathbb{R}$ -- with twice differentiable probability density function $f_X(\\, \\cdot \\,; \\theta)$. Let $\\hat{\\theta} = h(\\underline{X})$ denote an estimator for $\\theta$. Conditional on $\\underline{X}$, which of the following correctly specifies the Fisher information at $\\hat{\\theta}$?

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Start by considering the Fisher information at some fixed (i.e., deterministic) value $\\theta'$ and then plug in the random variable $\\hat{\\theta} = h(\\underline{X})$ at the very end.

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Which statement best describes the intuitive meaning of Fisher information $\\mathcal{I}(\\theta)$ where $\\theta$ is the true parameter value?

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Consider some probabilistic model parameterised by $\\underline{\\theta} \\in \\Theta$ and assume that you wish to perform a likelihood-ratio test for some null hypothesis $H_0: \\underline{\\theta} \\in \\Theta_0$.

What is the number of degrees of freedom of the asymptotic chi-square distribution of the likelihood-ratio test statistic if $\\Theta = \\mathbb{R}^3$ and $\\Theta_0 = \\{0\\} \\times \\mathbb{R} \\times \\{1\\}$?

Think about how many free parameters you need to represent arbitrary elements of $\\Theta_0$.

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What is the number of degrees of freedom of the asymptotic chi-square distribution of the likelihood-ratio test statistic if $\\Theta = \\mathbb{R}^5$ and $\\Theta_0 = \\{(\\theta_1,\\dotsc,\\theta_5)^{\\mathrm{T}} \\in \\Theta \\mid \\theta_1 = - 10 \\theta_2\\}$?

Think about how many free parameters you need to represent arbitrary elements of $\\Theta_0$.

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Think about how many free parameters you need to represent arbitrary elements of $\\Theta_0$.

"}], "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showBlankOption": true, "showCellAnswerState": true, "choices": ["1", "2", "3", "4", "5"], "matrix": [0, "3", 0, 0, 0], "distractors": ["", "", "", "", ""]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Which of the following best describes the (realised) p-value in a hypothesis test for a given realisation of the data?

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Assuming that the null hypothesis is correct, what is the distribution of the p-value before we have observed the realisation of the data? For simplicity, you may assume that the test statistic is continuous.

For some random variable $X$ with continuous cumulative distribution function $F$ and $u \\in (0,1)$, compute the probability $\\mathbb{P}(F(X) \\leq u)$ to determine the distribution of $F(X)$ which then immediately gives you the distribution of $1 - F(X)$.

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