Regression Analysis Quiz

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Questions and Answers

What is the primary objective of regression analysis?

To estimate the unknown conditional density $f^*(y|x)$. (correct)

To summarize data with descriptive statistics.

To find the exact distribution of response variables.

To predict future outcomes using historical data.

In linear location regression, how is the response variable expressed?

As an exponential function of explanatory variables.

As a quadratic function of explanatory variables.

As an interaction of multiple explanatory variables.

As a linear function of explanatory variables plus an error term. (correct)

What characterizes the random variables $ ext{ℰ}_i$ in linear location regression?

They depend on previous observations and are not identically distributed.

They are independent and identically distributed symmetric random variables. (correct)

They are uniformly distributed random variables.

They follow a normal distribution with positive mean.

When concerned with conditional distributions, which measures are typically focused on in regression analysis?

Measures of location such as mean and median. Signup and view all the answers

What is the common assumption about the explanatory variable measurements in regression analysis?

They are fixed or controlled at defined values. Signup and view all the answers

What is one limitation of estimating the whole conditional distribution in regression analysis?

It is difficult without a substantial amount of data. Signup and view all the answers

Which of the following statements is true regarding the vector of parameters $ heta^*(1)$ in linear location regression?

It consists of unknown parameters that need to be estimated. Signup and view all the answers

What is the relationship between $ ext{ℰ}_i$ and $- ext{ℰ}_i$ in regression analysis?

They are identically distributed random variables. Signup and view all the answers

What additional characteristic must be specified to uniquely determine a conditional probability distribution?

Variance Signup and view all the answers

Which parameter is represented by 𝜇 in the context of conditional distributions?

Expectation Signup and view all the answers

If the response variable can only take positive values, which form does the conditional mean take?

$exp(x^T heta^*)$ Signup and view all the answers

Which of the following statements is true regarding the joint parameter $ heta^*$?

It can be represented as $( heta^(1), heta^(2))$. Signup and view all the answers

To map the real line onto the unit interval (0, 1), which form does the conditional mean take?

$rac{exp(x^T heta^)}{1 + exp(x^T heta^)}$ Signup and view all the answers

What is the role of $ heta(2)$ in the definition of the conditional density function?

It provides variance characterization. Signup and view all the answers

Which distribution is uniquely characterized when both its mean and variance are specified?

Gaussian (Normal) distribution Signup and view all the answers

In the context of conditional probability distribution, what is the significance of the conditional mean?

It provides the expected value of the response variable. Signup and view all the answers

In mean regression, what is the objective related to the conditional mean function?

To estimate the unknown conditional mean function. Signup and view all the answers

Which assumption is made about the response random variables in generalised linear mean regression?

They are independent. Signup and view all the answers

What does the function ℎ represent in the context of generalised linear mean regression?

The link function that connects the conditional mean with the linear predictor. Signup and view all the answers

What is the implication of the normality assumption for the response distribution in linear models?

Key quantities have closed-form expressions. Signup and view all the answers

What is a characteristic of the linear predictor in generalised linear mean regression?

It is a linear function of the explanatory variables. Signup and view all the answers

Which statement best describes the conditional mean function in generalised linear mean regression?

It depends on a vector of unknown parameters and the covariates. Signup and view all the answers

In the context of linear models, what happens when the response distribution is Logistic?

Key quantities can be computed numerically and iteratively. Signup and view all the answers

What is the notation used to denote the expected value of the response conditional on the covariates?

μ(𝜽∗ , 𝐱) Signup and view all the answers

What is the definition of the logit function?

$logit(u) = log \left( \frac{u}{1 - u} \right)$ for $u \in (0, 1)$ Signup and view all the answers

Which statement correctly describes the relationship between the mean and the cumulant function for a random variable that follows an exponential family distribution?

$E[Y] = \Psi'(\eta)$ Signup and view all the answers

For the Poisson distribution, what is the canonical parameter?

$\eta = log(\mu)$ Signup and view all the answers

What expression defines the density function for an exponential family distribution?

$fEF-GLM(y|\eta, \delta) = exp \left( \frac{\eta y - \Psi(\eta)}{\delta} + c(y, \delta) \right)$ Signup and view all the answers

In generalized linear models, which assumption is made about the transformed conditional mean function?

It is assumed to be a linear function of the explanatory variables. Signup and view all the answers

Which distribution has a cumulant function defined as $\Psi(\eta) = -N log(N(1 + exp(\eta))^{-1})$?

Binomial distribution Signup and view all the answers

What is the dispersion parameter ($\delta$) for the normal distribution?

It can vary based on the specific instance. Signup and view all the answers

Which of the following statements about the density function for the exponential family is false?

It always has a constant dispersion parameter. Signup and view all the answers

What does the notation 𝜽𝑛̂ (𝐲) represent in the context of optimisation?

The estimated value of 𝜽 that maximises the likelihood function Signup and view all the answers

Why might optimisation software default to minimisation instead of maximisation?

Most optimisation problems are framed in terms of loss. Signup and view all the answers

What is the relationship between the loglikelihood function and the negative loglikelihood function?

Maximising the loglikelihood is the same as minimising the negative loglikelihood. Signup and view all the answers

In the context of the given data, which of the following can be inferred about the relationship between x and y?

y consistently increases as x increases. Signup and view all the answers

What defines the function 𝜙(𝜽|𝐲) in the minimisation problem?

The negative loglikelihood function Signup and view all the answers

What relationship exists between the median of the conditional distribution and the residuals?

The median is equal to $oldsymbol{x}_i^T oldsymbol{ heta}^{(1)*}$. Signup and view all the answers

What is the scale parameter in a location-scale linear regression model?

A parameter that must be greater than zero ($ u > 0$). Signup and view all the answers

In the context of estimating the unknown joint parameter $oldsymbol{ heta}^*$, what does $oldsymbol{ heta}^{(1)}$ represent?

The vector related to the conditional distribution medians. Signup and view all the answers

What is the probability density function for a random variable $Y$ in a location-scale model?

$f_{oldsymbol{ u}}(y|oldsymbol{ heta}) = rac{1}{ u} oldsymbol{ u}(rac{y - oldsymbol{ heta}}{ u})$. Signup and view all the answers

What does the random variable $Z$ represent in the context of a location-scale linear regression?

A standard normal random variable. Signup and view all the answers

In the estimation of symmetric residuals, what condition must hold true regarding the conditional expectation?

The conditional expectation must exist and match the vector's multiplication. Signup and view all the answers

What does the notation $ heta^{(2)}$ indicate in the context of estimating the joint parameter?

The vector related to the scale parameters. Signup and view all the answers

What condition does the symmetry of residuals imply regarding the conditional distribution?

The median is located at $oldsymbol{x}_i^T oldsymbol{ heta}^{(1)*}$. Signup and view all the answers

Study Notes

Lecture notes for MA40198 (Applied Statistical Inference)

Course is about Applied Statistical Inference, based on notes by Simon N. Wood
Date of notes: 2025-11-10
Course content is organized into chapters and sections, see table of contents for details.

Chapter 1: Applied Statistical Inference
- Overview of Applied Statistical Inference
- Objective
- Learning Outcomes
- Summative Assessment
- Moodle Page
Chapter 2: Optimisation in Statistics
- Regression Analysis
  - Linear Location Regression
  - Generalised Linear Mean Regression
  - Likelihood Function
  - Maximum Likelihood Estimation
- Unconstrained Optimisation Theory
  - Global and local minima
  - Conditions for local minima
- Optimisation Algorithms
  - Line-search algorithms
  - Step-length selection
  - Stopping criteria
  - Raw Newton's algorithm
  - Fisher's scoring algorithm
  - Quasi-Newton algorithms (BFGS algorithm)
Chapter 3: Likelihood Theory
- Large sample properties of the MLE
  - Consistency
  - Asymptotic Normality
- Likelihood as a random variable
- Estimators of the asymptotic variance
- Reparametrisations
- Delta Method
- Generalised likelihood ratio test (GLRT)
Chapter 4: Bayesian Inference
- Example: Bernoulli distribution
  - Prior distributions (Beta)
  - Posterior distributions
- Example: Poisson distribution
  - Prior distributions (Gamma)
  - Posterior distributions
Appendices
- Prerequisites
  - Numerical
  - Linear Algebra
  - Vector calculus

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Description

Test your understanding of regression analysis concepts, focusing on linear location regression. This quiz covers key objectives, assumptions, and characteristics related to response variables and random variables in regression contexts. Challenge yourself with questions on conditional distributions and parameter relationships.