Recall the linear regression model as introduced above in the previous question. This model is parametric, although it is not written in the standard notation previously introduced for parametric statistical models. In this problem, you will explicitly write the linear regression model as a parametric statistical model.
We will represent the linear regression model as an ordered pair (E, \{ P_{\boldsymbol \beta }\} _{{\boldsymbol \beta }\in \Theta } ). Here E denotes the sample space associated to the distribution P_{\boldsymbol \beta }, where P_{\boldsymbol \beta } is defined as follows for {\boldsymbol \beta }\in \mathbb {R}^ d:
The random ordered pair (\mathbf X, Y) \subset \mathbb {R}^ d \times \mathbb {R} is distributed as P_{\boldsymbol \beta } if:
\mathbf X\sim \mathcal{N}(0, I_ d),
Y \sim {\boldsymbol \beta }^ T X + \varepsilon, where \varepsilon \sim \mathcal{N}(0,1) and \varepsilon is independent of \mathbf X.
The set \Theta in the ordered pair (E, \{ P_{\boldsymbol \beta }\} _{{\boldsymbol \beta }\in \Theta } ) denotes the parameter space for this model.
The sample space for the linear regression model can be written E = \mathbb {R}^{k} for some integer k. What is k? (Express your answer in terms of d.)
Hint: You should use the fact that \mathbb {R}^{m + n} = \mathbb {R}^ m \times \mathbb {R}^ n for all integers m, n \geq 0.
k=\quad
unanswered
The parameter space for the model can be written as \Theta = \mathbb {R}^ j for some integer j. What is j? (Express your answer in terms of d.)
j=\quad
For the linear regression model, the sample space E is defined as E = ℝ^k, where k represents the dimensionality of the input variable vector X.
Since X ∈ ℝ^d, where d is the dimensionality of X, we can conclude that k = d.
Therefore, k = d.
The parameter space Θ is defined as Θ = ℝ^j, where j represents the dimensionality of the parameter vector β.
Since β ∈ ℝ^d, we can conclude that j = d.
Therefore, j = d.
To determine the value of k, we need to consider the dimensionality of the random vector (X, Y).
Since X is a random vector in \mathbb{R}^d, it has d dimensions.
Y is a scalar and is defined as Y = \boldsymbol{\beta}^TX + \varepsilon, where \varepsilon is a scalar. Therefore, Y has 1 dimension.
Hence, the dimensionality of the random vector (X, Y) is d + 1.
Therefore, k = d + 1.
To determine the value of j, we need to consider the parameter space \Theta.
From the given information, we know that \boldsymbol{\beta}\in \mathbb{R}^d.
Since \boldsymbol{\beta} is a d-dimensional vector, the parameter space \Theta will also have d dimensions.
Hence, j = d.