Question

Consider the linear regression model \[ y_i = \beta_0 + \beta_1 x_i + \epsilon_i, i = 1, 2, \dots, n, \text{where} \epsilon_i \text{ are i.i.d. standard normal random variables. Given that} \] \[ \frac{1}{n} \sum_{i=1}^n x_i = 3.2, \frac{1}{n} \sum_{i=1}^n y_i = 4.2, \frac{1}{n} \sum_{j=1}^n \left( x_j - \frac{1}{n} \sum_{i=1}^n x_i \right)^2 = 1.5, \] \[ \frac{1}{n} \sum_{j=1}^n \left( x_j - \frac{1}{n} \sum_{i=1}^n x_i \right) \left( y_j - \frac{1}{n} \sum_{i=1}^n y_i \right) = 1.7, \] the maximum likelihood estimates of \( \beta_0 \) and \( \beta_1 \), respectively, are

• A. \( \frac{17}{15} \) and \( \frac{32}{75} \)
• B. \( \frac{32}{75} \) and \( \frac{17}{15} \)
• C. \( \frac{17}{15}\) and \(\frac{43}{75} \)
• D. \( \frac{43}{75} \) and \( \frac{17}{15} \)

Hint:

For linear regression, the maximum likelihood estimates of \( \beta_0 \) and \( \beta_1 \) are the ordinary least squares estimates, which can be derived using the covariance and variance of the data.

Answer 1

The Correct Option is D

Step 1: Recall the maximum likelihood estimation in linear regression. 
In the linear regression model \( y_i = \beta_0 + \beta_1 x_i + \epsilon_i \), the maximum likelihood estimates of \( \beta_0 \) and \( \beta_1 \) are the ordinary least squares estimates given by: \[ \hat{\beta_1} = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^n (x_i - \bar{x})^2}, \hat{\beta_0} = \bar{y} - \hat{\beta_1} \bar{x}. \]

Step 2: Apply the given values. 
We are given the following sums: \[ \frac{1}{n} \sum_{i=1}^n x_i = 3.2, \frac{1}{n} \sum_{i=1}^n y_i = 4.2, \frac{1}{n} \sum_{j=1}^n \left( x_j - \frac{1}{n} \sum_{i=1}^n x_i \right)^2 = 1.5, \] \[ \frac{1}{n} \sum_{j=1}^n \left( x_j - \frac{1}{n} \sum_{i=1}^n x_i \right) \left( y_j - \frac{1}{n} \sum_{i=1}^n y_i \right) = 1.7. \] These values correspond to the sample means \( \bar{x} \), \( \bar{y} \), the sum of squared deviations \( S_x^2 \), and the covariance \( S_{xy} \).

Step 3: Calculate the estimates. 
From the provided values, we compute: \[ \hat{\beta_1} = \frac{S_{xy}}{S_x^2} = \frac{1.7}{1.5} = \frac{17}{15}, \] \[ \hat{\beta_0} = \bar{y} - \hat{\beta_1} \bar{x} = 4.2 - \left( \frac{17}{15} \times 3.2 \right) = \frac{43}{75}. \]

Step 4: Conclusion. 
The maximum likelihood estimates are \( \hat{\beta_0} = \dfrac{43}{75} \) and \( \hat{\beta_1} = \dfrac{17}{15} \), so the correct answer is (D).