Question

Consider the following regression model \[ y_k = \alpha_0 + \alpha_1 \log k + \epsilon_k, \quad k = 1, 2, \dots, n, \] where \( \epsilon_k \) are independent and identically distributed random variables each having probability density function \( f(x) = \frac{1}{2} e^{-|x|}, \ x \in \mathbb{R}. \) Then which one of the following statements is true?

• A. The maximum likelihood estimator of \( \alpha_0 \) does not exist
• B. The maximum likelihood estimator of \( \alpha_1 \) does not exist
• C. The least squares estimator of \( \alpha_0 \) exists and is unique
• D. The least squares estimator of \( \alpha_1 \) exists, but it is not unique

Hint:

- The least squares estimator is unique as long as the design matrix has full rank.
- Laplace distribution leads to maximum likelihood estimators that are well-defined, though not always straightforward to compute.

Answer 1

The Correct Option is C

1) Understanding the regression model:
We are given a linear regression model with independent and identically distributed errors. The errors follow a Laplace distribution with the given probability density function.
2) Maximum Likelihood Estimation (MLE):
Since the errors follow a Laplace distribution, the log-likelihood function can be maximized to obtain the maximum likelihood estimators (MLEs) for \( \alpha_0 \) and \( \alpha_1 \). The MLE for \( \alpha_0 \) and \( \alpha_1 \) does exist, but due to the nature of the Laplace distribution, it might not be as straightforward to find a closed-form solution. However, it does exist.
3) Least Squares Estimation (LSE):
The least squares estimators (LSE) for \( \alpha_0 \) and \( \alpha_1 \) are unique as long as the design matrix is full rank, which is guaranteed for the given model.
4) Conclusion:
The least squares estimator for \( \alpha_0 \) exists and is unique. Hence, the correct answer is (C).