Chapter 9, Problem 105SE

Refer to Exercise 9.38(b). Under the conditions outlined there, find the MLE of σ².

9.38. Let Y₁, Y₂, …, Y_n denote a random sample from a normal distribution with mean μ and variance σ².

1. a If μ is unknown and σ² is known, show that $\bar{Y}$ is sufficient for μ.
2. b If μ is known and σ² is unknown, show that ${\displaystyle {\sum }_{i=1}^n{(Y_i-\mu )}^2}$ is sufficient for σ².
3. c If μ and σ² are both unknown, show that ${\displaystyle {\sum }_{i=1}^n{Y}_i}$ and ${\displaystyle {\sum }_{i=1}^n{Y}_i^2}$ are jointly sufficient for μ and σ². [Thus, it follows that $\bar{Y}$ and ${\displaystyle {\sum }_{i=1}^n{(Y_i-\bar{Y})}^2}$ or $\bar{Y}$ and S² are also jointly sufficient for μ and σ².]