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Estimator of $\sigma^2$: \\ $\hat{\sigma}^2 = \frac{SS_e}{n - p - 1}$, \\ where $p$ is the number of covariates in the model and $SS_e = \sum_{i=1}^n (Y_i - \hat{Y}_i)^2$. \\ (a) Write down the likelihood function of $Y = (Y_1, ..., Y_n)$ given $X = \begin{pmatrix} 1 & X_{1,1} & \dots & X_{1,p} \\ \vdots & \vdots & \vdots \\ 1 & X_{n,1} & \dots & X_{n,p} \end{pmatrix}$ \\ using the matrix notation. Refer to the multivariate normal distribution form for this. \\ (b) Find the maximum likelihood estimator of $\sigma^2$. For this you may write the log-likelihood function $\sum_{i=1}^n \log f(Y_i | X_{i1}, ..., X_{ip}, \beta_0, \beta_1, ..., \beta_p, \sigma^2)$. \\ (c) Use the result that $E[SS_e] = (n - p - 1)\sigma^2$ to show that $\hat{\sigma}_{mle}^2$ is not an unbiased estimator of $\sigma^2$. \\ (d) Show that $E[\hat{\sigma}_{mle}^2]$ is asymptotically $\sigma^2$, that is, tends to $\sigma^2$ as the sample size increases.

          Estimator of $\sigma^2$: \\
$\hat{\sigma}^2 = \frac{SS_e}{n - p - 1}$, \\
where $p$ is the number of covariates in the model and $SS_e = \sum_{i=1}^n (Y_i - \hat{Y}_i)^2$. \\
(a) Write down the likelihood function of $Y = (Y_1, ..., Y_n)$ given $X = \begin{pmatrix} 1 & X_{1,1} & \dots & X_{1,p} \\ \vdots & \vdots & \vdots \\ 1 & X_{n,1} & \dots & X_{n,p} \end{pmatrix}$ \\
using the matrix notation. Refer to the multivariate normal distribution form for this. \\
(b) Find the maximum likelihood estimator of $\sigma^2$. For this you may write the log-likelihood function $\sum_{i=1}^n \log f(Y_i | X_{i1}, ..., X_{ip}, \beta_0, \beta_1, ..., \beta_p, \sigma^2)$. \\
(c) Use the result that $E[SS_e] = (n - p - 1)\sigma^2$ to show that $\hat{\sigma}_{mle}^2$ is not an unbiased estimator of $\sigma^2$. \\
(d) Show that $E[\hat{\sigma}_{mle}^2]$ is asymptotically $\sigma^2$, that is, tends to $\sigma^2$ as the sample size increases.

Estimator of σ^2:

σ̂^2 = (SSe)/(n - p - 1),

where p is the number of covariates in the model and SSe = ∑i=1^n (Yi - Ŷi)^2.

(a) Write down the likelihood function of Y = (Y1, ..., Yn) given X =
< p m a t r i x >

using the matrix notation. Refer to the multivariate normal distribution form for this.

(b) Find the maximum likelihood estimator of σ^2. For this you may write the log-likelihood function ∑i=1^n log f(Yi | Xi1, ..., Xip, β0, β1, ..., , σ^2).

(c) Use the result that E[SSe] = (n - p - 1)σ^2 to show that σ̂mle^2 is not an unbiased estimator of σ^2.

(d) Show that E[σ̂mle^2] is asymptotically σ^2, that is, tends to σ^2 as the sample size increases.

Added by Robert C.

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