Consider the linear model where $\epsilon \sim N(0, \sigma = 1)$. $Y = \beta x + \epsilon$ (a) Calculate the log-likelihood function associated to this model. (b) Setup the equation to calculate the Fisher information for $\beta$. Is it possible to calculate $I(\beta)$? 3
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Since the errors are normally distributed, the likelihood function is: $$L(\beta) = \prod_{i=1}^{n} \frac{1}{\sqrt{2\pi}} exp(-\frac{(y_i - \beta x_i)^2}{2})$$ Show more…
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Consider the computer output below. The regression equation is $Y=12.9+2.34 x$ $$\begin{array}{lrrll}\text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\\text { Constant } & 12.857 & 1.032 & ? & ? \\\text { X } & 2.3445 & 0.1150 & ? & \text { ? }\end{array}$$ $\begin{array}{ll}\mathrm{S}=1.48111 & \mathrm{R}-\mathrm{Sq}=98.1 \% & \mathrm{R}-\mathrm{Sq}(\mathrm{adj})=97.9 \%\end{array}$ Analysis of Variance $$\begin{array}{lrrrl}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } \\\text { Regression } & 1 & 912.43 & 912.43 & ? \\\text { Residual Error } & 8 & 17.55 & ? & \\\text { Total } & 9 & 929.98 & &\end{array}$$ (a) Fill in the missing information. You may use bounds for the $P$ -values (b) Can you conclude that the model defines a useful linear relationship? (c) What is your estimate of $\sigma^{2}$ ?
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The regression equation is Y = 12.9 + 2.34x Predictor Coef SE Coef T P Constant 12.857 1.032 ? ? X 2.3445 0.1150 ? ? S = 1.48111 R-sq = 98.1% R-sq(adj) = 97.9% Analysis of Variance Source DF SS MS F P Regression 1 912.43 912.43 ? ? Residual error 8 17.55 ? Total 9 929.98 (a) Fill in the missing information. You may use bounds for the P-values. (b) Can you conclude that the model defines a useful linear relationship? (c) What is your estimate of σ²?
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