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We have a correctly specified linear model $z = x\beta + \varepsilon$ with 100 independent observations. Given: $y = e^z$, $\sigma^2 = 4$ (assumed to be known), sample means $\bar{x} = -3$ and $\bar{z} = 0.5$, sample variances/covariances $\sigma^2_x = 1$, $cov(x,z) = \frac{2}{3}$. Also, $\gamma = 0.95$.

a) Construct Wald's $\gamma$-confidence interval for parameter $\theta = \textbf{E}[y|x=\frac{1}{2}]$ using the fact that $\textbf{E}[e^{\lambda Z}] = e^{a\lambda + \sigma^2\lambda^2/2}$, $\lambda \in \mathbb{R}$ when $Z \sim N(a, \sigma^2)$

b) Construct : i) exact $\gamma$-confidence interval for prediction when $x=\frac{1}{2}$ for the variable $y$; ii) approximate prediction $\gamma$-confidence interval using Delta method when $x=\frac{1}{2}$ for the variable $y$; iii) approximate prediction $\gamma$-confidence interval using methods from part a).

Full solution would obviously help me the most but I would also be very grateful for references on how to tackle this problem.

N.B. The problem is for preparing for exam and so, is not a homework problem. The course is actually drawn more towards mathematical understanding of statistical problems than the other way around and that is why I am having problems. The professor like unusual formulations and does not really provide us with sufficient material so I am stuck.

EDIT: here's an attempt:

denote $\hat{\mu} = \hat{a} + \hat{b}x_0$, where $\hat{a}, \hat{b}$ are the OLS estimates: $\hat{a} = \hat{z} -\hat{b} \bar{x}$ and $\hat{b} = cov(x,z)/\sigma^2_x$. So $$exp \left( \hat{\mu} \pm t_{N-2}((1+\gamma) /2)\sigma\sqrt{\frac{1}{N} + \frac{(\bar{x}-x_0)^2}{N\sigma^2_x}} \right)$$

would constitute the exact CI for predicting $y$ given $x_0$.

for the approximate CI using Delta method:

since $z|_x \sim N(a+bx, \sigma^2)$, and where interested in $\theta = \textbf{E}[e^z|x=x_0]$, we have $\theta = e^{a+bx_0+\sigma^2/2}$. Suppose we use ML to estimate $a,b, \sigma^2$. Then our parameter of interest $theta$ is a function of these values, that is: $\hat{\theta} = \theta(\hat{x}, \hat{b}, \hat{\sigma^2})$. Using properties of ML estimators, we have asymptotic normality: $$\hat{\theta} \sim AN(e^{a+bx_0+\sigma^2/2}, V_N)$$ where $$V_N = \theta '(a,b,\sigma^2)^TJ^{-1}(a,b,\sigma^2)\theta'(a,b,\sigma^2)$$ Note: dash denotes differentiation (partial, with respect to each parameter), $T$ denotes transpose and $J$ denotes fisher information matrix. Basically we have an application of the Delta method. Then the confidence interval of the b) ii) part would be:

$$exp\left(\hat{a}+\hat{b}x_0 + \hat{\sigma}^2/2 \ \pm z((1+\gamma)/2) V(\hat{\sigma}^2)\right)$$

Note: $z$ here denotes the z-value while $\hat{a},\hat{b},\hat{\sigma}^2$ denote ML estimates.


Now I realize my 'attempt' is very messy but that is what I managed to scrap-up using the material that the professor provided us. I hope at least that the idea is clear.

And that is what I actually need. I am still missing the a) and b) iii) parts. He wants us to construct confidence intervals based on Wald's statistic. But I don't see how then a) and b) iii) is different. Also, would the 'Delta method CI' be different from the CI resulting from CI using Wald's statistic. To my eyes it would be absolutely the same?

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    $\begingroup$ A question in preparation for an exam which is effectively routine bookwork (as might be set for coursework or exam) still falls under the guidelines of self-study problems, which includes homework, textbook exercises, assigned work and even personal study without any connection to coursework. Please add the self-study tag and read its tag wiki, and amend your question as suggested. Note that it explains that in return for proper following of the suggested guidelines, you should not expect to get full solutions, but hints and guidance. $\endgroup$ – Glen_b Jan 18 '15 at 5:49

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