Section 6b

6.2 Straight Line through the Origin, 6.3 Weighted Least Squares for the Straight Line: 6.3.1 Known Weights, 6.3.2 Unknown Weights

[3/100%]

Show

LaTeX

Show LaTeX

Problem 6b.1

(a)(b)

Let $Y_{1}, Y_{2}, \ldots , Y_{n}$ be independent random variables such that for $i = 1,2, \ldots , n$ ,

\mathbb {E}\left[Y_{i} \mid X = x_{i}\right] = \beta _{1} x_{i}

and

\operatorname {Var}\left[Y_{i} \mid X = x_{i}\right] = \sigma ^{2} w_{i}^{-1} \quad \left(w_{i} > 0\right)

(a)

If the conditional distribution of $Y$ given $x$ is the Type III (scaled gamma) distribution,

f(y \mid x) = \frac{1}{a_{x}^{p} \Gamma (p)} y^{p-1} \exp \left(-y / a_{x}\right) \quad 0 \leq y < \infty , \quad p > 0

where $\alpha_{x}$ a function of $x$ , and $w_{i}^{-1} = x_{i}^{2}$ , prove that the maximum likeithood estimate of $\beta_{1}$ is also the weighted least squares estimate.

(b)

If the conditional distribution of $Y$ given $x$ is Poisson, and $w_{i}^{-1} = x_{i}$ , show that the maximum likelihood estimate is just the weighted least squares estimate.

Problem 6b.2

Given the model $Y_{i} = \beta_{1} x_{i}+\varepsilon_{i}(i = 1,2, \ldots , n)$ , where the $\varepsilon_{i}$ are independently distributed as $N\left(0, \sigma^{2} w_{i}^{-1}\right), w_{i} > 0$ , show how to predict $x_{0}$ for a given value $Y_{0}$ of $Y$ . Describe briefly a method for constructing a confidence interval for $x_{0}$ .

Problem 6b.3

Given the regression line

Y_{i} = \beta _{0}+\beta _{1} x_{i}+\varepsilon _{i} \quad (i = 1,2, \ldots , n)

where the $\varepsilon_{i}$ are independent with $\mathbb {E}\left[\varepsilon_{i}\right] = 0$ and $\operatorname {Var}\left[\varepsilon_{i}\right] = \sigma^{2} x_{i}^{2}$ , show that weighted least squares estimation is equivalent to ordinary least squares estimation for the model

\frac{Y_{i}}{x_{i}} = \beta _{1}+\frac{\beta _{0}}{x_{i}}+\delta _{i} .