Math Problems Vault

Problem 3e.1

Prove the statement above that the minimum is given by $\lambda_{j} = c(j = 1,2, \ldots , p-1$ ).

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Problem 3e.2

It is required to fit a regression model of the form

\mathbb {E}\left[Y_{i}\right] = \beta _{0}+\beta _{1} x_{i}+\beta _{2} \phi \left(x_{i}\right) \quad (i = 1,2,3)

where $\phi (x)$ is a second-degree polynomial. If $x_{1} = -1, x_{2} = 0$ , and $x_{3} = 1$ , find $\phi$ such that the design matrix $\mathbf{X}$ has mutually orthogonal columns.

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Problem 3e.3

(a)(b)

Suppose that $\mathbf{X} = \left(\mathbf{x}^{(0)}, \mathbf{x}^{(1)}, \ldots , \mathbf{x}^{(p-1)}, \mathbf{x}^{(p)}\right) = \left(\mathbf{W}, \mathbf{x}^{(p)}\right)$ has linearly independent columns.

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(a)

Using A.9.5, prove that

\operatorname {det}\left(\mathbf{X}^{T} \mathbf{X}\right) = \operatorname {det}\left(\mathbf{W}^{T} \mathbf{W}\right)\left(\mathbf{x}^{(p) \prime } \mathbf{x}^{(p)}-\mathbf{x}^{(p) \prime } \mathbf{W}\left(\mathbf{W}^{T} \mathbf{W}\right)^{-1} \mathbf{W}^{T} \mathbf{x}^{(p)}\right)

(b)

Deduce that

\frac{\operatorname {det}\left(\mathbf{W}^{T} \mathbf{W}\right)}{\operatorname {det}\left(\mathbf{X}^{T} \mathbf{X}\right)} \geq \frac{1}{\mathbf{x}^{(\mathbf{p}) / \mathbf{x}^{(\mathbf{p})}}}

and hence show that $\operatorname {Var}\left[\widehat{\beta }_{p}\right] \geq \sigma^{2}\left(\mathbf{x}^{(p) \prime } \mathbf{x}^{(p)}\right)^{-1}$ with equality if and only if $\mathbf{x}^{(p) \prime } \mathbf{x}^{(j)} = 0(j = 0,1, \ldots , p-1)$ .

Problem 3e.4

What modifications in the statement of Example 3.3 proved above can be made if the term $\beta_{0}$ is omitted?

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Problem 3e.5

(a)(b)

Suppose that we wish to find the weights $\beta_{i}(i = 1,2, \ldots , k)$ of $k$ objects. One method is to weigh each object $r$ times and take the average; this requires a total of $k r$ weighings, and the variance of each average is $\sigma^{2} / r$ ( $\sigma^{2}$ being the variance of the weighing error). Another method is to weigh the objects in combinations; some of the objects are distributed between the two pans and weights are placed in one pan to achieve equilibrium. The regression model for such a scheme is

Y = \beta _{1} x_{1}+\beta _{2} x_{2}+\cdots +\beta _{k} x_{k}+\varepsilon

where $x_{i} = 0,1$ , or -1 according as the $i$ th object is not used, placed in the left pan or in the right pan, $\varepsilon$ is the weighing error (assumed to be the same for all weighings), and $Y$ is the weight required for equilibrium ( $Y$ is regarded as negative if placed in the left pan). After $n$ such weighing operations we can find the least squares estimates $\widehat{\beta }_{i}$ of the weights.

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(a)

Show that the estimates of the weights have maximum precision (i.e., minimum variance) when each entry in the design matrix $\mathbf{X}$ is $\pm 1$ and the columns of $\mathbf{X}$ are mutually orthogonal.

(b)

If the objects are weighed individually, show that $k n$ weighings are required to achieve the same precision as that given by the optimal design with $n$ weighings.