Math Problems Vault

Problem 3.4.1

(a)(b)

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(a)

Suppose $X_{i}: N\left(\xi_{i}, \sigma^{2}\right)$ with $\xi_{i}=\alpha +\beta t_{i}$ . If the first column of the matrix $C$ leading to the canonical form (4.7) is $(1 / \sqrt{n}, \ldots , 1 / \sqrt{n})^{\prime }$ , find the second column of $C$ .

(b)

If $X_{i}: N\left(\xi_{i}, \sigma^{2}\right)$ with $\xi_{i},=\alpha +\beta t_{i}+\gamma t_{i}^{2}$ , and the first two columns of $C$ are those of (a), find the third column under the simplifying assumptions $\Sigma t_{i}=0, \Sigma t_{i}^{2}=1$ .

Remark.

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(b)

The orthogonal polynomials that are progressively built up in this way are frequently used to simplify regression analysis.

Problem 3.4.2

Write out explicit expressions for the transformations (4.10) when $\Pi_{\Omega }$ is given by

(a) $\xi_{i}=\alpha +\beta t_{i}$ and (b) $\xi_{i}=\alpha +\beta t_{i}+\gamma t_{i}^{2}$ .

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Problem 3.4.3

Use Problem 3.10 to prove (iii) of Theorem 4.3.

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Problem 3.4.4

(a)(b)

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(a)

In Example 4.7, determine $\hat{\alpha }, \hat{\beta }$ , and hence $\hat{\xi }_{i}$ by minimizing $\Sigma \left(X_{i}-\alpha -\beta t_{i}\right)^{2}$ .

(b)

Verify the expressions (4.12) for $\alpha$ and $\beta$ , and the corresponding expressions for $\hat{\alpha }$ and $\hat{\beta }$ .

Problem 3.4.5

In Example 4.2, find the UMVU estimators of $\alpha , \beta , \gamma$ , and $\sigma^{2}$ when $\Sigma t_{i}=0$ and $\Sigma t_{i}^{2}=1$ .

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Problem 3.4.6

Let $X_{i j}$ be independent $N\left(\xi_{i j}, \sigma^{2}\right)$ with $\xi_{i j}=\alpha_{i}+\beta t_{i j}$ . Find the UMVU estimators of the $\alpha_{i}$ and $\beta$ .

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Problem 3.4.7

(a)(b)

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(a)

In Example 4.9, show that the vectors of the coefficients in the $\hat{\alpha }_{i}$ are not orthogonal to the vector of the coefficients of $\hat{\mu }$ .

(b)

Show that the conclusion of (a) is reversed if $\hat{\alpha }_{i}$ and $\hat{\mu }$ are replaced by $\hat{\hat{\alpha }}_{i}$ and $\hat{\hat{\mu }}$ .

Problem 3.4.8

In Example 4.9, find the UMVU estimator of $\mu$ when the $\alpha_{i}$ are known to be zero and compare it with $\hat{\mu }$ .

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Problem 3.4.9

The coefficient vectors of the $X_{i j k}$ given by (4.32) for $\hat{\mu }, \hat{\alpha }_{i}$ , and $\hat{\beta }_{j}$ are orthogonal to the coefficient vectors for the $\hat{\gamma }_{i j}$ given by (4.33).

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Problem 3.4.10

In the model defined by (4.26) and (4.27), determine the UMVU estimators of $\alpha_{i}$ , $\beta_{j}$ , and $\sigma^{2}$ under the assumption that the $\gamma_{i j}$ are known to zero.

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Problem 3.4.11

(a)(b)(c)

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(a)

In Example 4.11, show that

\Sigma \Sigma \Sigma \left(X_{i j k}-\mu -\alpha _{i}-\beta _{j}-\gamma _{i j}\right)^{2}=S^{2}+S_{\mu }^{2}+S_{\alpha }^{2}+S_{\beta }^{2}+S_{\gamma }^{2}

where $S^{2}=\Sigma \Sigma \Sigma \left(X_{i j k}-X_{i j.}\right)^{2}, S_{\mu }^{2}=\operatorname {IJm}\left(X_{\ldots }-\mu \right)^{2}, S_{\alpha }^{2}=\operatorname {Jm} \Sigma \left(X_{1. .}-X_{\ldots }-\alpha_{i}\right)^{2}$ , and $S_{\beta }^{2}, S^{2} \gamma$ are defined analogously.

(b)

Use the decomposition of (a) to show that the least squares estimators of $\mu , \alpha_{i}, \ldots$ are given by (4.32) and (4.33).

(c)

Show that the error sum of squares $S^{2}$ is equal to $\Sigma \Sigma \Sigma \left(X_{i j k}-\hat{\xi }_{i j}\right)^{2}$ and hence in the canonical form to $\Sigma_{j=s+1}^{n} Y_{j}^{2}$ .

Problem 3.4.12

(a)(b)

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(a)

Show how the decomposition in Problem 4.11(a) must be modified when it is known that the $\gamma_{i j}$ are zero.

(b)

Use the decomposition of (a) to solve Problem 4.10.

Problem 3.4.13

Let $X_{i j k}(i=1, \ldots , I, j=1, \ldots , J, k=1, \ldots , K)$ be $N\left(\xi_{i j k}, \sigma^{2}\right)$ with

\xi _{i j k}=\mu +\alpha _{i}+\beta _{j}+\gamma _{k}

where $\Sigma \alpha_{i}=\Sigma \beta_{j}=\Sigma \gamma_{k}=0$ . Express $\mu , \alpha_{i}, \beta_{j}$ , and $\gamma_{k}$ in terms of the $\xi$ 's and find their UMVU estimators. Viewed as a special case of (4.4), what is the value of $s$ ?

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Problem 3.4.14

Extend the results of the preceding problem to the model

\xi _{i j k}=\mu +\alpha _{i}+\beta _{j}+\gamma _{k}+\delta _{i j}+\varepsilon _{i k}+\lambda _{j k}

where

\sum _{i} \delta _{i j}=\sum _{j} \delta _{i j}=\sum _{i} \varepsilon _{i k}=\sum _{k} \varepsilon _{i k}=\sum _{j} \lambda _{j k}=\sum _{k} \lambda _{j k}=0.

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Problem 3.4.15

In the preceding problem, if it is known that the $\lambda$ 's are zero, determine whether the UMVU estimators of the remaining parameters remain unchanged.

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Problem 3.4.16

(a)(b)

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(a)

Show that under assumptions (4.35), if $\xi =\theta A$ , then the least squares estimate of $\theta$ is $\mathbf{x} A\left(A A^{\prime }\right)^{-1}$ .

(b)

If $(X, A)$ is multivariate normal with all parameters unknown, show that the least squares estimator of part (a) is a function of the complete sufficient statistic and, hence, prove part (a) of Theorem 4.14.

Problem 3.4.17

(a)(b)(c)

A generalization of the order statistics, to vectors, is given by the following definition.

Definition 8.1 The $c_{j}$ -order statistics of a sample of vectors are the vectors arranged in increasing order according to their $j$ th components.

Let $\mathbf{X}_{i}, i=1, \ldots , n$ , be an iid sample of $p \times 1$ vectors, and let $X=\left(X_{1}, \ldots , X_{n}\right)$ be a $p \times n$ matrix.

Theorem 4.14 states: under assumptions (4.35), with $\xi =\theta A$ , the following hold.

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(a)

If $(X, A)$ are jointly multivariate normal with all parameters unknown, then the least squares estimator $\gamma_{i}\hat{\xi }_{i}$ is the UMVU estimator of $\gamma_{i}\xi_{i}$ .

(b)

If the distribution of $A$ is unknown, then the least squares estimator $\gamma_{i}\hat{\xi }_{i}$ is UMVU among all linear estimators of $\gamma_{i}\xi_{i}$ .

(c)

If $\mathbb {E}\left[AA'\right]$ is known, no best linear unbiased estimator of $\gamma_{i}\xi_{i}$ exists.

If the distribution of $\mathbf{X}_{i}$ is completely unknown, show that, for any $j, j=1, \ldots , p$ , the $c_{j}$ -order statistics of $(X_{1}, \ldots , X_{n})$ are complete sufficient. (That is, the vectors $X_{1}, \ldots , X_{n}$ are ordered according to their $j$ th coordinate.)
Let $Y_{1 \times n}$ be a random variable with unknown distribution (possibly different from $\mathbf{X}_{i}$ ). Form the $(p-1) \times n$ matrix $\binom {\mathbf{x}}{\mathbf{y}}$ , and for any $j=1, \ldots , p$ , calculate the $c_{j}$ -order statistics based on the columns of $\binom {\mathbf{x}}{\mathbf{y}}$ . Show that these $c_{j}$ -order statistics are sufficient.
Use parts (a) and (b) to prove Theorem 4.14(b).

Problem 3.4.18

(a)(b)(c)

The proof of Theorem 4.14(c) is based on two results. Establish that:

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(a)

For large values of $\theta$ , the unconditional variance of a linear unbiased estimator will be greater than that of the least squares estimator.

(b)

For $\theta =0$ , the variance of $X A\left(A A^{\prime }\right)^{-1}$ is greater than that of $X A\left[\mathbb {E}\left[A A^{\prime }\right]\right]^{-1}$ .

(c)

Parts (a) and (b) imply that no best linear unbiased estimator of $\Sigma \gamma_{i} \xi_{i}$ exists if $\mathbb {E}\left[A A^{\prime }\right]$ is known.

Problem 3.4.19

(a)(b)

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(a)

Under the assumptions of Example 4.15, find the variance of $\Sigma \lambda_{i} S_{i}^{2}$ .

(b)

Show that the variance of (a) is minimized by the values stated in the example.

Problem 3.4.20

In the linear model (4.4), a function $\Sigma c_{i} \xi_{i}$ with $\Sigma c_{i}=0$ is called a contrast. Show that a linear function $\Sigma d_{i} \xi_{i}$ is a contrast if and only if it is translation invariant, that is, satisfies $\Sigma d_{i}\left(\xi_{i}+a\right)=\Sigma d_{i} \xi_{i}$ for all $a$ , and hence if and only if it is a function of the differences $\xi_{i}-\xi_{j}$ .

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Problem 3.4.21

(a)(b)(c)

Determine which of the following are contrasts:

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(a)

The regression coefficients $\alpha , \beta$ , or $\gamma$ of (4.2).

(b)

The parameters $\mu , \alpha_{i}, \beta_{j}$ , or $\gamma_{i j}$ of (4.27).

(c)

The parameters $\mu$ or $\alpha_{i}$ of (4.23) and (4.24).