Math Problems Vault

Problem 1.6.1

Extend Example 6.2 to the case that $X_{1}, \ldots , X_{r}$ are independently distributed with Poisson distributions $P\left(\lambda_{i}\right)$ where $\lambda_{i}=a_{i} \lambda \left(a_{i}>0 \text{, known}\right)$ .

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Problem 1.6.2

Let $X_{1}, \ldots , X_{n}$ be iid according to a distribution $F$ and probability density $f$ . Show that the conditional distribution given $X_{(i)}=a$ of the $i-1$ values to the left of $a$ and the $n-i$ values to the right of $a$ is that of $i-1$ variables distributed independently according to the probability density $f(x) / F(a)$ and $n-i$ variables distributed independently with density $f(x) /[1-F(a)]$ , respectively, with the two sets being (conditionally) independent of each other.

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Problem 1.6.3

Let $f$ be a positive integrable function over $(0, \infty )$ , and let $p_{\theta }(x)$ be the density over $(0, \theta )$ defined by $p_{\theta }(x)=c(\theta ) f(x)$ if $0<x<\theta$ , and 0 otherwise. If $X_{1}, \ldots , X_{n}$ are iid with density $p_{\theta }$ , show that $X_{(n)}$ is sufficient for $\theta$ .

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Problem 1.6.4

Let $f$ be a positive integrable function defined over $(-\infty , \infty )$ and let $p_{\xi , \eta }(x)$ be the probability density defined by $p_{\xi , \eta }(x)=c(\xi , \eta ) f(x)$ if $\xi <x<\eta$ , and 0 otherwise. If $X_{1}, \ldots , X_{n}$ are iid with density $p_{\xi , \eta }$ , show that $\left(X_{(1)}, X_{(n)}\right)$ is sufficient for $(\xi , n)$ .

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Problem 1.6.5

Show that each of the statistics $T_{1}-T_{4}$ of Example 6.11 is sufficient.

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Problem 1.6.6

Theorem 6.5 (Factorization Criterion) states: a necessary and sufficient condition for a statistic $T$ to be sufficient for a family $\mathcal{P}=\left\{ P_{\theta }, \theta \in \Omega \right\}$ of distributions of $X$ dominated by a $\sigma$ -finite measure $\mu$ is that there exist non-negative functions $g_{\theta }$ and $h$ such that the densities $p_{\theta }$ of $P_{\theta }$ satisfy $p_{\theta }(x)=g_{\theta }[T(x)] h(x)$ (a.e. $\mu$ ).

Corollary 6.13 states: under the assumptions of Theorem 6.5, a necessary and sufficient condition for a statistic $U$ to be sufficient is that for any fixed $\theta$ and $\theta_{0}$ , the ratio $p_{\theta }(x) / p_{\theta_{0}}(x)$ is a function only of $U(x)$ .

Prove Corollary 6.13.

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Problem 1.6.7

(a)(b)(c)

Let $X_{1}, \ldots , X_{m}$ and $Y_{1}, \ldots , Y_{n}$ be independently distributed according to $N\left(\xi , \sigma^{2}\right)$ and $N\left(\eta , \tau^{2}\right)$ , respectively. Find the minimal sufficient statistics for these cases:

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(a)

$\xi , \eta , \sigma , \tau$ are arbitrary: $-\infty <\xi , \eta <\infty , 0<\sigma , \tau$ .

(b)

$\sigma =\tau$ and $\xi , \eta , \sigma$ are arbitrary.

(c)

$\xi =\eta$ and $\xi , \sigma , \tau$ are arbitrary.

Problem 1.6.8

Let $X_{1}, \ldots , X_{n}$ be iid according to $N\left(\sigma , \sigma^{2}\right), 0<\sigma$ . Find a minimal set of sufficient statistics.

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Problem 1.6.9

(a)(b)

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(a)

If $\left(x_{1}, \ldots , x_{n}\right)$ and $\left(y_{1}, \ldots , y_{n}\right)$ , have the same elementary symmetric functions $\Sigma x_{i}=\Sigma y_{i}, \Sigma_{i \neq j} x_{i} y_{j}=\Sigma_{i \neq j} y_{i} y_{j}, \ldots , x_{1} \cdots x_{n}=y_{1} \cdots y_{n}$ , then the $y$ 's are a permutation of the $x$ 's.

(b)

In the notation of Example 6.10, show that $U$ is equivalent to $V$ . [Hint: Compare the coefficients and the roots of the polynomials $P(x)=\Pi \left(x-u_{i}\right)$ and $Q(x)= \Pi \left(x-v_{i}\right)$ .]

Problem 1.6.10

(a)(b)

Show that the order statistics are minimal sufficient for the location family (6.7) when $f$ is the density of

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(a)

the double exponential distribution $D(0,1)$ .

(b)

the Cauchy distribution $C(0,1)$ .

Problem 1.6.11

Prove the following generalization of Theorem 6.12 to families without common support.

Theorem 9.1 Let $\mathcal{P}$ be a finite family with densities $p_{i}, i=0, \ldots , k$ , and for any $x$ , let $S(x)$ be the set of pairs of subscripts $(i, j)$ for which $p_{i}(x)+p_{j}(x)>0$ . Then, the statistic

T(X)=\left\{ \frac{p_{j}(X)}{p_{i}(X)}, \quad i<j \text{ and }(i, j) \in S(X)\right\}

is minimal sufficient. Here, $p_{j}(x) / p_{i}(x)=\infty$ if $p_{i}(x)=0$ and $p_{j}(x)>0$ .

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Problem 1.6.12

In Problem 6.11 it is not enough to replace $p_{i}(X)$ by $p_{0}(X)$ . To see this let $k=2$ and $p_{0}=U(-1,0), p_{1}=U(0,1)$ , and $p_{2}(x)=2 x, 0<x<1$ .

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Problem 1.6.13

(a)(b)(c)

Let $k=1$ and $P_{i}=U(i, i+1), i=0,1$ .

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(a)

Show that a minimal sufficient statistic for $\mathcal{P}=\left\{ P_{0}, P_{1}\right\}$ is $T(X)=i$ if $i<X<i+1$ , $i=0,1$ .

(b)

Let $X_{1}$ and $X_{2}$ be iid according to a distribution from $\mathcal{P}$ . Show that each of the two statistics $T_{1}=T\left(X_{1}\right)$ and $T_{2}=T\left(X_{2}\right)$ is sufficient for $\left(X_{1}, X_{2}\right)$ .

(c)

Show that $T\left(X_{1}\right)$ and $T\left(X_{2}\right)$ are equivalent.

Problem 1.6.14

In Lemma 6.14, show that the assumption of common support can be replaced by the weaker assumption that every $\mathcal{P}_{0}$ -null set is also a $\mathcal{P}$ -null set so that (a.e. $\mathcal{P}_{0}$ ) is equivalent to (a.e. $\mathcal{P}$ ).

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Problem 1.6.15

Let $X_{1}, \ldots , X_{n}$ be iid according to a distribution from $\mathcal{P}=\left\{ U(0, \theta ), \theta >0\right\}$ , and let $\mathcal{P}_{0}$ be the subfamily of $\mathcal{P}$ for which $\theta$ is rational. Show that every $\mathcal{P}_{0}$ -null set in the sample space is also a $\mathcal{P}$ -null set.

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Problem 1.6.16

(a)(b)(c)

Let $X_{1}, \ldots , X_{n}$ be iid according to a distribution from a family $\mathcal{P}$ . Show that $T$ is minimal sufficient in the following cases:

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(a)

$\mathcal{P}=\left\{ U(0, \theta ), \theta >0\right\} ; T=X_{(n)}$ .

(b)

$\mathcal{P}=\left\{ U\left(\theta_{1}, \theta_{2}\right),-\infty <\theta_{1}<\theta_{2}<\infty \right\} ; T=\left(X_{(1)}, X_{(n)}\right)$ .

(c)

$\mathcal{P}=\left\{ U(\theta -1 / 2, \theta +1 / 2),-\infty <\theta <\infty \right\} ; T=\left(X_{(1)}, X_{(n)}\right)$ .

Problem 1.6.17

(a)(b)(c)

Solve the preceding problem for the following cases:

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(a)

$\mathcal{P}=\left\{ E(\theta , 1),-\infty <\theta <\infty \right\} ; T=X_{(1)}$ .

(b)

$\mathcal{P}=\left\{ E(0, b), 0<b\right\} ; T=\Sigma X_{i}$ .

(c)

$\mathcal{P}=\left\{ E(a, b),-\infty <a<\infty , 0<b\right\} ; T=\left(X_{(1)}, \Sigma \left[X_{i}-X_{(1)}\right]\right)$ .

Problem 1.6.18

Show that the statistics $X_{(1)}$ and $\Sigma \left[X_{i}-X_{(1)}\right]$ of Problem 6.17(c) are independently distributed as $E(a, b / n)$ and $b$ Gamma $(n-2,1)$ respectively.

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Problem 1.6.19

Show that the sufficient statistics of

Problem 6.3 and
Problem 6.4

are minimal sufficient.

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Problem 1.6.20

(a)(b)

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(a)

Show that in the $N(\theta , \theta )$ curved exponential family, the sufficient statistic $T= (\sum x_{i}, \sum x_{i}^{2})$ is not minimal.

(b)

For the density of Example 6.19, show that $T=\left(\sum x_{i}, \sum x_{i}^{2}, \sum x_{i}^{3}\right)$ is a minimal sufficient statistic.

Problem 1.6.21

For the situation of Example 6.25(ii), find an unbiased estimator of $\xi$ based on $\sum X_{i}$ , and another based on $\sum X_{i}^{2}$ ); hence, deduce that $T=\left(\sum X_{i}, \sum X_{i}^{2}\right)$ is not complete.

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Problem 1.6.22

For the situation of Example 6.26, show that $\mathbf{X}$ is minimal sufficient and complete.

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Problem 1.6.23

(a)(b)

For the situation of Example 6.27:

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(a)

Show that $\mathbf{X}=\left(X_{1}, X_{2}\right)$ is minimal sufficient for the family (6.16) with restriction (6.17).

(b)

Establish (6.18), and hence that the minimal sufficient statistic of part (a) is not complete.

Problem 1.6.24

Show that for the general dose-response model

p_{\theta }(\mathbf{x})=\prod _{i=1}^{m}\binom {n_{i}}{x_{i}}\left[\eta _{\theta }\left(d_{i}\right)\right]^{x_{i}}\left[1-\eta _{\theta }\left(d_{i}\right)\right]^{n_{i}-x_{i}},

the statistic $\mathbf{X}=\left(X_{1}, X_{2}, \ldots , X_{m}\right)$ is minimal sufficient if there exist vectors $\left.\theta_{1}, \theta_{2}, \cdots , \theta_{m}\right)$ such that the $m \times m$ matrix

P=\left\{ \log \left(\frac{\eta _{\theta _{j}}\left(d_{i}\right)\left[1-\eta _{\theta _{0}}\left(d_{i}\right)\right]}{\eta _{\theta _{0}}\left(d_{i}\right)\left[1-\eta _{\theta _{j}}\left(d_{i}\right)\right]}\right)\right\}

is invertible. (Hint: Theorem 6.12.)

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Problem 1.6.25

(a)(b)(c)

Let $\left(X_{i}, Y_{i}\right), i=1, \ldots , n$ , be iid according to the uniform distribution over a set $R$ in the $(x, y)$ plane and let $\mathcal{P}$ be the family of distributions obtained by letting $R$ range over a class $\mathcal{R}$ of sets $R$ . Determine a minimal sufficient statistic for the following cases:

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(a)

$\mathcal{R}$ is the set of all rectangles $a_{1}<x<a_{2}, b_{1}<y<b_{2},-\infty <a_{1}<a_{2}<\infty$ , $-\infty <b_{1}<b_{2}<\infty$ .

(b)

$\mathcal{R}^{\prime }$ is the subset of $\mathcal{R}$ , for which $a_{2}-a_{1}=b_{2}-b_{1}$ .

(c)

$\mathcal{R}^{\prime \prime }$ is the subset of $\mathcal{R}^{\prime }$ for which $a_{2}-a_{1}=b_{2}-b_{1}=1$ .

Problem 1.6.26

(a)(b)

Solve the preceding problem if

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(a)

$\mathcal{R}$ is the set of all triangles with sides parallel to the $x$ axis, the $y$ axis, and the line $y=x$ , respectively.

(b)

$\mathcal{R}^{\prime }$ is the subset of $\mathcal{R}$ in which the sides parallel to the $x$ and $y$ axes are equal.

Problem 1.6.27

Formulate a general result of which Problems 6.25(a) and 6.26(a) are special cases.

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Problem 1.6.28

If $Y$ is distributed as $E(\eta , 1)$ , the distribution of $X=e^{-Y}$ is $U\left(0, e^{-\eta }\right)$ . (This result is useful in the computer generation of random variables; see Problem 4.4.14.)

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Problem 1.6.29

If a minimal sufficient statistic exists, a necessary condition for a sufficient statistic to be complete is for it to be minimal.

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Problem 1.6.30

Show that the minimal sufficient statistics $T=\left(X_{(1)}, X_{(n)}\right)$ of Problem 6.16(b) are complete.

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Problem 1.6.31

(a)(b)(c)

For each of the following problems, determine whether the minimal sufficient statistic is complete:

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(a)

Problem 6.7(a)-(c);

(b)

Problem 6.25(a)-(c);

(c)

Problem 6.26(a) and (b).

Problem 1.6.32

(a)(b)

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(a)

Show that if $\mathcal{P}_{0}, \mathcal{P}_{1}$ are two families of distributions such that $\mathcal{P}_{0} \in \mathcal{P}_{1}$ and every null set of $\mathcal{P}_{0}$ is also a null set of $\mathcal{P}_{1}$ , then a sufficient statistic $T$ that is complete for $\mathcal{P}_{0}$ is also complete for $\mathcal{P}_{1}$ .

(b)

Let $\mathcal{P}_{0}$ be the class of binomial distributions $b(p, n), 0<p<1, n=$ fixed, and let $\mathcal{P}_{1}=\mathcal{P}_{0} \cup \left\{ Q\right\}$ where $Q$ is the Poisson distribution with expectation 1. Then $\mathcal{P}_{0}$ is complete but $\mathcal{P}_{1}$ is not.

Problem 1.6.33

Let $X_{1}, \ldots , X_{n}$ be iid each with density $f(x)$ (with respect to Lebesgue measure), which is unknown. Show that the order statistics are complete.

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Problem 1.6.34

(a)(b)(c)

Suppose that $X_{1}, \ldots , X_{n}$ are an iid sample from a location-scale family with distribution function $F((x-a) / b)$ .

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(a)

If $b$ is known, show that the differences $\left(X_{1}-X_{i}\right) / b, i=2, \ldots , n$ , are ancillary.

(b)

If $a$ is known, show that the ratios $\left(X_{1}-a\right) /\left(X_{i}-a\right), i=2, \ldots , n$ , are ancillary.

(c)

If neither $a$ or $b$ are known, show that the quantities $\left(X_{1}-X_{i}\right) /\left(X_{2}-X_{i}\right), i= 3, \ldots , n$ , are ancillary.

Problem 1.6.35

(a)(b)

Use Basu's theorem to prove independence of the following pairs of statistics:

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(a)

$\bar{X}$ and $\Sigma \left(X_{i}-\bar{X}\right)^{2}$ where the $X$ 's are iid as $N\left(\xi , \sigma^{2}\right)$ .

(b)

$X_{(1)}$ and $\Sigma \left[X_{i}-X_{(1)}\right]$ in Problem 6.18.

Problem 1.6.36

(a)(b)

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(a)

Under the assumptions of Problem 6.18, the ratios $Z_{i}=\left[X_{(n)}-X_{(i)}\right] / \left[X_{(n)}-X_{(n-1)}\right], i=1, \ldots , n-2$ , are independent of $\left\{ X_{(1)}, \Sigma \left[X_{i}-X_{(1)}\right]\right\}$ .

(b)

Under the assumptions of Problems 6.16(b) and 6.30 the ratios $Z_{i}=\left[X_{(i)}-X_{(1)}\right] / \left[X_{(n)}-X_{(1)}\right], i=2, \ldots , n-1$ , are independent of $\left\{ X_{(1)}, X_{(n)}\right\}$ .

Problem 1.6.37

(a)(b)

Under the assumptions of Theorem 6.5, let $A$ be any fixed set in the sample space, $P_{\theta }^{*}$ the distribution $P_{\theta }$ truncated on $A$ , and $\mathcal{P}^{*}=\left\{ P_{\theta }^{*}, \theta \in \Omega \right\}$ . Then prove

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(a)

if $T$ is sufficient for $\mathcal{P}$ , it is sufficient for $\mathcal{P}^{*}$ .

(b)

if, in addition, $T$ is complete for $\mathcal{P}$ , it is also complete for $\mathcal{P}^{*}$ .

Generalizations of this result were derived by Tukey in the 1940s and also by Smith (1957). The analogous problem for observations that are censored rather than truncated is discussed by Bhattacharyya, Johnson, and Mehrotra (1977).

Problem 1.6.38

(a)(b)

If $X_{1}, \ldots , X_{n}$ are iid as $B(a, b)$ ,

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(a)

Show that $\left[\Pi X_{i}, \Pi \left(1-X_{i}\right)\right]$ is minimal sufficient for $(a, b)$ .

(b)

Determine the minimal sufficient statistic when $a=b$ .