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For every Borel subset \( A \) of \( {\left\lbrack 0,1\right\rbrack }^{d} \), let \( {X}_{A} \) be a Gaussian random variable. We want the covariance of \( {X}_{A} \) and \( {X}_{B} \) to be the Lebesgue measure of \( A \cap B \) . This is known as a set-indexed process. If we let \( T \) be the collection of all Borel... | In order to get a continuous process \( X \) one must restrict \( T \) to be a subcollection of sets whose boundaries are sufficiently smooth; see Dudley (1973). | No |
For what subsets \( T \) of \( {L}^{2}\left( \left\lbrack {0,1}\right\rbrack \right) \) can one define a process \( {X}_{f} \) that has continuous paths with respect to \( d \) ? This means that the map \( f \rightarrow {X}_{f}\left( \omega \right) \) is continuous for almost all \( \omega \), where we use the pseudo-m... | It turns out \( T = \left\{ {f \in {L}^{2}\left( \left\lbrack {0,1}\right\rbrack \right) : \parallel f{\parallel }_{2} \leq 1}\right\} \) is too large to obtain a continuous Gaussian process, but, for example, \( T = \left\{ {f \in {C}^{2}\left( \left\lbrack {0,1}\right\rbrack \right) }\right. \) : \( {\begin{Vmatrix}f... | Yes |
Lemma 34.1 There exists \( {\varepsilon }_{0} \) such that\n\n\[ \rho \left( {f, g}\right) \leq {2d}\left( {f, g}\right) \]\n\nif \( d\left( {f, g}\right) < {\varepsilon }_{0} \) . | Proof Since \( \log \left( {1 + {2x}}\right) /\left( {2x}\right) \rightarrow 1 \) as \( x \rightarrow 0 \), we have\n\n\[ \log \left( {1 - {2\varepsilon }}\right) < - \varepsilon < \varepsilon < \log \left( {1 + {2\varepsilon }}\right) \]\n\nif \( \varepsilon \) is small enough. Suppose \( d\left( {f, g}\right) < \vare... | Yes |
Lemma 34.2 Suppose \( \delta < 1/4 \) . Let \( f \in D \) . If \( \rho \left( {f, g}\right) \leq {\delta }^{2} \), then \( d\left( {f, g}\right) \leq {4\delta } + {\xi }_{f}\left( \delta \right) \) . | Proof Choose \( {t}_{i} \) ’s such that \( {t}_{i} - {t}_{i - 1} > \delta \) and \( {\theta }_{f}\left\lbrack {{t}_{i - 1},{t}_{i}}\right) < {\xi }_{f}\left( \delta \right) + \delta \) for each \( i \) . Pick \( \mu \in \Lambda \) such that \( \mathop{\sup }\limits_{t}\left| {f\left( t\right) - g\left( {\mu \left( t\ri... | Yes |
Proposition 34.3 The metrics \( d \) and \( \rho \) are equivalent, i.e., they generate the same topology. In particular, \( \left( {D, d}\right) \) is separable. | Proof Let \( {B}_{\rho }\left( {f, r}\right) \) denote the ball with center \( f \) and radius \( r \) with respect to the metric \( \rho \) and define \( {B}_{d}\left( {f, r}\right) \) analogously. Let \( \varepsilon > 0 \) and let \( f \in D \) . If \( d\left( {f, g}\right) < \varepsilon /2 \) and \( \varepsilon \) i... | Yes |
Proposition 34.5 Suppose \( {f}_{n} \rightarrow f \) in the topology of \( D\left\lbrack {0,1}\right\rbrack \) with respect to \( d \) and \( f \in \) \( C\left\lbrack {0,1}\right\rbrack \) . Then \( \mathop{\sup }\limits_{{t \in \left\lbrack {0,1}\right\rbrack }}\left| {{f}_{n}\left( t\right) - f\left( t\right) }\righ... | Proof Let \( \varepsilon > 0 \) . Since \( f \) is uniformly continuous on \( \left\lbrack {0,1}\right\rbrack \), there exists \( \delta \) such that \( \left| {f\left( t\right) - f\left( s\right) }\right| < \varepsilon /2 \) if \( \left| {t - s}\right| < \delta \) . For \( n \) sufficiently large there exists \( {\lam... | Yes |
Proposition 34.9 Suppose \( {X}_{n} \) is a sequence of real-valued strong Markov processes and there exists \( c, p \), and \( \gamma > 0 \) such that\n\n\[ \n{\mathbb{E}}^{x}{\left| {X}_{n}\left( t\right) - {X}_{n}\left( 0\right) \right| }^{p} \leq c{t}^{\gamma },\;x \in \mathbb{R},\;t \in \left\lbrack {0,1}\right\rb... | Proof Fix \( x \) . For each \( t \) ,\n\n\[ \n{\mathbb{P}}^{x}\left( {\left| {{X}_{n}\left( t\right) }\right| \geq R + \left| x\right| }\right) \leq {\mathbb{P}}^{x}\left( {\left| {{X}_{n}\left( t\right) - {X}_{n}\left( 0\right) }\right| \geq R}\right)\n\]\n\n\[ \n\leq \frac{{\mathbb{E}}^{x}{\left| {X}_{n}\left( t\rig... | Yes |
Corollary 35.2 Let \( M = \mathop{\sup }\limits_{{s \leq 1}}{W}_{s} \) and \( {M}_{n} = \mathop{\sup }\limits_{{s \leq 1}}{Z}_{n}\left( s\right) \), where \( W \) is a Brownian motion. Then \( {M}_{n} \) converges weakly to \( \bar{M} \) . | Proof Let \( g \) be a bounded and continuous function on the reals and define a function \( F \) on \( C\left\lbrack {0,1}\right\rbrack \) by\n\n\[ F\left( f\right) = g\left( {\mathop{\sup }\limits_{{s \leq 1}}f\left( s\right) }\right) \]\n\nNotice \( \left| {\mathop{\sup }\limits_{{s \leq 1}}{f}_{2}\left( s\right) - ... | Yes |
Lemma 35.3 Suppose we have a sequence \( {Y}_{i} \) of i.i.d. random variables with mean zero and variance one and \( {S}_{n} = \mathop{\sum }\limits_{{i = 1}}^{n}{Y}_{i} \) . Suppose \( \lambda > 4 \) . Then\n\n\[ \mathbb{P}\left( {\mathop{\max }\limits_{{i \leq n}}\left| {S}_{i}\right| \geq \lambda \sqrt{n}}\right) \... | Proof Let \( N = \min \left\{ {i : \left| {S}_{i}\right| \geq \lambda \sqrt{n}}\right\} \), the first time \( {S}_{i} \) is bigger than \( \lambda \sqrt{n}.N \) is a stopping time and \( \left( {N = i}\right) \) is in the \( \sigma \) -field generated by \( {Y}_{1},\ldots ,{Y}_{i} \) . We have\n\n\[ \mathbb{P}\left( {\... | Yes |
Lemma 35.4 For each \( \varepsilon ,\eta > 0 \), there exist \( {n}_{0} \) and \( \delta \) such that if \( n \geq {n}_{0} \) and \( s \in \left\lbrack {0,1 - \delta }\right\rbrack \) , then\n\n\[ \mathbb{P}\left( {\mathop{\sup }\limits_{{s \leq t \leq s + \delta }}\left| {{Z}_{n}\left( t\right) - {Z}_{n}\left( s\right... | Proof Let \( \varepsilon ,\eta > 0 \), and choose \( \delta \) small enough that \( 2{e}^{-{\varepsilon }^{2}/{128\delta }} \leq {\delta \eta }/2 \) . Then choose \( {j}_{0} \) large enough so that, using (35.2),\n\n\[ \mathbb{P}\left( {\left| {S}_{j}\right| > \frac{\varepsilon \sqrt{j}}{8\sqrt{\delta }}}\right) \leq 2... | Yes |
Lemma 35.5 For each \( \varepsilon ,\eta > 0 \) there exist \( {n}_{0} \) and \( \delta \) such that if \( n \geq {n}_{0} \) , \[ \mathbb{P}\left( {{\omega }_{{Z}_{n}}\left( \delta \right) \geq \varepsilon }\right) \leq {2\eta } \] | Proof We will take \( \delta = 1/K \) for some large \( K \) . If \( \left| {t - s}\right| \leq 1/K \), then either both \( s, t \) are in the same interval \( \left\lbrack {\left( {i - 1}\right) /K, i/K}\right\rbrack \) or they are in adjoining intervals. Thus they both lie in some interval of the form \( \left\lbrack... | Yes |
Proposition 35.6 Suppose the \( {Y}_{i} \) are independent with mean zero and variance one. The \( {\widetilde{Z}}_{n} \) converge weakly with respect to the topology of \( D\left\lbrack {0,1}\right\rbrack \) to Brownian motion. | Proof The \( {Z}_{n} \) converge weakly with respect to the topology of \( C\left\lbrack {0,1}\right\rbrack \) to a Brownian motion. By the Skorokhod representation (Theorem 31.2), we can find a probability space and random variables \( {Z}_{n}^{\prime } \) having the same law as \( {Z}_{n} \) that converge almost sure... | No |
Proposition 35.7 \( {\mathbb{P}}_{\varepsilon } \) converges weakly to \( {\mathbb{P}}_{0} \) with respect to the topology of \( C\left\lbrack {0,1}\right\rbrack \) as \( \varepsilon \rightarrow 0 \) . | Proof Since \( W \) is a jointly normal process and\n\n\[ \operatorname{Cov}\left( {{W}_{t} - t{W}_{1},{W}_{1}}\right) = \operatorname{Cov}\left( {{W}_{t},{W}_{1}}\right) - t\operatorname{Var}\left( {W}_{1}\right) = 0, \]\n\nthen the process \( {W}_{t}^{0} = {W}_{t} - t{W}_{1} \) and the random variable \( {W}_{1} \) a... | Yes |
Proposition 36.3 If \( f \) is bounded and Borel measurable, \( s, t > 0 \), and \( x \in \mathcal{S} \), then\n\n\[{\mathbb{E}}^{x}\left\lbrack {{\mathbb{E}}^{{X}_{t}}f\left( {X}_{s}\right) }\right\rbrack = {\mathbb{E}}^{x}f\left( {X}_{s + t}\right)\] | Proof The proof of (36.4) is mainly a matter of sorting out notation. Let \( \varphi \left( x\right) = {\mathbb{E}}^{x}f\left( {X}_{s}\right) = \) \( {P}_{s}f\left( x\right) \) . Hence \( {\mathbb{E}}^{{X}_{t}}f\left( {X}_{s}\right) = \varphi \left( {X}_{t}\right) = {P}_{s}f\left( {X}_{t}\right) \) . Then the left-hand... | Yes |
Proposition 36.4 Ifs \( < t \) with \( s, t \in \mathcal{D} \) and \( f \) is bounded and Borel measurable, then\n\n\[ \n{\mathbb{E}}^{x}\left\lbrack {f\left( {X}_{t}\right) \mid {\mathcal{F}}_{s}^{\prime }}\right\rbrack = {\mathbb{E}}^{{X}_{s}}f\left( {X}_{t - s}\right) ,\;{\mathbb{P}}^{x}\text{-a.s. }\n\] | Proof Take \( n \geq 1,{r}_{1} \leq {r}_{2} \leq \cdots \leq {r}_{n} \leq s \) with each \( {r}_{j} \) in \( \mathcal{D} \), and \( {A}_{1},\ldots ,{A}_{n} \) Borel subsets of \( \mathcal{S} \) . It suffices to show that\n\n\[ \n{\mathbb{E}}^{x}\left\lbrack {f\left( {X}_{t}\right) {1}_{{A}_{1}}\left( {X}_{{r}_{1}}\righ... | Yes |
Lemma 36.5 If \( f \geq 0 \) is bounded and Borel measurable and \( x \in \mathcal{S} \), then \( {M}_{t} = {e}^{-{\lambda t}}{R}_{\lambda }f\left( {X}_{t}\right) \) , \( t \in \mathcal{D} \), is a supermartingale with respect to the filtration \( \left\{ {{\mathcal{F}}_{t}^{\prime };t \in \mathcal{D}}\right\} \) and t... | Proof What we need to show is that if \( s < t \in \mathcal{D} \), then\n\n\[ \n{\mathbb{E}}^{x}\left\lbrack {{e}^{-{\lambda t}}{R}_{\lambda }f\left( {X}_{t}\right) \mid {\mathcal{F}}_{s}^{\prime }}\right\rbrack \leq {e}^{-{\lambda s}}{R}_{\lambda }f\left( {X}_{s}\right) ,\;{\mathbb{P}}^{x}\text{-a.s. }\n\]\n\nBy Propo... | Yes |
Example 36.6 Our first example is a Brownian motion. Let\n\n\[ \np\left( {t, x, y}\right) = {\left( 2\pi t\right) }^{d/2}{e}^{-{\left| x - y\right| }^{2}/{2t}}, \n\]\n\nand set\n\n\[ \n{P}_{t}\left( {x, A}\right) = {\int }_{A}p\left( {t, x, y}\right) {dy}. \n\]\n\nWe know\n\n\[ \n\int p\left( {t, x, z}\right) p\left( {... | We showed in Section 19.4 that Assumption 36.1 is satisfied, except for the fact that \( {P}_{t} \) maps \( {C}_{0} \) to \( {C}_{0} \) ; this is Exercise 36.2. Therefore we have a strong Markov process associated with \( {P}_{t} \). By Proposition 21.5, the paths of the strong Markov process can be taken to be continu... | No |
Example 36.7 We now use the machinery we have developed in this chapter to construct the Poisson process. Define transition probabilities by\n\n\[ \n{P}_{t}\left( {x, A}\right) = {e}^{-{\lambda t}}\mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{{\left( \lambda t\right) }^{k}}{k!}{1}_{A}\left( {x + k}\right) ,\n\]\n\nwhe... | We therefore have a strong Markov process \( X \) whose paths are right continuous with left limits. We want to show that the process \( {X}_{t} \) under the probability measure \( {\mathbb{P}}^{0} \) is a Poisson process. That \( {\mathbb{P}}^{0}\left( {{X}_{0} = 0}\right) = 1 \) is obvious. We need to show that Defin... | Yes |
Corollary 37.3 If \( \mu ,\lambda > 0 \) and \( \left| {\mu - \lambda }\right| < \lambda \), then\n\n\[ \n{R}_{\mu }f = {R}_{\lambda }f + \mathop{\sum }\limits_{{i = 1}}^{\infty }{\left( \lambda - \mu \right) }^{i}{R}_{\lambda }^{i + 1}f.\n\]\n\n(37.3)\n\nHere \( {R}_{\lambda }^{2}f = {R}_{\lambda }\left( {{R}_{\lambda... | Proof By Proposition 37.2, we have\n\n\[ \n{R}_{\mu }f = {R}_{\lambda }f + \left( {\lambda - \mu }\right) {R}_{\lambda }{R}_{\mu }f.\n\]\n\n(37.4)\n\nIf we substitute for \( {R}_{\mu }f \) in the last term on the right-hand side of (37.4), we have\n\n\[ \n{R}_{\mu }f = {R}_{\lambda }f + \left( {\lambda - \mu }\right) {... | Yes |
Proposition 37.5 Fix \( \lambda > 0 \) and let \( C = \left\{ {{R}_{\lambda }f : f \in \mathcal{B}}\right\} \) . Then \( C = \mathcal{D}\left( \mathcal{L}\right) \) and for \( f \in \mathcal{B} \) , \[ \mathcal{L}{R}_{\lambda }f = \lambda {R}_{\lambda }f - f. \] | Proof Suppose that \( g \in C \), so that \( g = {R}_{\lambda }f \) for some \( f \in \mathcal{B} \) . Then \[ {P}_{h}{R}_{\lambda }f = {\int }_{0}^{\infty }{e}^{-{\lambda t}}{P}_{h + t}{fdt} = {e}^{\lambda h}{\int }_{h}^{\infty }{e}^{-{\lambda t}}{P}_{t}{fdt}, \] (37.6) and so \[ {P}_{h}g - g = {P}_{h}{R}_{\lambda }f ... | Yes |
Let us compute the infinitesimal generator when \( \\left( {{X}_{t},{\\mathbb{P}}^{x}}\\right) \) is a one-dimensional Brownian motion. For our space \( \\mathcal{B} \) we take the continuous functions on \( \\mathbb{R} \) that vanish at infinity. Suppose \( f \\in {C}^{2} \) with compact support. By a Taylor series ex... | \[ {P}_{h}f\\left( x\\right) = {\\mathbb{E}}^{x}f\\left( {X}_{h}\\right) = f\\left( x\\right) + {f}^{\\prime }\\left( x\\right) {\\mathbb{E}}^{x}\\left( {{X}_{h} - x}\\right) + \\frac{1}{2}{f}^{\\prime \\prime }\\left( x\\right) {\\mathbb{E}}^{x}{\\left( {X}_{h} - x\\right) }^{2} + {R}_{h}, \] where \( {R}_{h} \) is th... | No |
Next we compute the generator for a Poisson process with parameter \( \lambda \) . We can let \( \mathcal{B} \) be as in Example 37.6. | \[ {P}_{h}f\left( x\right) = \mathop{\sum }\limits_{{i = 0}}^{\infty }{e}^{-{\lambda h}}\frac{{\left( \lambda h\right) }^{i}}{i!}f\left( {x + i}\right) \] \[ = {e}^{-{\lambda h}}f\left( x\right) + {e}^{-{\lambda h}}{\lambda hf}\left( {x + 1}\right) + \mathop{\sum }\limits_{{i = 2}}^{\infty }{e}^{-{\lambda h}}\frac{{\le... | Yes |
Theorem 37.8 Suppose \( {P}_{t} \) operating on the space \( \mathcal{B} \) of continuous functions vanishing at infinity is the semigroup of a Markov process \( \left( {{X}_{t},{\mathbb{P}}^{x}}\right), f \in \mathcal{D}\left( \mathcal{L}\right) \), and \( f \) and \( \mathcal{L}f \) are bounded. If \( x \in \mathcal{... | Proof If \( f \in \mathcal{D}\left( \mathcal{L}\right) \), then \( \mathcal{L}f \in \mathcal{B} \), and so \( {P}_{t}\mathcal{L}f \) is continuous in \( t \) . Moreover, as we saw in (37.5), \[ \frac{\partial }{\partial t}{P}_{t}f\left( y\right) = {P}_{t}\mathcal{L}f\left( y\right) \] By the fundamental theorem of calc... | Yes |
Corollary 38.2 If \( 0 \leq f \leq 1 \), m-a.e., then \( 0 \leq {P}_{t}f \leq 1 \), m-a.e. | Proof If \( 0 \leq f \leq 1 \) , \( m \) -a.e., then \( 0 \leq b{R}_{b}f \leq 1, m \) -a.e, by Theorem 38.1, and iterating, \( 0 \leq {\left( b{R}_{b}\right) }^{i}f \leq 1, m \) -a.e., for every \( i \) . Using the notation of the proof of Proposition 37.9,\n\n\[ \n{Q}_{t}^{b}f\left( x\right) = {e}^{-{bt}}\mathop{\sum ... | Yes |
Theorem 39.3 Suppose \( {X}_{t} \) is a solution to (39.1), \( \sigma \) and \( b \) are bounded and Borel measurable, and \( a = \sigma {\sigma }^{T} \) . Suppose \( f \in {C}^{2} \) . Then\n\n\[ f\left( {X}_{t}\right) = f\left( {X}_{0}\right) + {M}_{t} + {\int }_{0}^{t}\mathcal{L}f\left( {X}_{s}\right) {ds}, \] | Proof Since the components of the Brownian motion \( {W}_{t} \) are independent, we have \( d{\left\langle {W}^{k},{W}^{\ell }\right\rangle }_{t} = 0 \) if \( k \neq \ell \) ; see Exercise 9.4. Therefore\n\n\[ d{\left\langle {X}^{i},{X}^{j}\right\rangle }_{t} = \mathop{\sum }\limits_{k}\mathop{\sum }\limits_{\ell }{\si... | No |
Theorem 40.1 Suppose \( u \) is a \( {C}^{2} \) solution to (40.3) such that \( u \) and its first and second partial derivatives are bounded. Then\n\n\[ u\left( x\right) = {\mathbb{E}}^{x}{\int }_{0}^{\infty }{e}^{-{\lambda t}}f\left( {X}_{t}\right) {dt}. \] | Proof Let \( u \) be the solution to (40.3). By Theorem 39.3,\n\n\[ u\left( {X}_{t}\right) - u\left( {X}_{0}\right) = {M}_{t} + {\int }_{0}^{t}\mathcal{L}u\left( {X}_{s}\right) {ds} \]\n\nwhere \( {M}_{t} \) is a martingale. By the product formula,\n\n\[ {e}^{-{\lambda t}}u\left( {X}_{t}\right) - u\left( {X}_{0}\right)... | Yes |
Theorem 40.2 Suppose \( u \) is a solution to Poisson’s equation in a bounded domain \( D \) that is \( {C}^{2} \) in \( D \) and continuous on \( \bar{D} \) . Then\n\n\[ u\left( x\right) = {\mathbb{E}}^{x}{\int }_{0}^{{\tau }_{D}}{e}^{-{\lambda s}}f\left( {X}_{s}\right) {ds}. \] | Proof The proof is nearly identical to that of the previous theorem. We already mentioned that \( {\tau }_{D} < \infty \), a.s.; see Exercise 40.1. Let \( {S}_{n} = \inf \left\{ {t : \operatorname{dist}\left( {{X}_{t},\partial D}\right) < 1/n}\right\} \) . By Theorem 39.3,\n\n\[ u\left( {X}_{t \land {S}_{n}}\right) - u... | No |
Theorem 40.3 Suppose \( u \) is a solution to the Dirichlet problem specified by (40.4). Then \( u \) satisfies\n\n\[ u\left( x\right) = {\mathbb{E}}^{x}f\left( {X}_{{\tau }_{D}}\right) \] | Proof As we mentioned above, \( {\tau }_{D} < \infty \), a.s. Let \( {S}_{n} = \inf \left\{ {t : \operatorname{dist}\left( {{X}_{t},\partial D}\right) < 1/n}\right\} \) . By Theorem 39.3,\n\n\[ u\left( {X}_{t \land {S}_{n}}\right) = u\left( {X}_{0}\right) + \text{ martingale } + {\int }_{0}^{t \land {S}_{n}}\mathcal{L}... | Yes |
Theorem 40.4 Suppose there exists a solution to (40.5) that is \( {C}^{2} \) in \( x \) and \( {C}^{1} \) in \( t \) for \( t > 0 \) . Then \( u \) satisfies\n\n\[ u\left( {x, t}\right) = {\mathbb{E}}^{x}f\left( {X}_{t}\right) \] | Proof Fix \( {t}_{0} \) and let \( {M}_{t} = u\left( {{X}_{t},{t}_{0} - t}\right) \) . Note\n\n\[ \frac{\partial }{\partial t}u\left( {x,{t}_{0} - t}\right) = - {u}_{t}\left( {x,{t}_{0} - t}\right) . \]\n\nSimilarly to the proof of Theorem 39.3 (see Exercise 40.2) but using now the multivariate version of Itô's formula... | Yes |
Theorem 40.5 Let \( D, q, f \) be as above. Let \( u \) be a \( {C}^{2} \) function on \( \bar{D} \) that agrees with \( f \) on \( \partial D \) and satisfies \( \mathcal{L}u + {qu} = 0 \) in \( D \) . If\n\n\[ \n{\mathbb{E}}^{x}\exp \left( {{\int }_{0}^{{\tau }_{D}}{q}^{ + }\left( {X}_{s}\right) {ds}}\right) < \infty... | Proof Let \( {B}_{t} = {\int }_{0}^{t \land {\tau }_{D}}q\left( {X}_{s}\right) {ds} \) . By Itô’s formula and the product formula,\n\n\[ \n{e}^{B\left( {t \land {\tau }_{D}}\right) }u\left( {X}_{t \land {\tau }_{D}}\right) = u\left( {X}_{0}\right) + \text{ martingale } + {\int }_{0}^{t \land {\tau }_{D}}u\left( {X}_{r}... | Yes |
Theorem 41.1 The scale function \( s\left( x\right) \) is the solution to\n\n\[ \frac{1}{2}a\left( x\right) {s}^{\prime \prime }\left( x\right) + b\left( x\right) {s}^{\prime }\left( x\right) = 0, \]\n\nand for some constants \( {c}_{1},{c}_{2} \), and \( {x}_{0} \) is given by\n\n\[ s\left( x\right) = {c}_{1} + {c}_{2... | Proof To solve the differential equation, we write\n\n\[ \frac{{s}^{\prime \prime }\left( x\right) }{{s}^{\prime }\left( x\right) } = - 2\frac{b\left( x\right) }{a\left( x\right) } \]\n\nor \( {\left( \log {s}^{\prime }\left( x\right) \right) }^{\prime } = - {2b}\left( x\right) /a\left( x\right) \), from which (41.5) f... | Yes |
Theorem 41.3 There exists a continuous strictly increasing function \( s \) such that \( s\left( {X}_{t}\right) \) is on natural scale on \( s\left( \mathbb{R}\right) \) . | Proof Let \( {J}_{n} \) be closed intervals increasing up to \( \mathbb{R} \) . Pick two points in \( {J}_{1} \) ; label them \( a \) and \( b \) with \( a < b \) . Choose \( {A}_{n} \) and \( {B}_{n} \) so that if \( {s}_{n}\left( x\right) = {A}_{n}{p}_{{J}_{n}}\left( x\right) + {B}_{n} \), then \( {s}_{n}\left( a\rig... | Yes |
Lemma 41.4 If \( \left\lbrack {a, b}\right\rbrack \) is a finite interval, then \( \mathop{\sup }\limits_{x}{\mathbb{E}}^{x}{\tau }_{\left( a, b\right) }^{k} < \infty \) for each positive integer \( k \) . | Proof Pick \( y \in \left( {a, b}\right) \) . Since \( {X}_{t} \) is a regular diffusion, \( {\mathbb{P}}^{y}\left( {{T}_{a} < \infty }\right) = 1 \), and hence there exists \( {t}_{0} \) such that \( {\mathbb{P}}^{y}\left( {{T}_{a} > {t}_{0}}\right) < 1/2 \) . Similarly, taking \( {t}_{0} \) larger if necessary, \( {\... | Yes |
Lemma 41.5 If \( \left( {{X}_{t},{\mathbb{P}}^{x}}\right) \) has a speed measure \( m \) and \( \left\lbrack {a, b}\right\rbrack \) is a non-empty finite interval, then \( 0 < m\left( {a, b}\right) < \infty \) . | Proof If \( m\left( {a, b}\right) = 0 \), then for \( x \in \left( {a, b}\right) \), we have\n\n\[ \n{\mathbb{E}}^{x}{\tau }_{\left( a, b\right) } = \int {G}_{ab}\left( {x, y}\right) m\left( {dy}\right) = 0, \n\]\n\nwhich implies \( {\tau }_{\left( a, b\right) } = 0,{\mathbb{P}}^{x} \) -a.s., a contradiction to the con... | Yes |
Corollary 41.7 Suppose \( {X}_{t} \) is a diffusion on natural scale on \( \mathbb{R} \) . If \( f \) is bounded and measurable, for each \( a < b \) , \[ {\mathbb{E}}^{x}{\int }_{0}^{{\tau }_{\left( a, b\right) }}f\left( {X}_{s}\right) {ds} = \int {G}_{ab}\left( {x, y}\right) f\left( y\right) m\left( {dy}\right) . \] | Proof Suppose that \( f \) is continuous and bounded on \( \left\lbrack {a, b}\right\rbrack \) . Let \( {x}_{i},{S}_{j}, B\left( {x}_{i}\right) ,{N}_{i} \), and \( {m}_{n} \) be as in the proof of Theorem 41.6. Let \[ {\varepsilon }_{n} = \sup \left\{ {\left| {f\left( x\right) - f\left( y\right) }\right| : \left| {x - ... | Yes |
Theorem 41.10 Suppose \( {c}_{1} < \sigma \left( x\right) < {c}_{2} \) for all \( x \) and \( \sigma \) is continuous. The speed measure of \( {X}_{t} \) is given by\n\n\[ m\left( {dx}\right) = \frac{1}{a\left( x\right) }{dx}. \] | Proof Since \( d{X}_{t} = \sigma \left( {X}_{t}\right) d{W}_{t} \), then \( \langle X{\rangle }_{t} = {\int }_{0}^{t}a\left( {X}_{s}\right) {ds} \) . To obtain a Brownian motion \( {\bar{W}}_{t} \) by time-changing the martingale \( {X}_{t} \), we must time-change by the inverse of \( \langle X{\rangle }_{t} \) . On th... | Yes |
Lemma 42.3 If \( {X}_{t} \) is a Lévy process with bounded jumps and with \( {X}_{0} = 0 \), then \( {X}_{t} \) has moments of all orders, that is, \( \mathbb{E}{\left| {X}_{t}\right| }^{p} < \infty \) for all positive integers \( p \) . | Proof Suppose the jumps of \( {X}_{t} \) are bounded in absolute value by \( K \) . Since \( {X}_{t} \) is right continuous with left limits, there exists \( M > K \) such that \( \mathbb{P}\left( {\mathop{\sup }\limits_{{s \leq t}}\left| {X}_{s}\right| \geq {2M}}\right) \leq 1/2 \) .\n\nLet \( {T}_{1} = \inf \left\{ {... | Yes |
Theorem 42.6 Suppose \( m \) is a measure on \( \mathbb{R} \) with \( m\left( {\{ 0\} }\right) = 0 \) and\n\n\[ \int \left( {1 \land {x}^{2}}\right) m\left( {dx}\right) < \infty .\n\]\n\nSuppose \( b \in \mathbb{R} \) and \( \sigma \geq 0 \) . There exists a Lévy process \( {X}_{t} \) such that\n\n\[ \mathbb{E}{e}^{{iu... | Proof Let \( m\left( {dx}\right) \) be a measure supported on \( (0,1\rbrack \) with \( \int {x}^{2}m\left( {dx}\right) < \infty \) . Let \( {m}_{n}\left( {dx}\right) \) be the measure \( m \) restricted to \( \left( {{2}^{-n},{2}^{-n + 1}}\right\rbrack \) . Let \( {Y}_{t}^{n} \) be independent Lévy processes whose cha... | Yes |
Lemma 42.7 If \( {X}_{t} \) is a Lévy process and \( A \) is a Borel subset of \( \mathbb{R} \) that is a positive distance from 0 , then\n\n\[ \n{N}_{t}\left( A\right) = \mathop{\sum }\limits_{{s \leq t}}{1}_{A}\left( {\Delta {X}_{s}}\right)\n\]\n\nis a Poisson process. | Proof Since \( {X}_{t} \) has paths that are right continuous with left limits and \( A \) is a positive distance from 0, then there can only be finitely many jumps of \( X \) that lie in \( A \) in any finite time interval, and so \( {N}_{t}\left( A\right) \) is finite and has paths that are right continuous with left... | Yes |
Theorem 42.8 Let \( {X}_{t} \) be a Lévy process with \( {X}_{0} = 0 \) and let \( {A}_{1},\ldots ,{A}_{n} \) be disjoint bounded Borel subsets of \( \left( {0,\infty }\right) \), each a finite distance from 0 . Set\n\n\[ \n{N}_{t}\left( {A}_{k}\right) = \mathop{\sum }\limits_{{s \leq t}}{1}_{{A}_{k}}\left( {\Delta {X}... | Proof Define \( \lambda \left( A\right) = \mathbb{E}{N}_{1}\left( A\right) \) . The previous lemma shows that if \( \lambda \left( A\right) < \infty \), then \( {N}_{t}\left( A\right) \) is a Poisson process, and clearly its parameter is \( \lambda \left( A\right) \) . The result now follows from Theorem 18.3. | No |
Example 1.2 (The arrays \( \bar{x},{S}_{n} \), and \( R \) for bivariate data) Consider the data introduced in Example 1.1. Each receipt yields a pair of measurements, total dollar sales, and number of books sold. Find the arrays \( \overline{\mathbf{x}},{\mathbf{S}}_{n} \), and \( \mathbf{R} \) . | Since there are four receipts, we have a total of four measurements (observations) on each variable.\n\nThe sample means are\n\n\[ \n{\bar{x}}_{1} = \frac{1}{4}\mathop{\sum }\limits_{{j = 1}}^{4}{x}_{j1} = \frac{1}{4}\left( {{42} + {52} + {48} + {58}}\right) = {50} \n\] \n\n\[ \n{\bar{x}}_{2} = \frac{1}{4}\mathop{\sum ... | Yes |
Example 1.4 (A scatter plot for baseball data) In a July 17, 1978, article on money in sports, Sports Illustrated magazine provided data on \( {x}_{1} = \) player payroll for National League East baseball teams. We have added data on \( {x}_{2} = \) won-lost percentage for 1977 . The results are given in Table 1.1. The... | To construct the scatter plot in Figure 1.4, we have regarded the six paired observations in Table 1.1 as the coordinates of six points in two-dimensional space. The figure allows us to examine visually the grouping of teams with respect to the variables total payroll and won-lost percentage. | No |
Example 1.6 (Looking for lower-dimensional structure) A zoologist obtained measurements on \( n = {25} \) lizards known scientifically as Cophosaurus texanus. The weight, or mass, is given in grams while the snout-vent length (SVL) and hind limb span (HLS) are given in millimeters. The data are displayed in Table 1.3. ... | To help answer questions regarding reduced dimensionality, we construct the three-dimensional scatter plot in Figure 1.6. Clearly most of the variation is scatter about a one-dimensional straight line. Knowing the position on a line along the major axes of the cloud of points would be almost as good as knowing the thre... | Yes |
Example 1.7 (Looking for group structure in three dimensions) Referring to Example 1.6, it is interesting to see if male and female lizards occupy different parts of the three-dimensional space containing the size data. The gender, by row, for the lizard data in Table 1.3 are\n\n$ {fmffmfmfmfmf} $\n\n$ {mmmfmmmff} $ | Figure 1.8 repeats the scatter plot for the original variables but with males marked by solid circles and females by open circles. Clearly, males are typically larger than females.\n\n\n\nFigure 1.8 3D scatter plot of ... | No |
Example 1.9 (Rotated plots in three dimensions) Four different measurements of lumber stiffness are given in Table 4.3, page 186. In Example 4.14, specimen (board) 16 and possibly specimen (board) 9 are identified as unusual observations. Figures \( {1.12}\left( \mathrm{a}\right) \) ,(b), and (c) contain perspectives o... | Additional insights can sometimes be gleaned from visual inspection of the slowly spinning data. It is this dynamic aspect that statisticians are just beginning to understand and exploit. Plots like those in Figure 1.12 allow one to identify readily observations that do not conform to the rest of the data and that may ... | No |
Example 1.11 (Utility data as stars) Stars representing the first 5 of the 22 public utility firms in Table 12.4, page 688, are shown in Figure 1.16. There are eight variables; consequently, the stars are distorted octagons. | The observations on all variables were standardized. Among the first five utilities, the smallest standardized observation for any variable was -1.6. Treating this value as zero, the variables are plotted on identical scales along eight equiangular rays originating from the center of the circle. The variables are order... | Yes |
A set of paired measurements \( \left( {{x}_{1},{x}_{2}}\right) \) on two variables yields \( {\bar{x}}_{1} = {\bar{x}}_{2} = 0,{s}_{11} = 4 \), and \( {s}_{22} = 1 \) . Suppose the \( {x}_{1} \) measurements are unrelated to the \( {x}_{2} \) measurements; that is, measurements within a pair vary independently of one ... | \[ {d}^{2}\left( {O, P}\right) = \frac{{x}_{1}^{2}}{4} + \frac{{x}_{2}^{2}}{1} \] All points \( \left( {{x}_{1},{x}_{2}}\right) \) that are a constant distance 1 from the origin satisfy the equation \[ \frac{{x}_{1}^{2}}{4} + \frac{{x}_{2}^{2}}{1} = 1 \] The coordinates of some points a unit distance from the origin ar... | Yes |
Compute the mean vector \( \overline{\mathbf{x}} \) from the data matrix. | \[ \mathbf{X} = \left\lbrack \begin{array}{rr} 4 & 1 \\ - 1 & 3 \\ 3 & 5 \end{array}\right\rbrack \] The first point, \( {\mathbf{x}}_{1} \), has coordinates \( {\mathbf{x}}_{1}^{\prime } = \left\lbrack {4,1}\right\rbrack \) . Similarly, the remaining two points are \( {\mathbf{x}}_{2}^{\prime } = \left\lbrack {-1,3}\r... | Yes |
Example 3.2 (Data as \( p \) vectors in \( n \) dimensions) Plot the following data as \( p = 2 \) vectors in \( n = 3 \) space:\n\n\[ \mathbf{X} = \left\lbrack \begin{array}{rr} 4 & 1 \\ - 1 & 3 \\ 3 & 5 \end{array}\right\rbrack \] | Figure 3.2 A plot of the data matrix \( \mathbf{X} \) as \( p = 2 \) vectors in \( n = 3 \) space.\n\nHere \( {\mathbf{y}}_{1}^{\prime } = \left\lbrack {4, - 1,3}\right\rbrack \) and \( {\mathbf{y}}_{2}^{\prime } = \left\lbrack {1,3,5}\right\rbrack \) . These vectors are shown in Figure 3.2. ∎ | Yes |
Example 3.3 (Decomposing a vector into its mean and deviation components) Let us carry out the decomposition of \( {\mathbf{y}}_{i} \) into \( {\bar{x}}_{i}\mathbf{1} \) and \( {\mathbf{d}}_{i} = {\mathbf{y}}_{i} - {\bar{x}}_{i}\mathbf{1}, i = 1,2 \), for the data given in Example 3.2: | Here, \( {\bar{x}}_{1} = \left( {4 - 1 + 3}\right) /3 = 2 \) and \( {\bar{x}}_{2} = \left( {1 + 3 + 5}\right) /3 = 3 \), so\n\n\[ {\bar{x}}_{1}\mathbf{1} = 2\left\lbrack \begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right\rbrack = \left\lbrack \begin{array}{l} 2 \\ 2 \\ 2 \end{array}\right\rbrack \;{\bar{x}}_{2}\mathbf{1} =... | Yes |
Given the deviation vectors in Example 3.3, let us compute the sample variance-covariance matrix \( {\mathbf{S}}_{n} \) and sample correlation matrix \( \mathbf{R} \) using the geometrical concepts just introduced. | From Example 3.3,\n\n\[ \n{\mathbf{d}}_{1} = \left\lbrack \begin{array}{r} 2 \\ - 3 \\ 1 \end{array}\right\rbrack \text{ and }{\mathbf{d}}_{2} = \left\lbrack \begin{array}{r} - 2 \\ 0 \\ 2 \end{array}\right\rbrack \n\] \n\nThese vectors, translated to the origin, are shown in Figure 3.5. Now,\n\n\[ \n{\mathbf{d}}_{1}^{... | Yes |
Example 3.5 (Selecting a random sample) As a preliminary step in designing a permit system for utilizing a wilderness canoe area without overcrowding, a natural-resource manager took a survey of users. The total wilderness area was divided into subregions, and respondents were asked to give information on the regions v... | The method followed was to select persons randomly (perhaps using a random number table) from all those who entered the wilderness area during a particular week. All persons were equally likely to be in the sample, so the more popular entrances were represented by larger proportions of canoeists. | No |
Evaluate the generalized variance. | In this case, we compute\n\n\[ \left| \mathbf{S}\right| = \left( {252.04}\right) \left( {123.67}\right) - \left( {-{68.43}}\right) \left( {-{68.43}}\right) = {26},{487} \] | Yes |
Example 3.8 (Interpreting the generalized variance) Figure 3.7 gives three scatter plots with very different patterns of correlation.\n\nAll three data sets have \( {\overline{\mathbf{x}}}^{\prime } = \left\lbrack {2,1}\right\rbrack \), and the covariance matrices are\n\n\[ \mathbf{S} = \left\lbrack \begin{array}{ll} 5... | Each covariance matrix \( \mathbf{S} \) contains the information on the variability of the component variables and also the information required to calculate the correlation coefficient. In this sense, \( \mathbf{S} \) captures the orientation and size of the pattern of scatter.\n\nThe eigenvalues and eigenvectors extr... | Yes |
Example 3.9 (A case where the generalized variance is zero) Show that \( \\left| \\mathbf{S}\\right| = 0 \) for\n\n\[ \n\\underset{\\left( 3 \\times 3\\right) }{\\mathbf{X}} = \\left\\lbrack \\begin{array}{lll} 1 & 2 & 5 \\ 4 & 1 & 6 \\ 4 & 0 & 4 \\end{array}\\right\\rbrack \n\]\n\nand determine the degeneracy. | Here \( {\\overline{\\mathbf{x}}}^{\\prime } = \\left\\lbrack {3,1,5}\\right\\rbrack \), so\n\n\[ \n\\mathbf{X} - \\mathbf{1}{\\overline{\\mathbf{x}}}^{\\prime } = \\left\\lbrack \\begin{array}{lll} 1 - 3 & 2 - 1 & 5 - 5 \\ 4 - 3 & 1 - 1 & 6 - 5 \\ 4 - 3 & 0 - 1 & 4 - 5 \\end{array}\\right\\rbrack = \\left\\lbrack \\be... | Yes |
Example 3.11 (Illustrating the relation between \( \left| S\right| \) and \( \left| R\right| \) ) Let us illustrate the relationship in (3-21) for the generalized variances \( \left| \mathbf{S}\right| \) and \( \left| \mathbf{R}\right| \) when \( p = 3 \) . Suppose\n\n\[ \n\underset{\left( 3 \times 3\right) }{\mathbf{S... | Using Definition 2A.24, we obtain\n\n\[ \n\left| \mathbf{S}\right| = 4\left| \begin{array}{ll} 9 & 2 \\ 2 & 1 \end{array}\right| {\left( -1\right) }^{2} + 3\left| \begin{array}{ll} 3 & 2 \\ 1 & 1 \end{array}\right| {\left( -1\right) }^{3} + 1\left| \begin{array}{ll} 3 & 9 \\ 1 & 2 \end{array}\right| {\left( -1\right) }... | Yes |
Calculate the total sample variance for the variance-covariance matrices \( \mathbf{S} \) in Examples 3.7 and 3.9. | From Example 3.7.\n\n\[ \mathbf{S} = \left\lbrack \begin{array}{rr} {252.04} & - {68.43} \\ - {68.43} & {123.67} \end{array}\right\rbrack \]\n\nand\n\n\[ \text{Total sample variance} = {s}_{11} + {s}_{22} = {252.04} + {123.67} = {375.71} \]\n\nFrom Example 3.9,\n\n\[ \mathbf{S} = \left\lbrack \begin{array}{rrr} 3 & - \... | Yes |
Example 3.13 (Means and covariances for linear combinations) We shall consider two linear combinations and their derived values for the \( n = 3 \) observations given in Example 3.9 as\n\n\[ \n\mathbf{X} = \left\lbrack \begin{array}{lll} {x}_{11} & {x}_{12} & {x}_{13} \\ {x}_{21} & {x}_{22} & {x}_{23} \\ {x}_{31} & {x}... | Observations on these linear combinations are obtained by replacing \( {X}_{1},{X}_{2} \) , and \( {X}_{3} \) with their observed values. For example, the \( n = 3 \) observations on \( {\mathbf{b}}^{\prime }\mathbf{X} \) are\n\n\[ \n{\mathbf{b}}^{\prime }{\mathbf{x}}_{1} = 2{x}_{11} + 2{x}_{12} - {x}_{13} = 2\left( 1\... | Yes |
Given the vectors \( {\mathbf{x}}^{\prime } = \left\lbrack {1,3,2}\right\rbrack \) and \( {\mathbf{y}}^{\prime } = \left\lbrack {-2,1, - 1}\right\rbrack \), find \( 3\mathbf{x} \) and \( \mathbf{x} + \mathbf{y} \). Next, determine the length of \( \mathbf{x} \), the length of \( \mathbf{y} \), and the angle between \( ... | First,\n\n\[ 3\mathbf{x} = 3\left\lbrack \begin{array}{l} 1 \\ 3 \\ 2 \end{array}\right\rbrack = \left\lbrack \begin{array}{l} 3 \\ 9 \\ 6 \end{array}\right\rbrack \]\n\n\[ \mathbf{x} + \mathbf{y} = \left\lbrack \begin{array}{l} 1 \\ 3 \\ 2 \end{array}\right\rbrack + \left\lbrack \begin{array}{r} - 2 \\ 1 \\ - 1 \end{a... | Yes |
Consider the set of vectors\n\n\[ \n{\mathbf{x}}_{1} = \left\lbrack \begin{array}{l} 1 \\ 2 \\ 1 \end{array}\right\rbrack \;{\mathbf{x}}_{2} = \left\lbrack \begin{array}{r} 1 \\ 0 \\ - 1 \end{array}\right\rbrack \;{\mathbf{x}}_{3} = \left\lbrack \begin{array}{r} 1 \\ - 2 \\ 1 \end{array}\right\rbrack \n\]\n\nSetting\n\... | with the unique solution \( {c}_{1} = {c}_{2} = {c}_{3} = 0 \) . As we cannot find three constants \( {c}_{1},{c}_{2} \) , and \( {c}_{3} \), not all zero, such that \( {c}_{1}{\mathbf{x}}_{1} + {c}_{2}{\mathbf{x}}_{2} + {c}_{3}{\mathbf{x}}_{3} = \mathbf{0} \), the vectors \( {\mathbf{x}}_{1},{\mathbf{x}}_{2} \), and \... | Yes |
Example 2.3 (The transpose of a matrix) If\n\n\\[ \n\\underset{\\left( 2 \\times 3\\right) }{\\mathbf{A}} = \\left\\lbrack \\begin{array}{rrr} 3 & - 1 & 2 \\\\ 1 & 5 & 4 \\end{array}\\right\\rbrack \n\\]\n\nthen | \\[ \n\\underset{\\left( 3 \\times 2\\right) }{{\\mathbf{A}}^{\\prime }} = \\left\\lbrack \\begin{array}{rr} 3 & 1 \\\\ - 1 & 5 \\\\ 2 & 4 \\end{array}\\right\\rbrack \n\\] | Yes |
Example 2.4 (The sum of two matrices and multiplication of a matrix by a constant) If\n\n\\[ \n\\underset{\\left( 2 \\times 3\\right) }{\\mathbf{A}} = \\left\\lbrack \\begin{array}{rrr} 0 & 3 & 1 \\ 1 & - 1 & 1 \\end{array}\\right\\rbrack \\text{ and }\\underset{\\left( 2 \\times 3\\right) }{\\mathbf{B}} = \\left\\lbra... | \n\n\\[ \n\\underset{\\left( 2 \\times 3\\right) }{4\\mathbf{A}} = \\left\\lbrack \\begin{matrix} 0 & {12} & 4 \\ 4 & - 4 & 4 \\end{matrix}\\right\\rbrack \\text{ and } \n\\]\n\n\\[ \n\\underset{\\left( 2 \\times 3\\right) }{\\mathbf{A}} + \\underset{\\left( 2 \\times 3\\right) }{\\mathbf{B}} = \\left\\lbrack \\begin{a... | Yes |
Example 2.5 (Matrix multiplication) If\n\n\\[ \n\\mathbf{A} = \\left\\lbrack \\begin{array}{rrr} 3 & - 1 & 2 \\ 1 & 5 & 4 \\end{array}\\right\\rbrack ,\\;\\mathbf{B} = \\left\\lbrack \\begin{array}{r} - 2 \\ 7 \\ 9 \\end{array}\\right\\rbrack ,\\;\\text{ and }\\;\\mathbf{C} = \\left\\lbrack \\begin{array}{rr} 2 & 0 \\ ... | \\[ \n\\underset{\\left( {2 \\times 3}\\right) \\left( {3 \\times 1}\\right) }{\\mathbf{A}} = \\left\\lbrack \\begin{array}{rrr} 3 & - 1 & 2 \\ 1 & 5 & 4 \\end{array}\\right\\rbrack \\left\\lbrack \\begin{array}{r} - 2 \\ 7 \\ 9 \\end{array}\\right\\rbrack = \\left\\lbrack \\begin{array}{r} 3\\left( {-2}\\right) + \\le... | Yes |
Example 2.6 (Some typical products and their dimensions) Let\n\n\[ \mathbf{A} = \left\lbrack \begin{array}{rrr} 1 & - 2 & 3 \\ 2 & 4 & - 1 \end{array}\right\rbrack \;\mathbf{b} = \left\lbrack \begin{array}{r} 7 \\ - 3 \\ 6 \end{array}\right\rbrack \;\mathbf{c} = \left\lbrack \begin{array}{r} 5 \\ 8 \\ - 4 \end{array}\r... | \[ \mathbf{{Ab}} = \left\lbrack \begin{array}{rrr} 1 & - 2 & 3 \\ 2 & 4 & - 1 \end{array}\right\rbrack \left\lbrack \begin{array}{r} 7 \\ - 3 \\ 6 \end{array}\right\rbrack = \left\lbrack \begin{array}{r} {31} \\ - 4 \end{array}\right\rbrack \]\n\nThe product \( \mathbf{{Ab}} \) is a vector with dimension equal to the n... | Yes |
Example 2.8 (The existence of a matrix inverse) For\n\n\[ \mathbf{A} = \left\lbrack \begin{array}{ll} 3 & 2 \\ 4 & 1 \end{array}\right\rbrack \] | you may verify that\n\n\[ \left\lbrack \begin{array}{rr} - {.2} & {.4} \\ {.8} & - {.6} \end{array}\right\rbrack \left\lbrack \begin{array}{ll} 3 & 2 \\ 4 & 1 \end{array}\right\rbrack = \left\lbrack \begin{matrix} \left( {-{.2}}\right) 3 + \left( {.4}\right) 4 & \left( {-{.2}}\right) 2 + \left( {.4}\right) 1 \\ \left( ... | Yes |
Example 2.12 (Computing expected values for discrete random variables) Suppose \( p = 2 \) and \( n = 1 \), and consider the random vector \( {\mathbf{X}}^{\prime } = \left\lbrack {{X}_{1},{X}_{2}}\right\rbrack \) . Let the discrete random variable \( {X}_{1} \) have the following probability function:\n\n\n\nThen \( E\left( {X}_{2}\right) = \mathop{\sum }\limits_{{\text{all }{x}_{2}}}{x}_{2}{p}_{2}\left( {x}_{2}\right) = \left(... | Yes |
Example 2.14 (Computing the correlation matrix from the covariance matrix) Suppose\n\n\\[ \n\\mathbf{\\sum } = \\left\\lbrack \\begin{array}{rrr} 4 & 1 & 2 \\ 1 & 9 & - 3 \\ 2 & - 3 & {25} \\end{array}\\right\\rbrack = \\left\\lbrack \\begin{array}{lll} {\\sigma }_{11} & {\\sigma }_{12} & {\\sigma }_{13} \\ {\\sigma }_... | Here\n\n\\[ \n{\\mathbf{V}}^{1/2} = \\left\\lbrack \\begin{matrix} \\sqrt{{\\sigma }_{11}} & 0 & 0 \\ 0 & \\sqrt{{\\sigma }_{22}} & 0 \\ 0 & 0 & \\sqrt{{\\sigma }_{33}} \\end{matrix}\\right\\rbrack = \\left\\lbrack \\begin{array}{lll} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \\end{array}\\right\\rbrack \n\\]\n\nand\n\n\\[ \... | Yes |
Find the mean vector and covariance matrix for the linear combinations\n\n\[ \n{Z}_{1} = {X}_{1} - {X}_{2} \]\n\n\[ \n{Z}_{2} = {X}_{1} + {X}_{2} \]\n\nor\n\n\[ \n\mathbf{Z} = \left\lbrack \begin{array}{l} {Z}_{1} \\ {Z}_{2} \end{array}\right\rbrack = \left\lbrack \begin{array}{rr} 1 & - 1 \\ 1 & 1 \end{array}\right\rb... | Here\n\n\[ \n{\mathbf{\mu }}_{\mathbf{Z}} = E\left( \mathbf{Z}\right) = \mathbf{C}{\mathbf{\mu }}_{\mathbf{X}} = \left\lbrack \begin{array}{rr} 1 & - 1 \\ 1 & 1 \end{array}\right\rbrack \left\lbrack \begin{array}{l} {\mu }_{1} \\ {\mu }_{2} \end{array}\right\rbrack = \left\lbrack \begin{array}{l} {\mu }_{1} - {\mu }_{2... | Yes |
Example 4.1 (Bivariate normal density) Let us evaluate the \( p = 2 \) -variate normal density in terms of the individual parameters \( {\mu }_{1} = E\left( {X}_{1}\right) ,\;{\mu }_{2} = E\left( {X}_{2}\right) \) , \( {\sigma }_{11} = \operatorname{Var}\left( {X}_{1}\right) ,{\sigma }_{22} = \operatorname{Var}\left( {... | Using Result 2A.8, we find that the inverse of the covariance matrix\n\n\[ \mathbf{\sum } = \left\lbrack \begin{array}{ll} {\sigma }_{11} & {\sigma }_{12} \\ {\sigma }_{12} & {\sigma }_{22} \end{array}\right\rbrack \]\n\nis\n\n\[ {\mathbf{\sum }}^{-1} = \frac{1}{{\sigma }_{11}{\sigma }_{22} - {\sigma }_{12}^{2}}\left\l... | Yes |
Example 4.2 (Contours of the bivariate normal density) We shall obtain the axes of constant probability density contours for a bivariate normal distribution when \( {\sigma }_{11} = {\sigma }_{22} \) . From (4-7), these axes are given by the eigenvalues and eigenvectors of \( \mathbf{\sum } \) . Here \( \left| {\mathbf... | \[ 0 = \left| \begin{matrix} {\sigma }_{11} - \lambda & {\sigma }_{12} \\ {\sigma }_{12} & {\sigma }_{11} - \lambda \end{matrix}\right| = {\left( {\sigma }_{11} - \lambda \right) }^{2} - {\sigma }_{12}^{2} \] \[ = \left( {\lambda - {\sigma }_{11} - {\sigma }_{12}}\right) \left( {\lambda - {\sigma }_{11} + {\sigma }_{12... | Yes |
Example 4.3 (The distribution of a linear combination of the components of a normal random vector) Consider the linear combination \( {\mathbf{a}}^{\prime }\mathbf{X} \) of a multivariate normal random vector determined by the choice \( {\mathbf{a}}^{\prime } = \left\lbrack {1,0,\ldots ,0}\right\rbrack \) . Since | \[ {\mathbf{a}}^{\prime }\mathbf{X} = \left\lbrack {1,0,\ldots ,0}\right\rbrack \left\lbrack \begin{matrix} {X}_{1} \\ {X}_{2} \\ \vdots \\ {X}_{p} \end{matrix}\right\rbrack = {X}_{1} \] and \[ {\mathbf{a}}^{\prime }\mathbf{\mu } = \left\lbrack {1,0,\ldots ,0}\right\rbrack \left\lbrack \begin{matrix} {\mu }_{1} \\ {\mu... | Yes |
Example 4.4 (The distribution of two linear combinations of the components of a normal random vector) For \( \mathbf{X} \) distributed as \( {N}_{3}\left( {\mathbf{\mu },\mathbf{\sum }}\right) \), find the distribution of\n\n\[ \left\lbrack \begin{array}{l} {X}_{1} - {X}_{2} \\ {X}_{2} - {X}_{3} \end{array}\right\rbrac... | By Result 4.3, the distribution of \( \mathbf{{AX}} \) is multivariate normal with mean\n\n\[ \mathbf{A}\mathbf{\mu } = \left\lbrack \begin{array}{rrr} 1 & - 1 & 0 \\ 0 & 1 & - 1 \end{array}\right\rbrack \left\lbrack \begin{array}{l} {\mu }_{1} \\ {\mu }_{2} \\ {\mu }_{3} \end{array}\right\rbrack = \left\lbrack \begin{... | Yes |
Example 4.6 (The equivalence of zero covariance and independence for normal variables) Let \( \mathbf{X} \) be \( {N}_{3}\left( {\mathbf{\mu },\mathbf{\sum }}\right) \) with \( \left( {3 \times 1}\right) \)\n\n\[ \mathbf{\sum } = \left\lbrack \begin{array}{lll} 4 & 1 & 0 \\ 1 & 3 & 0 \\ 0 & 0 & 2 \end{array}\right\rbra... | Since \( {X}_{1} \) and \( {X}_{2} \) have covariance \( {\sigma }_{12} = 1 \), they are not independent. However, partitioning \( \mathbf{X} \) and \( \mathbf{\sum } \) as\n\n\[ \mathbf{X} = \left\lbrack \begin{matrix} {X}_{1} \\ {X}_{2} \\ \cdots \cdots \\ {X}_{3} \end{matrix}\right\rbrack ,\;\mathbf{\sum } = \left\l... | Yes |
Find the mean vector and covariance matrix for each linear combination of vectors and also the covariance between them. | By Result 4.8 with \( {c}_{1} = {c}_{2} = {c}_{3} = {c}_{4} = 1/2 \), the first linear combination has mean vector\n\n\[ \left( {{c}_{1} + {c}_{2} + {c}_{3} + {c}_{4}}\right) \mathbf{\mu } = 2\mathbf{\mu } = \left\lbrack \begin{array}{r} 6 \\ - 2 \\ 2 \end{array}\right\rbrack \]\n\nand covariance matrix\n\n\[ \left( {{... | Yes |
Example 4.9 (Constructing a Q-Q plot) A sample of \( n = {10} \) observations gives the values in the following table: | Let us now construct the \( Q - Q \) plot and comment on its appearance. The \( Q - Q \) plot for the foregoing data, which is a plot of the ordered data \( {x}_{\left( j\right) } \) against the normal quantiles \( {q}_{\left( j\right) } \), is shown in Figure 4.5. The pairs of points \( \left( {{q}_{\left( j\right) },... | No |
Example 4.10 (A Q-Q plot for radiation data) The quality-control department of a manufacturer of microwave ovens is required by the federal government to monitor the amount of radiation emitted when the doors of the ovens are closed. Observations of the radiation emitted through closed doors of \( n = {42} \) randomly ... | A computer was used to assemble the pairs \( \left( {{q}_{\left( j\right) },{x}_{\left( j\right) }}\right) \) and construct the \( Q - Q \) plot, pictured in Figure 4.6 on page 181. It appears from the plot that the data as a whole are not normally distributed. The points indicated by the circled locations in the figur... | Yes |
Let us calculate the correlation coefficient \( {r}_{Q} \) from the \( Q - Q \) plot of Example 4.9 (see Figure 4.5) and test for normality. | Using the information from Example 4.9, we have \( \bar{x} = {.770} \) and\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{{10}}\left( {{x}_{\left( j\right) } - \bar{x}}\right) {q}_{\left( j\right) } = {8.584},\mathop{\sum }\limits_{{j = 1}}^{{10}}{\left( {x}_{\left( j\right) } - \bar{x}\right) }^{2} = {8.472}\text{, and}\matho... | Yes |
Example 4.13 (Constructing a chi-square plot) Let us construct a chi-square plot of the generalized distances given in Example 4.12. The ordered distances and the corresponding chi-square percentiles for \( p = 2 \) and \( n = {10} \) are listed in the following table: | A graph of the pairs \( \left( {{q}_{c,2}\left( {\left( {j - \frac{1}{2}}\right) /{10}}\right) ,{d}_{\left( j\right) }^{2}}\right) \) is shown in Figure 4.7. The points in Figure 4.7 are reasonably straight. Given the small sample size it is difficult to reject bivariate normality on the evidence in this graph. If furt... | No |
Example 4.15 (Detecting outliers in the data on lumber) Table 4.4 contains the data in Table 4.3, along with the standardized observations. These data consist of four different measures of stiffness \( {x}_{1},{x}_{2},{x}_{3} \), and \( {x}_{4} \), on each of \( n = {30} \) boards. Recall that the first measurement inv... | <table><thead><tr><th colspan=\ | No |
Example 4.16 (Determining a power transformation for univariate data) We gave readings of the microwave radiation emitted through the closed doors of \( n = {42} \) ovens in Example 4.10. The \( Q - Q \) plot of these data in Figure 4.6 indicates that the observations deviate from what would be expected if they were no... | The pairs \( \left( {\lambda ,\ell \left( \lambda \right) }\right) \) are listed in the following table for several values of \( \lambda \) :\n\n<table><thead><tr><th>\( \lambda \)</th><th>\( \ell \left( \lambda \right) \)</th><th>\( \lambda \)</th><th>\( \ell \left( \lambda \right) \)</th></tr></thead><tr><td>\( - {1.... | Yes |
Example 4.17 (Determining power transformations for bivariate data) Radiation measurements were also recorded through the open doors of the \( n = {42} \) microwave ovens introduced in Example 4.10. The amount of radiation emitted through the open doors of these ovens is listed in Table 4.5. | In accordance with the procedure outlined in Example 4.16, a power transformation for these data was selected by maximizing \( \ell \left( \lambda \right) \) in (4-35). The approximate maximizing value was \( \widehat{\lambda } = {.30} \) . Figure 4.14 on page 199 shows \( Q - Q \) plots of the untransformed and transf... | Yes |
Evaluate the observed \( {T}^{2} \) for \( {\mathbf{\mu }}_{0}^{\prime } = \left\lbrack {9,5}\right\rbrack \) . What is the sampling distribution of \( {T}^{2} \) in this case? | We find\n\n\[ \overline{\mathbf{x}} = \left\lbrack \begin{array}{l} {\bar{x}}_{1} \\ {\bar{x}}_{2} \end{array}\right\rbrack = \left\lbrack \begin{matrix} \frac{6 + {10} + 8}{3} \\ \frac{9 + 6 + 3}{3} \end{matrix}\right\rbrack = \left\lbrack \begin{array}{l} 8 \\ 6 \end{array}\right\rbrack \]\n\nand\n\n\[ {s}_{11} = \fr... | Yes |
Test the hypothesis \( {H}_{0} : {\mathbf{\mu }}^{\prime } = \left\lbrack {4,{50},{10}}\right\rbrack \) against \( {H}_{1} : {\mathbf{\mu }}^{\prime } \neq \left\lbrack {4,{50},{10}}\right\rbrack \) at level of significance \( \alpha = {.10} \) . | Computer calculations provide\n\n\[ \overline{\mathbf{x}} = \left\lbrack \begin{array}{r} {4.640} \\ {45.400} \\ {9.965} \end{array}\right\rbrack ,\;\mathbf{S} = \left\lbrack \begin{array}{rrr} {2.879} & {10.010} & - {1.810} \\ {10.010} & {199.788} & - {5.640} \\ - {1.810} & - {5.640} & {3.628} \end{array}\right\rbrack... | Yes |
Example 5.3 (Constructing a confidence ellipse for \( \mu \) ) Data for radiation from microwave ovens were introduced in Examples 4.10 and 4.17. Let\n\n\[ \n{x}_{1} = \sqrt[4]{\\text{ measured radiation with door closed }} \n\]\n\nand\n\n\[ \n{x}_{2} = \sqrt[4]{\\text{ measured radiation with door open }} \n\]\n\nFor ... | or, since \( {F}_{2,{40}}\\left( {.05}\\right) = {3.23} \) ,\n\n\[ \n{42}\\left( {203.018}\\right) {\\left( {.564} - {\\mu }_{1}\\right) }^{2} + {42}\\left( {200.228}\\right) {\\left( {.603} - {\\mu }_{2}\\right) }^{2} \n\]\n\n\[ \n- {84}\\left( {163.391}\\right) \\left( {{.564} - {\\mu }_{1}}\\right) \\left( {{.603} -... | Yes |
In Example 5.3, we obtained the 95% confidence ellipse for the means of the fourth roots of the door-closed and door-open microwave radiation measurements. The 95% simultaneous \( {T}^{2} \) intervals for the two component means are, from (5-24), | \[ \left( {{\bar{x}}_{1} - \sqrt{\frac{p\left( {n - 1}\right) }{\left( n - p\right) }{F}_{p, n - p}\left( {.05}\right) }\sqrt{\frac{{s}_{11}}{n}},\;{\bar{x}}_{1} + \sqrt{\frac{p\left( {n - 1}\right) }{\left( n - p\right) }{F}_{p, n - p}\left( {.05}\right) }\sqrt{\frac{{s}_{11}}{n}}}\right) \] \[ = \left( {{.564} - \sqr... | Yes |
Let us compute the \( {95}\% \) simultaneous confidence intervals for \( {\mu }_{1},{\mu }_{2} \), and \( {\mu }_{3} \) . | We have\n\n\[ \frac{p\left( {n - 1}\right) }{n - p}{F}_{p, n - p}\left( \alpha \right) = \frac{3\left( {{87} - 1}\right) }{\left( {87} - 3}\right) }{F}_{3,{84}}\left( {.05}\right) = \frac{3\left( {86}\right) }{84}\left( {2.7}\right) = {8.29} \]\n\nand we obtain the simultaneous confidence statements [see (5-24)]\n\n\[ ... | Yes |
Example 5.6 (Constructing Bonferroni simultaneous confidence intervals and comparing them with \( {T}^{2} \) -intervals) Let us return to the microwave oven radiation data in Examples 5.3 and 5.4. We shall obtain the simultaneous 95% Bonferroni confidence intervals for the means, \( {\mu }_{1} \) and \( {\mu }_{2} \), ... | \n\[
{\bar{x}}_{1} \pm {t}_{41}\left( {.0125}\right) \sqrt{\frac{{s}_{11}}{n}} = {.564} \pm {2.327}\sqrt{\frac{.0144}{42}}\text{ or }{.521} \leq {\mu }_{1} \leq {.607}
\]
\n\[
{\bar{x}}_{2} \pm {t}_{41}\left( {.0125}\right) \sqrt{\frac{{s}_{22}}{n}} = {.603} \pm {2.327}\sqrt{\frac{.0146}{42}}\text{ or }{.560} \leq {\mu... | Yes |
Let us construct \( {90}\% \) simultaneous confidence intervals for the individual mean components \( {\mu }_{i}, i = 1,2,\ldots ,7 \) . | From Result 5.5, simultaneous 90% confidence limits are given by \( {\bar{x}}_{i} \pm \sqrt{{\chi }_{7}^{2}\left( {.10}\right) }\sqrt{\frac{{s}_{ii}}{n}}, i = 1,2,\ldots ,7 \), where \( {\chi }_{7}^{2}\left( {.10}\right) = {12.02} \) . Thus, with approximately \( {90}\% \) confidence,\n\n\[{28.1} \pm \sqrt{12.02}\frac{... | Yes |
Example 5.8 (Creating a univariate control chart) The Madison, Wisconsin, police department regularly monitors many of its activities as part of an ongoing quality improvement program. Table 5.8 gives the data on five different kinds of overtime hours. Each observation represents a total for 12 pay periods, or about ha... | \[ \mathrm{{UCL}} = {\overline{\bar{x}}}_{1} + 3\left( \sqrt{{s}_{11}}\right) = {3558} + 3\left( {607}\right) = {5379} \]\n\n\[ \mathrm{{LCL}} = {\overline{\bar{x}}}_{1} - 3\left( \sqrt{{s}_{11}}\right) = {3558} - 3\left( {607}\right) = {1737} \] | Yes |
Example 5.9 (An ellipse format chart for overtime hours) Let us refer to Example 5.8 and create a quality ellipse for the pair of overtime characteristics (legal appearances, extraordinary event) hours. A computer calculation gives\n\n\[ \overline{\mathbf{x}} = \left\lbrack \begin{array}{l} {3558} \\ {1478} \end{array}... | Here \( p = 2 \), so \( {\chi }_{2}^{2}\left( {.01}\right) = {9.21} \), and the ellipse becomes\n\n\[ \frac{{s}_{11}{s}_{22}}{{s}_{11}{s}_{22} - {s}_{12}^{2}}\left( {\frac{{\left( {x}_{1} - {\bar{x}}_{1}\right) }^{2}}{{s}_{11}} - 2{s}_{12}\frac{\left( {{x}_{1} - {\bar{x}}_{1}}\right) \left( {{x}_{2} - {\bar{x}}_{2}}\ri... | Yes |
Example 5.10 (A \( {T}^{2} \) -chart for overtime hours) Using the police department data in Example 5.8, we construct a \( {T}^{2} \) -plot based on the two variables \( {X}_{1} = \) legal appearances hours and \( {X}_{2} = \) extraordinary event hours. \( {T}^{2} \) -charts with more than two variables are considered... | The \( {T}^{2} \) -chart in Figure 5.8 reveals that the pair (legal appearances, extraordinary event) hours for period 11 is out of control. Further investigation, as in Example 5.9 , confirms that this is due to the large value of extraordinary event overtime during that period. | No |
Example 5.12 (A control ellipse for future overtime hours) In Example 5.9, we checked the stability of legal appearances and extraordinary event overtime hours. Let's use these data to determine a control region for future pairs of values. | From Example 5.9 and Figure 5.6, we find that the pair of values for period 11 were out of control. We removed this point and determined the new \( {99}\% \) ellipse. All of the points are then in control, so they can serve to determine the \( {95}\% \) prediction region just defined for \( p = 2 \) . This control elli... | Yes |
Do the two laboratories' chemical analyses agree? If differences exist, what is their nature? | The \( {T}^{2} \) -statistic for testing \( {H}_{0} : {\mathbf{\delta }}^{\prime } = \left\lbrack {{\delta }_{1},{\delta }_{2}}\right\rbrack = \left\lbrack {0,0}\right\rbrack \) is constructed from the differences of paired observations:\n\n<table><thead><tr><th>\( {d}_{j1} = {x}_{1j1} - {x}_{2j1} \)</th><th>-19</th><t... | Yes |
Example 6.2 (Testing for equal treatments in a repeated measures design) Improved anesthetics are often developed by first studying their effects on animals. In one study, 19 dogs were initially given the drug pentobarbitol. Each dog was then administered carbon dioxide \( {\mathrm{{CO}}}_{2} \) at each of two pressure... | There are three treatment contrasts that might be of interest in the experiment. Let \( {\mu }_{1},{\mu }_{2},{\mu }_{3} \), and \( {\mu }_{4} \) correspond to the mean responses for treatments \( 1,2,3 \), and 4 , respectively. Then\n\n\[ \left( {{\mu }_{3} + {\mu }_{4}}\right) - \left( {{\mu }_{1} + {\mu }_{2}}\right... | Yes |
Let us find \( {95}\% \) simultaneous confidence intervals for the differences in the mean components. | Although there appears to be somewhat of a discrepancy in the sample variances, for illustrative purposes we proceed to a calculation of the pooled sample covariance matrix. Here\n\n\[{\mathbf{S}}_{\text{pooled }} = \frac{{n}_{1} - 1}{{n}_{1} + {n}_{2} - 2}{\mathbf{S}}_{1} + \frac{{n}_{2} - 1}{{n}_{1} + {n}_{2} - 2}{\m... | Yes |
The 95% simultaneous confidence intervals for the linear combinations ${\\mathbf{a}}^{\\prime }\\left( {{\\mathbf{\\mu }}_{1} - {\\mathbf{\\mu }}_{2}}\\right) = \\left\\lbrack {1,0}\\right\\rbrack \\left\\lbrack \\begin{array}{l} {\\mu }_{11} - {\\mu }_{21} \\\\ {\\mu }_{12} - {\\mu }_{22} \\end{array}\\right\\rbrack =... | ${\\mu }_{11} - {\\mu }_{21} : \\;{74.4} \\pm \\sqrt{5.99}\\sqrt{464.17}\\;\\text{ or }\\;\\left( {{21.7},{127.1}}\\right)$ and ${\\mu }_{12} - {\\mu }_{22} : {201.6} \\pm \\sqrt{5.99}\\sqrt{2642.15}\\;\\mathrm{{or}}\\;\\left( {{75.8},{327.4}}\\right)$ | Yes |
Example 6.8 (A univariate ANOVA table and F-test for treatment effects) Using the information in Example 6.7, we have the following ANOVA table: | \[ F = \frac{{\mathrm{{SS}}}_{\mathrm{{tr}}}/\left( {g - 1}\right) }{{\mathrm{{SS}}}_{\mathrm{{res}}}/\left( {\sum {n}_{\ell } - g}\right) } = \frac{{78}/2}{{10}/5} = {19.5} \] Since \( F = {19.5} > {F}_{2,5}\left( {.01}\right) = {13.27} \), we reject \( {H}_{0} : {\tau }_{1} = {\tau }_{2} = {\tau }_{3} = 0 \) (no trea... | Yes |
Example 6.9 (A MANOVA table and Wilks' lambda for testing the equality of three mean vectors) Suppose an additional variable is observed along with the variable introduced in Example 6.7. The sample sizes are \( {n}_{1} = 3,{n}_{2} = 2 \), and \( {n}_{3} = 3 \) . Arranging the observation pairs \( {\mathbf{x}}_{\ell j}... | We have already expressed the observations on the first variable as the sum of an overall mean, treatment effect, and residual in our discussion of univariate ANOVA. We found that\n\n\[ \left( \begin{array}{lll} 9 & 6 & 9 \\ 0 & 2 & \\ 3 & 1 & 2 \end{array}\right) = \left( \begin{array}{lll} 4 & 4 & 4 \\ 4 & 4 & \\ 4 &... | Yes |
Example 6.11 (Simultaneous intervals for treatment differences-nursing homes) We saw in Example 6.10 that average costs for nursing homes differ, depending on the type of ownership. We can use Result 6.5 to estimate the magnitudes of the differences. A comparison of the variable \( {X}_{3} \), costs of plant operation ... | Using (6-39) and the information in Example 6.10, we have\n\n\[ \n{\widehat{\mathbf{\tau }}}_{1} = \left( {{\overline{\mathbf{x}}}_{1} - \overline{\mathbf{x}}}\right) = \left\lbrack \begin{matrix} - {.070} \\ - {.039} \\ - {.020} \\ - {.020} \end{matrix}\right\rbrack ,\;{\widehat{\mathbf{\tau }}}_{3} = \left( {{\overli... | Yes |
We test the hypothesis \( {H}_{0} : {\mathbf{\sum }}_{1} = {\mathbf{\sum }}_{2} = {\mathbf{\sum }}_{3} = \mathbf{\sum } \) . | Using the information in Example 6.10, we have \( {n}_{1} = {271},{n}_{2} = {138} \) , \( {n}_{3} = {107} \) and \( \left| {\mathbf{S}}_{1}\right| = {2.783} \times {10}^{-8},\left| {\mathbf{S}}_{2}\right| = {89.539} \times {10}^{-8},\left| {\mathbf{S}}_{3}\right| = {14.579} \times {10}^{-8} \), and \( \left| {\mathbf{S... | Yes |
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