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Theorem 8.6.2. Let \( f\left( \mathbf{H}\right) \) be a function whose discontinuities form a set of probability zero when \( \mathbf{H} \) is distributed as \( \mathop{\sum }\limits_{{t = 1}}^{{q}_{1}}{\mathbf{Y}}_{t}{\mathbf{Y}}_{t}^{\prime } \) with the \( {\mathbf{Y}}_{t} \) independent, each with distribution \( N... | Proof. This is a straightforward application of a general theorem [for example, Theorem 2 of Chernoff (1956)] to the effect that if the cdf of \( {X}_{n} \) converges to that of \( X \) (at every continuity point of the latter) and if \( g\left( x\right) \) is a function whose discontinuities form a set of probability ... | Yes |
Corollary 8.6.1. The limiting distribution of \( N \) tr \( H{G}^{-1} \) or \( n \) tr \( H{G}^{-1} \) is the \( {\chi }^{2} \) -distribution with \( p{q}_{1} \) degrees of freedom. | This follows from Theorem 8.6.2, because\n\n(10)\n\n\[ \operatorname{tr}\mathbf{H} = \mathop{\sum }\limits_{{l = 1}}^{p}{h}_{ll} = \mathop{\sum }\limits_{{l = 1}}^{p}\mathop{\sum }\limits_{{v = 1}}^{{q}_{1}}{Y}_{vv}^{2} \] | No |
Lemma 8.7.1. For positive definite matrices \( A \) and \( G \) ,\n\n\[ \left| {\operatorname{tr}{\Phi }^{\prime }\mathbf{Y}}\right| \leq \sqrt{\operatorname{tr}{A}^{-1}{\Phi }^{\prime }G}\Phi \sqrt{\operatorname{tr}A{Y}^{\prime }{G}^{-1}Y}. \] | Proof. Let \( b = \operatorname{tr}{\Phi }^{\prime }Y/\operatorname{tr}{A}^{-1}{\Phi }^{\prime }{G\Phi } \) . Then\n\n\[ 0 \leq \operatorname{tr}A{\left( \mathbf{Y} - b\mathbf{G}\mathbf{\Phi }{\mathbf{A}}^{-1}\right) }^{\prime }{\mathbf{G}}^{-1}\left( {\mathbf{Y} - b\mathbf{G}\mathbf{\Phi }{\mathbf{A}}^{-1}}\right) \]\... | Yes |
Theorem 5.6.5 (Stein's theorem) will be used because we can write the distribution of \( \\left( {\\mathbf{X},\\mathbf{Y},\\mathbf{Z}}\\right) \) in exponential form. Let \( \\mathbf{U} = \\mathbf{X}{\\mathbf{X}}^{\\prime } + \\mathbf{Y}{\\mathbf{Y}}^{\\prime } + \\mathbf{Z}{\\mathbf{Z}}^{\\prime } = \\left( {u}_{ij}\\... | \[ \text{(9)}f\\left( {\\mathbf{X},\\mathbf{Y},\\mathbf{Z}}\\right) = K\\left( {\\mathbf{\\Xi },\\mathbf{H},\\mathbf{\\sum }}\\right) \\exp \\left\\{ {\\operatorname{tr}{\\mathbf{\\Xi }}^{\\prime }{\\mathbf{\\sum }}^{-1}\\mathbf{X} + \\operatorname{tr}{\\mathbf{H}}^{\\prime }{\\mathbf{\\sum }}^{-1}\\mathbf{Y} - \\frac{... | Yes |
\[ \lambda \left\lbrack {M\left( {p{V}_{1} + g{V}_{2}}\right) }\right\rbrack \succ {}_{n}{p\lambda }\left\lbrack {M\left( {V}_{1}\right) }\right\rbrack + {q\lambda }\left\lbrack {M\left( {V}_{2}\right) }\right\rbrack . \] | The proof of Theorem 8.10.3 (Figure 8.4) follows from the pair of majorizations\n\n\[ \lambda \left\lbrack {M\left( {p{V}_{1} + q{V}_{2}}\right) }\right\rbrack { \succ }_{w}\lambda \left\lbrack {{pM}\left( {V}_{1}\right) + {qM}\left( {V}_{2}\right) }\right\rbrack \]\n\n\[ { \succ }_{w}p\mathbf{\lambda }\left\lbrack {M\... | Yes |
Lemma 8.10.1. For \( A \) and \( B \) symmetric,\n\n\[ \lambda \left( {A + B}\right) \succ {}_{\mu }\lambda \left( A\right) + \lambda \left( B\right) . \] | Proof. By Corollary A.4.2 of the Appendix,\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{k}{\lambda }_{i}\left( {A + B}\right) = \mathop{\max }\limits_{{{R}^{\prime }R = {l}_{k}}}\operatorname{tr}{R}^{\prime }\left( {A + B}\right) R \]\n\n\[ \leq \mathop{\max }\limits_{{{R}^{\prime }R = {l}_{k}}}\operatorname{tr}{R}^{\prime }... | Yes |
Lemma 8.10.2\n\n\[ p{U}_{1} + q{U}_{2} - \left( {p{Y}_{1} + q{Y}_{2}}\right) {\left( p{Y}_{1} + q{Y}_{2}\right) }^{\prime } \]\n\n\[ \geq p\left( {{U}_{1} - {Y}_{1}{Y}_{1}^{\prime }}\right) + q\left( {{U}_{2} - {Y}_{2}{Y}_{2}^{\prime }}\right) . \] | Proof. The left-hand side minus the right-hand side is\n\n(16)\n\n\[ p{\mathbf{Y}}_{1}{\mathbf{Y}}_{1}^{\prime } + q{\mathbf{Y}}_{2}{\mathbf{Y}}_{2}^{\prime } - {p}^{2}{\mathbf{Y}}_{1}{\mathbf{Y}}_{1}^{\prime } - {q}^{2}{\mathbf{Y}}_{2}{\mathbf{Y}}_{2}^{\prime } - {pq}\left( {{\mathbf{Y}}_{1}{\mathbf{Y}}_{2}^{\prime } ... | Yes |
Lemma 8.10.3. If \( A \geq B > \mathbf{0} \), then \( {A}^{-1} \leq {B}^{-1} \) . | Proof. See Problem 8.31. | No |
Lemma 8.10.4. If \( A > 0 \), then \( f\left( {x, A}\right) = {x}^{\prime }{A}^{-1}x \) is convex in \( \left( {x, A}\right) \) . | Proof. See Problem 5.17. | No |
Lemma 8.10.5. If \( {A}_{1} > \mathbf{0},{A}_{2} > \mathbf{0} \), then\n\n(17)\n\n\[{\left( p{B}_{1} + q{B}_{2}\right) }^{\prime }{\left( p{A}_{1} + q{A}_{2}\right) }^{-1}\left( {p{B}_{1} + q{B}_{2}}\right) \leq p{B}_{1}^{\prime }{A}_{1}^{-1}{B}_{1} + q{B}_{2}^{\prime }{A}_{2}^{-1}{B}_{2}.\] | Proof. From Lemma 8.10.4 we have for all \( \mathbf{y} \n\n(18)\n\n\[p{y}^{\prime }{B}_{1}^{\prime }{A}_{1}^{-1}{B}_{1}y + q{y}^{\prime }{B}_{2}^{\prime }{A}_{2}^{-1}{B}_{2}y\]\n\n\[- {y}^{\prime }{\left( p{B}_{1} + q{B}_{2}\right) }^{\prime }{\left( p{A}_{1} + q{A}_{2}\right) }^{-1}\left( {p{B}_{1} + q{B}_{2}}\right) ... | Yes |
Lemma 8.10.6.\n\n\[ M\left( {p{V}_{1} + q{V}_{2}}\right) \leq {pM}\left( {V}_{1}\right) + {qM}\left( {V}_{2}\right) ,\] | Proof. Lemmas 8.10.2 and 8.10.3 show that\n\n\[ {\left\lbrack p{\mathbf{U}}_{1} + q{\mathbf{U}}_{2} - \left( p{\mathbf{Y}}_{1} + q{\mathbf{Y}}_{2}\right) {\left( p{\mathbf{Y}}_{1} + q{\mathbf{Y}}_{2}\right) }^{\prime }\right\rbrack }^{-1} \]\n\n\[ \leq {\left\lbrack p\left( {\mathbf{U}}_{1} - {\mathbf{Y}}_{1}{\mathbf{Y... | Yes |
Lemma 8.10.7. If \( A \leq B \), then \( \lambda \left( A\right) { \prec }_{w}\lambda \left( B\right) \) . | Proof. From Corollary A.4.2 of the Appendix,\n\n(23)\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{k}{\lambda }_{t}\left( A\right) = \mathop{\max }\limits_{{{R}^{\prime }R = {I}_{k}}}\operatorname{tr}{R}^{\prime }{AR} \leq \mathop{\max }\limits_{{{R}^{\prime }R = {I}_{k}}}\operatorname{tr}{R}^{\prime }{BR} = \mathop{\sum }\li... | Yes |
Lemma 8.10.9. \( A \subset {R}_{ < }^{m} \) is convex and monotone in majorization if and only if \( A \) is monotone and \( {A}^{ * } \) is convex. | Proof. Necessity. If \( A \) is monotone in majorization, then it is obviously monotone. \( {A}^{ * } \) is convex (see Problem 8.35).\n\nSufficiency. For \( \lambda \in {R}_{ < }^{m} \) let\n\n\[ C\left( \lambda \right) = \left\{ {x \mid x \in {R}_{ + }^{m}, x{ \succ }_{w}\lambda }\right\} \]\n\n\[ D\left( \mathbf{\la... | No |
Lemma 8.10.10. Let \( C \) be compact and convex, and let \( D \) be convex. If the extreme points of \( C \) are contained in \( D \), then \( C \subset D \) . | Proof. Obvious. | No |
Corollary 8.10.1. \( C\left( \lambda \right) \subset {A}^{ * } \) . | Proof. If \( A \) is monotone, then \( {A}^{ * } \) is monotone in the sense that if \( \mathbf{\lambda } = {\left( {\lambda }_{1},\ldots ,{\lambda }_{m}\right) }^{\prime } \in {A}^{ * },\mathbf{\nu } = {\left( {\nu }_{1},\ldots ,{\nu }_{m}\right) }^{\prime },{\nu }_{i} \leq {\lambda }_{i}, i = 1,\ldots, m \), then \( ... | No |
Corollary 8.10.2. Let \( g \) be continuous, nondecreasing, and convex in \( \lbrack 0,1) \) . Let\n\n\[ f\left( \lambda \right) = f\left( {{\lambda }_{1},\ldots ,{\lambda }_{m}}\right) = \mathop{\sum }\limits_{{t = 1}}^{m}g\left( {\lambda }_{i}\right) . \]\n\nThen a test with the acceptance region \( A = \{ \lambda \m... | Proof. Being a sum of convex functions \( f \) is convex, and hence \( A \) is convex. \( \mathcal{A} \) is closed because \( f \) is continuous. We want to show that if \( f\left( x\right) \leq c \) and \( y{ \prec }_{w}x\left( {x, y \in {R}_{ < }^{m}}\right) \), then \( f\left( y\right) \leq c \) . Let \( {\widetilde... | Yes |
Theorem 8.10.4. A necessary condition for an invariant test to be admissible is that the extended region in the space of \( \sqrt{{\lambda }_{1}},\ldots ,\sqrt{{\lambda }_{t}} \) is convex and monotone. | We shall only sketch the proof of this theorem [following Schwartz (1967)]. Let \( \sqrt{{\lambda }_{i}} = {d}_{i}, i = 1,\ldots, t \), and let the density of \( {d}_{1},\ldots ,{d}_{t} \) be \( f\left( {\mathbf{d} \mid \mathbf{v}}\right) \), where \( \mathbf{v} = {\left( {\nu }_{1},\ldots ,{\nu }_{t}\right) }^{\prime ... | No |
Lemma 8.10.12. Let \( E, F \) be convex and symmetric about the origin. Then\n\n\[ V\{ \left( {E + {ky}}\right) \cap F\} \geq V\{ \left( {E + y}\right) \cap F\} \]\n\nwhere \( 0 \leq k \leq 1 \) and \( V \) denotes the \( n \) -dimensional volume. | Proof. Consider the set \( \alpha \left( {E + y}\right) + \left( {1 - \alpha }\right) \left( {E - y}\right) = {\alpha E} + \left( {1 - \alpha }\right) E + \left( {{2\alpha } - 1}\right) y \) which consists of points \( \alpha \left( {x + y}\right) + \left( {1 - \alpha }\right) \left( {z - y}\right) \) with \( x, z \in ... | Yes |
There exist matrices \( B\left( {p \times p}\right) \) and \( \mathbf{F}\left( {m \times m}\right) \) such that\n\n\[ \n{B\sum }{B}^{\prime } = {I}_{p},\;F{F}^{\prime } = {I}_{m}, \]\n\n(46)\n\n\[ \nB\Xi {F}^{\prime } = \left( {{D}_{\nu }^{\frac{1}{2}},0}\right) \]\n\n\( p \leq m \)\n\n\[ \n= \left( \begin{array}{l} {D... | We prove this for the case \( p \leq m \) and \( {\nu }_{p} > 0 \) . Other cases can be proved similarly. By Theorem A.2.2 of the Appendix there is a matrix \( B \) such that\n\n(47)\n\n\[ \n{B\sum }{B}^{\prime } = I,\;{B\Xi }{\Xi }^{\prime }{B}^{\prime } = {D}_{\nu }. \]\n\nLet\n\n(48)\n\n\[ \n{F}_{1} = {D}_{\nu }^{-\... | Yes |
Theorem 8.10.6. If the acceptance region of an invariant test is convex in the space of each column vector of \( U \) for each set of fixed values of \( V \) and of the other column vectors of \( \mathbf{U} \), then the power of the test increases monotonically in each \( {\nu }_{i} \) . | Proof. Since \( U{U}^{\prime } \) is unchanged when any column vector of \( U \) is multiplied by -1 , the acceptance region is symmetr'c about the origin in each of the column vectors of \( \mathbf{U} \) . Now the density of \( \mathbf{U} = \left( {n}_{ij}\right) ,\mathbf{V} = \left( {v}_{ij}\right) \) is\n\n(54) \( f... | Yes |
Corollary 8.10.3. If the acceptance region \( A \) of an invariant test is convex in \( U \) for each fixed \( V \), then the power of the test increases monotonically in each \( {\mathbf{v}}_{t} \) . | From this we see that Roy’s maximum root test \( A : {l}_{1} \leq K \) and the Lawley-Hotelling trace test \( A : \operatorname{tr}{U}^{\prime }{\left( V{V}^{\prime }\right) }^{-1}U \leq K \) have power functions that are monotonically increasing in each \( {\nu }_{t} \) . To see that the acceptance region of the likel... | Yes |
Lemma 8.11.1. Under the conditions of Theorem 8.11.1 the limiting distribution of \( \mathbf{H} \) is \( W\left( {\mathbf{\sum }, q}\right) \) . | Proof. Write \( \mathbf{H} \) as\n\n(5)\n\n\[ H = \sqrt{N}\left( {B - \beta }\right) \frac{1}{N}A\sqrt{N}{\left( B - \beta \right) }^{\prime }.\]\n\nThen the lemma follows from Theorem 8.11.1 and (4) of Section 8.4. | No |
Theorem 8.11.2. Under the conditions of Theorem 8.11.1, when the null hypothesis is true,\n\n\[ - 2\log \lambda \overset{d}{ \rightarrow }{\chi }_{pq}^{2} \] | Proof. We use the fact that \( N\log \left| {I + {N}^{-1}C}\right| = \operatorname{tr}C + {O}_{p}\left( {N}^{-1}\right) \) when \( N \rightarrow \) \( \infty \), since \( \left| {I + {xC}}\right| = 1 + x\operatorname{tr}C + O\left( {x}^{2}\right) \) (Theorem A.4.8).\n\nWe have\n\n\[ \operatorname{tr}{\left( \frac{1}{N}... | Yes |
Theorem 9.2.1. Let \( {x}_{1},\ldots ,{x}_{N} \) be a sample of \( N \) observations drawn from \( N\left( {\mathbf{\mu },\mathbf{\sum }}\right) \), where \( {x}_{\alpha },\mathbf{\mu } \), and \( \mathbf{\sum } \) are partitioned into \( {p}_{1},\ldots ,{p}_{q} \) rows (and columns in the case of \( \mathbf{\sum } \) ... | Since \( {r}_{ij} = {a}_{ij}/\sqrt{{a}_{ij}{a}_{ij}} \), we have\n\n(20)\n\n\[ \left| \mathbf{A}\right| = \left| \mathbf{R}\right| \mathop{\prod }\limits_{{i = 1}}^{p}{a}_{ii} \]\n\nwhere\n\n(21)\n\n\[ \mathbf{R} = \left( {\mathbf{r}}_{ij}\right) = \left( \begin{matrix} {\mathbf{R}}_{11} & {\mathbf{R}}_{12} & \cdots & ... | Yes |
Theorem 9.3.2. The distribution of \( V \) under the null hypothesis is the distribution of \( {V}_{2}{V}_{3}\cdots {V}_{q} \), where \( {V}_{2},\ldots ,{V}_{q} \) are independently distributed with \( {V}_{i} \) having the distribution of \( {U}_{{p}_{i},{\bar{p}}_{i}, n - {\bar{p}}_{i}} \), where \( {\bar{p}}_{i} = {... | Proof. From the proof of Theorem 9.3.1, we see that the distribution of \( {V}_{i} \) is that of \( {U}_{{p}_{i},{\bar{p}}_{i}, n - {\bar{p}}_{i}} \) not depending on the conditioning \( {z}_{\alpha }^{\left( k\right) }, k = 1,\ldots, i - 1 \) , \( \alpha = 1,\ldots, n \) . Hence the distribution of \( {V}_{i} \) does ... | Yes |
Theorem 9.3.3. Under the null hypothesis \( V \) is distributed as \( \mathop{\prod }\limits_{{i = 2}}^{q}\mathop{\prod }\limits_{{j = 1}}^{{p}_{i}}{X}_{ij} \) , where the \( {X}_{ij} \) ’s are independent and \( {X}_{ij} \) has the density \( \beta \left\lbrack {x\left| {\;\frac{1}{2}\left( {n - {\bar{p}}_{i} + 1 - }\... | Proof. This theorem follows from Theorems 9.3.2 and 8.4.1. | No |
Theorem 9.3.4. When the null hypothesis is true, the \( h \) th moment of the criterion is\n\n\[ \mathcal{E}{V}^{h} = \mathop{\prod }\limits_{{i = 2}}^{q}\left\{ {\mathop{\prod }\limits_{{j = 1}}^{{p}_{i}}\frac{\Gamma \left\lbrack {\frac{1}{2}\left( {n - {\bar{p}}_{i} + 1 - j}\right) + h}\right\rbrack \Gamma \left\lbra... | Proof. Because \( {V}_{2},\ldots ,{V}_{q} \) are independent,\n\n\[ \mathcal{E}{V}^{h} = \mathcal{E}{V}_{2}^{h}\mathcal{E}{V}_{3}^{h}\cdots \mathcal{E}{V}_{q}^{h} \]\n\nTheorem 9.3.2 implies \( \mathcal{E}{V}_{i}^{h} = \mathcal{E}{U}_{{p}_{i},{\bar{p}}_{i}, n - {\bar{p}}_{i}}^{h} \). Then the theorem follows by substit... | Yes |
Lemma 9.10.1. There exist matrices \( {B}_{1}\left( {{p}_{1} \times {p}_{1}}\right) ,{B}_{2}\left( {{p}_{2} \times {p}_{2}}\right) \) such that\n\n\[ \n{B}_{1}{\sum }_{11}{B}_{1}^{\prime } = {I}_{{p}_{1}},\;{B}_{2}{\sum }_{22}{B}_{2}^{\prime } = {I}_{{p}_{2}},\;{B}_{1}{\sum }_{12}{B}_{2}^{\prime } = \Delta .\n\] | Proof. Let \( m = {p}_{2}, B = {B}_{1},{F}^{\prime } = {\sum }_{22}^{\frac{1}{2}}{B}_{2}^{\prime },\Xi = {\sum }_{12}{\sum }_{22}^{-\frac{1}{2}} \) in Lemma 8.10.13.\n\nThen \( {F}^{\prime }F = {B}_{2}{\sum }_{22}{B}_{2}^{\prime } = {I}_{{p}_{2}},{B}_{1}{\sum }_{12}{B}_{2}^{\prime } = {B}_{1}{\Xi F} = \Delta \) . | Yes |
Lemma 9.10.3. If \( A \geq B \), then \( {\lambda }_{i}\left( A\right) \geq {\lambda }_{t}\left( B\right) \) . | Proof. By the minimax property of the characteristic roots [see, e.g., Courant and Hilbert (1953)], \n\n\[ \n{\lambda }_{t}\left( \mathbf{A}\right) = \mathop{\max }\limits_{{S}_{t}}\mathop{\min }\limits_{{\mathbf{x} \in {S}_{t}}}\frac{{\mathbf{x}}^{\prime }\mathbf{A}\mathbf{x}}{{\mathbf{x}}^{\prime }\mathbf{x}} \geq \m... | Yes |
Theorem 9.11.1. When \( X \) has the density (1) and the null hypothesis is true. the limiting distribution of \( \mathbf{H} \), is \( W\left\lbrack {\left( {1 + \kappa }\right) {\sum }_{u},{\widetilde{p}}_{t}}\right\rbrack \), where \( {\bar{p}}_{t} = {p}_{1} + \cdots + {p}_{t - 1} \) and \( {p}_{j} \) is the number o... | Proof. Since \( {\widetilde{\mathbf{\sum }}}^{\left( t, t - 1\right) } = \mathbf{0} \), we have \( {\mathcal{E}}^{o}{\widetilde{\mathbf{S}}}^{\left( t, t - 1\right) } = \mathbf{0} \) and\n\n(9)\n\n\[ \mathcal{E}{s}_{jk}{s}_{lm} = \left( {\frac{\kappa }{N} + \frac{1}{N - 1}}\right) {\sigma }_{jl}{\sigma }_{km} \]\n\nif ... | Yes |
Theorem 9.11.3. Let \( f\left( X\right) \) be a vector-valued function of \( X\left( {p \times N}\right) \) such that\n\n(18)\n\n\[ f\left( {X + v{\varepsilon }_{N}^{\prime }}\right) = f\left( X\right) \]\nfor all \( \mathbf{v} \) and\n\n(19)\n\n\[ f\left( {KX}\right) = f\left( X\right) \]\n\nfor all \( \mathbf{K} = \o... | Proof. The proof is similar to the proof of Theorem 4.5.4. | No |
Lemma 10.3.1. Let \( y \) be an observation vector on a random vector with density \( f\left( {z,\mathbf{\theta }}\right) \), where \( \mathbf{\theta } \) is a parameter vector in a space \( \Omega \) . Let \( {H}_{a} \) be the hypothesis \( \mathbf{\theta } \in {\Omega }_{a} \subset \Omega \), let \( {H}_{b} \) be the... | Proof. The lemma follows from the definitions:\n\n\[{\lambda }_{a} = \frac{\mathop{\max }\limits_{{\mathbf{\theta } \in {\Omega }_{a}}}f\left( {y,\mathbf{\theta }}\right) }{\mathop{\max }\limits_{{\mathbf{\theta } \in \Omega }}f\left( {y,\mathbf{\theta }}\right) }\n\n{\lambda }_{b} = \frac{\mathop{\max }\limits_{{\math... | Yes |
Theorem 10.4.1. \( {V}_{12},{V}_{1};\ldots ,{V}_{1q} \) defined by (1) are independent when \( {\sum }_{1} = \cdots = {\sum }_{q} \) and \( {n}_{g} \geq p, g = 1,\ldots, q \) . | The theorem is a consequence of the following lemma:\n\nLemma 10.4.1. If \( A | No |
Lemma 10.4.1. If \( A \) and \( B \) are independently distributed according to \( W\left( {\mathbf{\sum }, m}\right) \) and \( W\left( {\mathbf{\sum }, n}\right) \), respectively, \( n \geq p, m \geq p \), and \( C \) is such that \( C(A + \) B) \( {C}^{\prime } = I \), then \( A + B \) and \( {CA}{C}^{\prime } \) are... | Proof of Lemma. The density of \( D = A + B \) and \( E = {CA}{C}^{\prime } \) is found by replacing \( A \) and \( B \) in their joint density by \( {C}^{-1}E{C}^{1 - 1} \) and \( D - {C}^{-1}E{C}^{1 - 1} = \) \( {C}^{-1}\left( {I - E}\right) {C}^{t - 1} \), respectively, and multiplying by the Jacobian, which is \( {... | Yes |
Lemma 10.4.2. For \( {B}_{i - 1} \) and \( {C}_{t - 1} \) positive definite\n\n\[ {b}_{\left( i\right) }^{\prime }{B}_{i - 1}^{-1}{b}_{\left( i\right) } + {c}_{\left( i\right) }^{\prime }{C}_{i - 1}^{-1}{c}_{\left( i\right) } - {\left( {b}_{\left( i\right) } + {c}_{\left( i\right) }\right) }^{\prime }{\left( {B}_{i - 1... | Proof. Use of \( {\left( {\mathbf{B}}^{-1} + {C}^{-1}\right) }^{-1} = {\left\lbrack {C}^{-1}\left( \mathbf{B} + C\right) {\mathbf{B}}^{-1}\right\rbrack }^{-1} = \mathbf{B}{\left( \mathbf{B} + C\right) }^{-1}\mathbf{C} \) shows the left-hand side of (8) is (omitting \( i \) and \( i - 1 \) )\n\n\[ {\mathbf{b}}^{\prime }... | Yes |
Theorem 10.4.2. \[ {V}_{1} = \mathop{\prod }\limits_{{g = 2}}^{q}\left\{ {\mathop{\prod }\limits_{{i = 1}}^{p}{X}_{ig}^{\frac{1}{2}\left( {{n}_{1} + \cdots + {n}_{g - 1}}\right) }{\left( 1 - {X}_{ig}\right) }^{\frac{1}{3}{n}_{g}} \cdot \mathop{\prod }\limits_{{i = 2}}^{p}{Y}_{ig}^{\frac{1}{2}\left( {{n}_{1} + \cdots + ... | Proof. The factors \( {V}_{12},\ldots ,{V}_{1q} \) are independent by Theorem 10.4.1. Each term \( {V}_{1g} \) is decomposed according to (7), and the factors are independent. | No |
Theorem 10.4.3\n\n\[ W = \mathop{\prod }\limits_{{g = 2}}^{q}\left\{ {\mathop{\prod }\limits_{{i = 1}}^{p}{X}_{ig}^{\frac{1}{2}\left( {{N}_{1} + \cdots + {N}_{g - 1}}\right) }{\left( 1 - {X}_{ig}\right) }^{\frac{1}{2}{N}_{g}}\mathop{\prod }\limits_{{i = 2}}^{p}{Y}_{ig}^{\frac{1}{2}\left( {{N}_{1} + \cdots + {N}_{g}}\ri... | Proof. The characterization of the first factor in (13) corresponds to that of \( {V}_{1} \) with the exponents of \( {X}_{tg} \) and \( 1 - {X}_{tg} \) modified by replacing \( {n}_{g} \) by \( {N}_{g} \) . The second term in \( {U}_{p, q - 1, n}^{\frac{1}{2}N} \), and its characterization follows from Theorem 8.4.1. | No |
Theorem 10.4.4. Let \( {V}_{1} \) be the criterion defined by (10) of Section 10.2 for testing the hypothesis that \( {H}_{1} : {\sum }_{1} = \cdots = {\sum }_{q} \), where \( {A}_{g} \) is \( {n}_{g} \) times the sample covariance matrix and \( {n}_{g} + 1 \) is the size of the sample from the gth population; let \( W... | This theorem was first proved by Wilks (1932). See Problem 10.5 for an alternative approach. | No |
Theorem 10.10.2. If \( {2p} < {N}_{g} + 1, g = 1,\ldots, q \), the likelihood ratio test and modified likelihood ratio test of the null hypothesis (8) are admissible. | For more details see Kiefer and Schwartz (1965). | No |
Theorem 10.11.1. When sampling from (1) and the null hypothesis is true,\n\n(8)\n\n\[ - 2\log {\lambda }_{1}\overset{d}{ \rightarrow }\left( {\kappa + 1}\right) {\chi }_{\left( {q - 1}\right) \left( {p - 1}\right) \left( {p + 2}\right) /2}^{2} + \left\lbrack {\left( {\kappa + 1}\right) + {p\kappa }/2}\right\rbrack {\ch... | When \( \kappa = 0, - 2\log {\lambda }_{1}\overset{d}{ \rightarrow }{\chi }_{\left( {q - 1}\right) p\left( {p + 1}\right) /2}^{2} \) is in agreement with (12) of Section 10.5. The validity of the distributions derived in Section 10.4 depend on the observations being normally distributed; Theorem 10.11.1 shows that even... | Yes |
Lemma 10.11.1. Let \( {A}_{1} = {n}_{1}{S}_{1} \) and \( {A}_{2} = {n}_{2}{S}_{2} \) be defined by (2) of Section 10.2 with \( {\sum }_{1} = {\sum }_{2} = I \) . Then \( {A}_{1}{\left( {A}_{1} + {A}_{2}\right) }^{-1} \) and \( {A}_{1} + {A}_{2} \) are asymptotically independent. | \[ \text{Proof. Let}\left( {1\sqrt{{n}_{g}}}\right) \left( {{A}_{g} - {n}_{g}I}\right) = {W}_{g}, g = 1,2\text{. Then} \]\n\n\[ \sqrt{{n}_{1}}\left\lbrack {{A}_{1}{\left( {A}_{1} + {A}_{2}\right) }^{-1} - \frac{{n}_{1}}{{n}_{1} + {n}_{2}}I}\right\rbrack = \frac{{n}_{1}{n}_{2}}{{\left( {n}_{1} + {n}_{2}\right) }^{2}}{W}... | Yes |
Theorem 10.11.2. When \( {\sum }_{1} = \cdots = {\sum }_{g} \) and \( {\mathbf{\mu }}^{\left( 1\right) } = \cdots = {\mathbf{\mu }}^{\left( g\right) } \) | \[ - 2\log {\lambda }_{1}{\lambda }_{2} = - 2\log {\lambda }_{1} - 2\log {\lambda }_{2} \] \[ \overset{d}{ \rightarrow }\left( {\kappa + 1}\right) {\chi }_{\left( {q - 1}\right) \left( {p - 1}\right) \left( {p + 2}\right) /2}^{2} + \left\lbrack {\left( {\kappa + 1}\right) + {p\kappa }/2}\right\rbrack {\chi }_{q - 1}^{2... | Yes |
Theorem 10.11.3. Let \( f\left( X\right) \) be a vector-valued function of \( X = \) \( \left( {{X}^{\left( 1\right) },\ldots ,{X}^{\left( q\right) }}\right) \left( {p \times N}\right) \) such that\n\n(16)\n\n\[ f\left( {{\mathbf{X}}^{\left( 1\right) } + {\mathbf{v}}^{\left( 1\right) }{\mathbf{\varepsilon }}_{{N}_{1}}^... | The proof of Theorem 10.11.3 is similar to the proof of Theorem 4.5.4. The theorem implies that the distribution of the criterion \( {V}_{1} \) of (10) of Section 10.2 when the density of \( \mathbf{X} \) is (15) with \( {\mathbf{\Lambda }}_{1} = \cdots = {\mathbf{\Lambda }}_{q} \) is the same as for normality. Hence t... | No |
Corollary 10.11.1. Let \( f\left( X\right) \) be a vector-valued function of \( X\left( {p \times N}\right) \) such that\n\n(18)\n\n\[ f\left( {X + v{\varepsilon }_{N}^{\prime }}\right) = f\left( X\right) \]\n\nfor every \( \mathbf{v} \) and (17) holds. Then the distribution of \( f\left( X\right) \), where \( X \) has... | If follows that the distribution of the criterion \( \lambda \) of (7) or \( V \) of (11) of Section 10.3 is the same for the density (15) as for \( X \) being normally distributed. | Yes |
Theorem 11.2.1. Let the p-component random vector \( X \) have \( \mathcal{E}X = 0 \) and \( \mathcal{E}\mathbf{X}{\mathbf{X}}^{\prime } = \mathbf{\sum } \). Then there exists an orthogonal linear transformation\n\n(20)\n\n\[ U = {\mathbf{B}}^{\prime }X \]\n\nsuch that the covariance matrix of \( \mathbf{U} \) is \( \m... | The vector \( \mathbf{U} \) is defined as the vector of principal components of \( \mathbf{X} \). It will be observed that we have proved Theorem A.2.1 of Appendix A for \( \mathbf{B} \) positive semidefinite, and indeed, the proof holds for any symmetric \( \mathbf{B} \). It might be noted that once the transformation... | Yes |
Corollary 11.2.1. Suppose \( {\lambda }_{r + 1} = \cdots = {\lambda }_{r + m} = v \) (i.e., \( v \) is a root of multiplicity \( m \) ); then \( \sum - \nu \mathbf{I} \) is of rank \( p - m \) . Furthermore \( {\mathbf{B}}^{ * } = \left( {\mathbf{\beta }}^{\left( r + 1\right) }\right. \ldots \) \( {\mathbf{\beta }}^{\l... | Proof. From the derivation of the theorem we have \( \left( {\sum - {\nu I}}\right) {\mathbf{\beta }}^{\left( i\right) } = \mathbf{0} \) , \( i = r + 1,\ldots, r + m \) ; that is, \( {\mathbf{B}}^{\left( r + 1\right) },\ldots ,{\mathbf{b}}^{\left( r + m\right) } \) are \( m \) linearly independent solutions of \( \left... | Yes |
Theorem 11.2.2. An orthogonal transformation \( V = {CX} \) of a random vector \( \mathbf{X} \) leaves invariant the generalized variance and the sum of the variances of the components. | Proof. Let \( \mathcal{E}X = 0 \) and \( \mathcal{E}X{X}^{\prime } = \sum \) . Then \( \mathcal{E}V = \mathbf{0} \) and \( \mathcal{E}V{V}^{\prime } = {C\sum }{C}^{\prime } \) . The generalized variance of \( V \) is\n\n(22)\n\n\[ \left| {{C\sum }{C}^{\prime }}\right| = \left| C\right| \cdot \left| \sum \right| \cdot \... | Yes |
Theorem 11.3.1. Let \( {x}_{1},\ldots ,{x}_{N} \) be \( N\left( { > p}\right) \) observations from \( N\left( {\mathbf{\mu },\mathbf{\sum }}\right) \) , where \( \sum \) is a matrix with \( p \) different characteristic roots. Then a set of maximum likelihood estimators of \( {\lambda }_{1},\ldots ,{\lambda }_{p} \) an... | Proof. When the roots of \( \left| {\sum - {\lambda I}}\right| = 0 \) are different, each vector \( {\mathbf{\beta }}^{\left( i\right) } \) is uniquely defined except that \( {\mathbf{\beta }}^{\left( i\right) } \) can be replaced by \( - {\mathbf{\beta }}^{\left( i\right) } \) . If we require that the first nonzero co... | Yes |
Lemma 11.6.1. For any positive definite matrix \( H \)\n\n\[{\operatorname{ch}}_{p}\left( H\right) \leq \frac{1}{{h}^{u}} \leq {\operatorname{ch}}_{1}\left( H\right)\]\n\nwhere \( {\mathbf{H}}^{-1} = \left( {h}^{ij}\right) \) and \( {\operatorname{ch}}_{p}\left( \mathbf{H}\right) \) and \( {\operatorname{ch}}_{1}\left(... | Proof. From Theorem A.2.4 in the Appendix we have \( {\operatorname{ch}}_{p}\left( H\right) \leq {h}_{H} \leq \) \( {\mathrm{{ch}}}_{1}\left( H\right) \) and\n\n\[{\operatorname{ch}}_{p}\left( {\mathbf{H}}^{-1}\right) \leq {h}^{\prime \prime } \leq {\operatorname{ch}}_{1}\left( {\mathbf{H}}^{-1}\right) ,\;l = 1\ldots, ... | Yes |
Theorem 11.6.2. A confidence interval for the characteristic roots of \( \sum \) with confidence at least \( 1 - \varepsilon \) is\n\n(19)\n\n\[ \frac{{l}_{p}}{{u}^{\prime }} \leq {\lambda }_{p} \leq {\lambda }_{1} \leq \frac{{l}_{1}}{{l}^{\prime }} \] \n\nwhere \( {l}^{\prime } \) and \( {u}^{\prime } \) satisfy (17). | Anderson (1965a, 1965b) showed that the above confidence bounds are optimal within the class of bounds\n\n(20)\n\n\[ f\left( {{l}_{1},\ldots ,{l}_{p}}\right) \leq {\lambda }_{p} \leq {\lambda }_{1} \leq g\left( {{l}_{1},\ldots ,{l}_{p}}\right) , \] \n\nwhere \( f \) and \( g \) are homogeneous of degree 1 and are monot... | Yes |
Theorem 12.2.1. Let \( X = {\left( {X}^{\left( 1\right) }{}^{\prime }{X}^{\left( 2\right) }{}^{\prime }\right) }^{\prime } \) be a random vector with covariance matrix \( \sum \) . The rth canonical correlation between \( {X}^{\left( 1\right) } \) and \( {X}^{\left( 2\right) } \) is the \( r \) th largest root of (14).... | We can now verify (without differentiation) that \( {U}_{1},{V}_{1} \) have maximum correlation. The linear combinations \( {a}^{\prime }U = \left( {{a}^{\prime }{\mathbf{A}}^{\prime }}\right) {X}^{\left( 1\right) } \) and \( {b}^{\prime }V = \left( {{b}^{\prime }{\Gamma }^{\prime }}\right) {X}^{\left( 2\right) } \) ar... | Yes |
Theorem 12.2.2. The canonical correlations are invariant with respect to transformations \( {\mathbf{X}}^{\left( 1\right) * } = {\mathbf{C}}_{t}{\mathbf{X}}^{\left( t\right) } \), where \( {\mathbf{C}}_{t} \) is nonsingular, \( i = 1,2 \), and any function of \( \sum \) that is invariant is a function of the canonical ... | Proof. Equation (14) is transformed to\n\n(54)\n\n\[ 0 = \left| \begin{array}{rr} - \lambda {C}_{1}{\sum }_{11}{C}_{1}^{\prime } & {C}_{1}{\sum }_{12}{C}_{2}^{\prime } \\ {C}_{2}{\sum }_{21}{C}_{1}^{\prime } & - \lambda {C}_{2}{\sum }_{22}{C}_{2}^{\prime } \end{array}\right| = \left| \begin{array}{ll} {C}_{1} & \mathbf... | Yes |
Theorem 12.3.1. Let \( {x}_{1},\ldots ,{x}_{N} \) be \( N \) observations from \( N\left( {\mathbf{\mu },\sum }\right) \) . Let \( \sum \) be partitioned into \( {p}_{1} \) and \( {p}_{2}\left( {{p}_{1} \leq {p}_{2}}\right) \) rows and columns as in (2) in Section 12.2, and let \( {x}_{\alpha } \) be similarly partitio... | In the population the canonical correlations and canonical variates were found in terms of maximizing correlations of linear combinations of two sets of variates. The entire argument can be carried out in terms of the sample. Thus \( {\widehat{\alpha }}^{\left( 1\right) }{}^{\prime }{\mathbf{x}}_{\alpha }^{\left( 1\rig... | Yes |
Lemma 12.8.1. Suppose \( \left( {1/T}\right) A \rightarrow {A}^{0} \), a positive definite matrix, as \( T \rightarrow \) \( \infty \) . Then \( \nu = {O}_{p}\left( {1/T}\right) \), where \( \nu \) is the smallest root of (26). | Proof. Let \( {\overline{\mathbf{P}}}_{12} = \sqrt{T}\left( {{\mathbf{P}}_{12} - {\mathbf{\Pi }}_{12}}\right) \) . Then because \( {\mathbf{\beta }}^{\prime }{\mathbf{\Pi }}_{12} = \mathbf{0} \)\n\n(51)\n\n\[ \frac{{\mathbf{\beta }}^{\prime }{P}_{12}{S}_{22} \cdot {}_{1}{P}_{12}\mathbf{\beta }}{{\mathbf{\beta }}^{\prim... | Yes |
Theorem 12.8.1. Under the conditions of Theorem 8.11.1\n\n\[ \sqrt{T}\left( {{\widehat{\mathbf{\beta }}}_{\mathrm{{LIML}}}^{ * } - {\mathbf{\beta }}^{ * }}\right) \overset{d}{ \rightarrow }N\left\lbrack {\mathbf{0},{\sigma }_{11}{\left( {\mathbf{\Pi }}_{12}{\mathbf{S}}_{22}^{0}{.}_{1}{\mathbf{\Pi }}_{12}^{\prime }\righ... | Proof. The theorem follows from (55), \( {S}_{22} \cdot {}_{1} \rightarrow {S}_{22}^{0} \cdot {}_{1} \), and \( {\mathbf{P}}_{12}\overset{p}{ \rightarrow }{\mathbf{\Pi }}_{12} \) . | Yes |
Theorem 13.2.2. If \( A \) and \( B \) are distributed independently according to \( W\left( {\sum, m}\right) \) and \( W\left( {\sum, n}\right) \) respectively \( \left( {m \geq p, n \geq p}\right) \), the joint density of the roots of \( \left| {A - {lB}}\right| = 0 \) is (50) where \( {C}_{2} \) is defined by (47). | The joint density of \( \mathbf{Y} \) can be found from (45) and the fact that the Jacobian is \( {\left| \mathbf{Y}\right| }^{-{2p}} \) . (See Theorem A.4.6 of the Appendix.) | No |
Lemma 13.3.1. If the density of \( Y\left( {p \times m}\right) \) is \( f\left( {Y{Y}^{\prime }}\right) \), then the density of \( B = Y{Y}^{\prime } \) is\n\n\[ \frac{{\left| B\right| }^{\frac{1}{2}\left( {m - p - 1}\right) }f\left( B\right) {\pi }^{\frac{1}{2}{pm}}}{{\Gamma }_{p}\left( {\frac{1}{2}m}\right) }.\] | The proof of this, like that of Theorem 13.3.1, depends on exhibiting a special case; let \( f\left( {Y{Y}^{\prime }}\right) = {\left( 2\pi \right) }^{-\frac{1}{2}{pm}}{e}^{-\frac{1}{2}\operatorname{tr}Y{Y}^{\prime }} \), then (9) is \( w\left( {B \mid I, m}\right) \) . | No |
Lemma 13.3.2. If the orthogonal matrix \( E \) has a distribution such that \( {e}_{t1} \geq 0 \) and if \( {E}^{* * } = J\left( {E{Q}^{\prime }}\right) E{Q}^{\prime } \) has the same distribution for every orthogonal \( Q \), then \( E \) has the conditional Haar invariant distribution. | Proof. Let the space \( V \) of orthogonal matrices be partitioned into the subspaces \( {V}_{1},\ldots ,{V}_{2} \), so that \( {J}_{i}{V}_{i} = {V}_{1} \), say, where \( {J}_{1} = I \) and \( {V}_{1} \) is the set for which \( {e}_{11} \geq 0 \) . Let \( {\mu }_{1} \) be the measure in \( {V}_{1} \) defined by the dis... | Yes |
Theorem 13.3.4. If the symmetric matrix \( \mathbf{B} \) has a density of the form \( g\left( {{l}_{1},\ldots ,{l}_{p}}\right) \), where \( {l}_{1} > \cdots > {l}_{p} \) are the characteristic roots of \( \mathbf{B} \), then the joint density of the roots is (2) and the matrix of normalized characteristic vectors \( \m... | Proof. The density of \( {QB}{Q}^{\prime } \), where \( Q{Q}^{\prime } = I \), is the same as that of \( \mathbf{B} \) (for the roots are invariant), and therefore the distribution of \( J\left( {{Y}^{\prime }{Q}^{\prime }}\right) {Y}^{\prime }{Q}^{\prime } \) is the same as that of \( {Y}^{\prime } \) . Then Theorem 1... | No |
Theorem 13.3.5. Let \( B = {B}^{\prime } \) have the density\n\n(25)\n\n\[ \n{\pi }^{-p\left( {p + 1}\right) /4}{2}^{-\frac{1}{2}p}{e}^{-\frac{1}{2}\operatorname{tr}{B}^{2}}.\n\]\n\nThen the characteristic roots \( {l}_{1} > \cdots > {l}_{p} \) of \( \mathbf{B} \) have the density\n\n(26)\n\n\[ \n{2}^{-\frac{1}{2}p}{\p... | Proof. Since the characteristic roots of \( {\mathbf{B}}^{2} \) are \( {l}_{1}^{2},\ldots ,{l}_{p}^{2} \) and \( \operatorname{tr}{\mathbf{B}}^{2} = \sum {l}_{t}^{2} \) , the theorem follows directly. | No |
Corollary 13.3.2. Let \( n \) S be distributed according to \( W\left( {I, n}\right) \), and define the diagonal matrix \( L \) and \( B \) by \( S = {C}^{\prime }{LC},{C}^{\prime }C = I,{l}_{1} > \cdots > {l}_{p} \), and \( {c}_{i1} \geq 0 \) , \( i = 1,\ldots, p \) . Then the density of the limiting distribution of \... | Proof. The density of the limiting distribution of \( \sqrt{n}\left( {S - I}\right) \) is (25), and the diagonal elements of \( D \) are the characteristic roots of \( \sqrt{n}\left( {S - I}\right) \) and the columns of \( {\mathbf{C}}^{\prime } \) are the characteristic vectors. | No |
Theorem 13.5.1. Suppose \( {nS} \) has the distribution \( W\left( {\sum, n}\right) \) . Define diagonal \( \mathbf{A} \) and \( \mathbf{L} \) and orthogonal \( \mathbf{B} \) and \( \mathbf{B} \) by\n\n(1)\n\n\[ \sum = {B\Lambda }{\beta }^{\prime }.\;S = {BL}{B}^{\prime }, \]\n\n\( {\lambda }_{1} > {\lambda }_{2} > \cd... | Proof. The matrix \( {nT} = n{\mathbf{B}}^{\prime }S\mathbf{B} \) is distributed according to \( W\left( {\mathbf{\Lambda }, n}\right) \) . Let\n\n(4)\n\n\[ T = {YL}{Y}^{\prime } \]\n\nwhere \( Y \) is orthogonal. In order that (4) determine \( Y \) uniquely, we require \( {y}_{u} \geq 0 \) . Let \( \sqrt{n}\left( {T -... | Yes |
Theorem 13.5.2. Under the conditions of Theorem 13.5.1 and \( \mathbf{\Lambda } = \) \( \operatorname{diag}\left( {{\mathbf{\Lambda }}_{1},{\lambda }^{ * }{\mathbf{I}}_{q}}\right) \), the density of the limiting distribution of \( {d}_{p - q + 1},\ldots ,{d}_{p} \) is\n\n(17)\n\n\[ \n{2}^{-\frac{1}{2}q}{\left( {\lambda... | To justify the preceding derivation we note that \( {D}_{2} \) and \( {Y}_{22} \) are functions of \( U \) depending on \( n \) that converge to the solution of \( {U}_{22}^{ * } = {Y}_{22}^{ * }{D}_{22}^{ * }{Y}_{22}^{*\prime } \) . We can use the following theorem given by Anderson (1963a) and due to Rubin. | No |
Theorem 13.5.3. Let \( {F}_{n}\left( u\right) \) be the cumulative distribution function of a random matrix \( {U}_{n} \) . Let \( {V}_{n} \) be a matrix-valued function of \( {U}_{n},{V}_{n} = {f}_{n}\left( {u}_{n}\right) \), and Let \( {G}_{n}\left( v\right) \) be the (induced) distribution of \( {V}_{n} \) . Suppose... | The details of verifying that \( \mathbf{U}\left( n\right) \) and\n\n(19)\n\n\[ \left( {{D}_{2}\left( n\right) ,{Y}_{22}\left( n\right) }\right) = {f}_{n}\left( {U\left( {nn}\right) }\right) \]\nsatisfy the conditions of the theorem have been given by Anderson (1963a). | Yes |
Theorem 13.6.1. Let \( m{S}^{ * } \) and \( n{T}^{ * } \) be independently distributed according to \( W\left( {\Phi, m}\right) \) and \( W\left( {\sum, n}\right) \), respectively. Let \( {\lambda }_{1} > {\lambda }_{2} > \cdots > {\lambda }_{p}\left( { > 0}\right) \) be the roots of (3), and let \( \mathbf{\Lambda } \... | Proof. Let (9) \( S = {\Gamma }^{\prime }{S}^{ * }\Gamma ,\;T = {\Gamma }^{\prime }{T}^{ * }\Gamma . \) Then \( {mS} \) and \( {nT} \) are distributed independently according to \( W\left( {\mathbf{\Lambda }, m}\right) \) and \( W\left( {\mathbf{I}, n}\right) \), respectively (Section 7.3.3). Then \( {l}_{1},\ldots ,{l... | Yes |
Theorem 13.7.1. Let \( {\left( {X}^{\left( 1\right) },{X}^{\left( 2\right) }\right) }^{\prime },\alpha = 1,\ldots, n \), be observations on the random vector \( {x}_{\alpha } \) with mean \( \mathbf{0} \) and covariance matrix \( \sum \) . Let \( \mathbf{B} = {\sum }_{12}{\sum }_{22}^{-1} \) . Let the columns of \( {\w... | Note that \( \mathbf{B} = \mathbf{\Omega }{\Pi }^{\prime } = \mathbf{\Omega }{M}^{\prime }{\left( \Pi {M}^{-1}\right) }^{\prime } \) for arbitrary nonsingular \( M \) ; however,(47) and (48) are invariant with respect to the transform ation \( \mathbf{\Omega } \rightarrow \mathbf{\Omega }M \) and \( \Pi \rightarrow \Pi... | Yes |
Define diagonal \( \mathbf{\Lambda } \) and \( \mathbf{L} \) and orthogonal \( \mathbf{B} \) and \( \mathbf{B} \) by (2), \( {\lambda }_{1} > \cdots > {\lambda }_{p},{l}_{1} > \cdots > {l}_{p},{\beta }_{i1} \geq 0,{b}_{i1} \geq 0, i = 1,\ldots, p \) . Define \( G = \sqrt{N}(B - \) B) and diagonal \( D = \sqrt{N}\left( ... | The proof is the same as for Theorem 13.5.1 except that (4) is used instead of (4) with \( \kappa = 0 \) . | No |
Theorem 13.8.2. Let \( {S}^{ * } \) be the sample covariance matrix of a sample of size \( M \) from (1), and let \( {T}^{ * } \) be the sample covariance matrix of a sample of size \( N \) from (1) with \( \mathbf{\Psi } \) replaced by \( \mathbf{\sum } \) . Let \( \mathbf{\Lambda } \) be the diagonal matrix with \( {... | Proof. Transform \( {S}^{ * } \) and \( {T}^{ * } \) to \( S = {\Gamma }^{\prime }{S}^{ * }\Gamma \) and \( T = {\Gamma }^{\prime }{T}^{ * }\Gamma ,\Phi \) and \( \sum \) to \( \Lambda = {\Gamma }^{\prime }{\Phi \Gamma } \) and \( I = {\Gamma }^{\prime }{\sum \Gamma } \), and \( {X}^{ * } \) to \( X = {\Gamma }^{-1}{X}... | Yes |
Theorem 15.2.2. The conditional independences\n\n(6)\n\n\[ \nX ⫫ Y \mid Z,\;X ⫫ Z \mid Y \]\n\nhold if and only if\n\n(7)\n\[ \nX ⫫ \left( {Y, Z}\right) \text{.} \]\n | Proof. The relations (6) imply that the density of \( \mathbf{X},\mathbf{Y} \), and \( \mathbf{Z} \) can be written as\n\n(8)\n\n\[ \nf\left( {x, y, z}\right) = f\left( {x \mid z}\right) g\left( {y \mid z}\right) h\left( z\right) \]\n\n\[ \n= k\left( {x \mid y}\right) l\left( {z \mid y}\right) m\left( y\right) . \]\n\n... | Yes |
Theorem 15.2.3. A locally Markov distribution on a graph is pairwise Markov. | Proof. Suppose the graph is locally Markov (Definition 15.2.3). Let \( u \) and \( v \) be nonadjacent vertices. Because \( v \) is not adjacent to \( u \), it is not in \( \operatorname{bd}\left( u\right) \) ; hence,\n\n(12)\n\n\[ \n{X}_{u} ⫫ {X}_{V \smallsetminus \mathrm{{cl}}\left( u\right) } \mid {X}_{\mathrm{{bd}}... | Yes |
Theorem 15.2.5. If \( S \) separates \( A \) and \( B \) in a graph with a globally Markov distribution, \( {\mathbf{\Lambda }}_{AB} = 0 \) . | Proof. Because \( S \) separates \( A \) and \( B \), every element \( u \) of \( A \) and every element \( v \) of \( B \) are nonadjacent, for otherwise the path \( \left( {u, v}\right) \) would connect \( A \) and \( B \) without intersecting \( S \) . The globally Markov property is that \( {X}_{1} \) and \( {X}_{B... | Yes |
Theorem 15.2.6. A distribution on a globally Markov graph is pairwise Markov. | Proof. Let the set \( B \) be \( i \), the set \( C \) be \( j \) not adjacent to \( i \) . and the set \( A \) the rest of the variables. Any path from \( B \) to \( C \) must include elements of \( A \) , Hence \( i \) is independent of \( j \) in the distribution conditioned on the other variables. | No |
Theorem 15.2.7. A globally Markov family of distributions on a graph is locally Markov. | Proof. The boundary of a set \( B \) separates \( B \) and \( V \smallsetminus \operatorname{cl}\left( B\right) \) . | No |
Theorem 15.2.8. A pairwise Markov family of distributions on a graph is globally Markov. | Proof. Let \( A, B \), and \( S \) be disjoint sets in a pairwise Markov graph such that \( S \) separates \( A \) and \( B \) . Let \( \# \left( S\right) \) and \( \# \left( V\right) \) denote the numbers of vertices in \( S \) and \( V \), respectively. If \( \# \left( V\right) = \# \left( S\right) + 2 \), that is, \... | Yes |
Theorem 15.2.9. Suppose \( A, B, S \) decomposes \( G = \left( {V, E}\right) \), Then the density of \( {X}_{V} \) factorizes with respect to \( G \) if and only if its marginal densities \( {f}_{A \cup S}\left( {x}_{A \cup S}\right) \) and \( {f}_{B \cup S}\left( {x}_{B \cup S}\right) \) factorize and the densities sa... | Proof. Suppose that \( {f}_{V}\left( {x}_{V}\right) \) factorizes as\n\n(30)\n\n\[ {f}_{V}\left( {\mathbf{x}}_{V}\right) = \mathop{\prod }\limits_{{c \in C}}{g}_{c}\left( {\mathbf{x}}_{c}\right) \]\n\nBecause \( A, B, S \) decomposes \( G \), every clique is either a subset of \( A \cup S \) or a subset of \( B \cup S ... | Yes |
Theorem 15.3.1. A locally Markov distribution on an acyclic directed graph is pairwise Markov. | Proof. The proof is the same as the proof of Theorem 15.2.3 for undirected graphs. | No |
Theorem 15.3.2. A pairwise Markov distribution on an acyclic directed graph is locally Markov. | Proof. The proof is the same as the proof of Theorem 15.2.4. | No |
Lemma 15.3.1. A finite, partially ordered set \( \left( {V, \leq }\right) \) has at least one maximal element \( {a}^{ * } \) . | Proof of Lemma. The proof is by induction with \( {a}^{ * } = a \) if \( \# \left( V\right) = 1 \) . Assume the lemma holds for \( \# \left( V\right) = n \), and consider \( \# \left( V\right) = n + 1 \) . Then \( V = a \cup \left( {V \smallsetminus a}\right) \) for any \( a \in V \) . Since \( \# \left( {V \smallsetmi... | Yes |
Theorem 15.3.4. A distribution on an acyclic directed graph that is well-numbered Markov is locally Markov, | Proof. \( \left( {{v}_{1},\ldots ,{v}_{i - 1}}\right) \in \operatorname{nd}\left( {v}_{t}\right) \smallsetminus \operatorname{pa}\left( {v}_{t}\right) \) . | No |
Theorem 15.3.5. A distribution on an acyclic directed graph that is globally Markov is locally Markov. | Proof. For any \( v \in V \) let \( \operatorname{pa}\left( v\right) = S \) in the definition of globally Markov. Let \( v = A \) and \( \operatorname{nd}\left( v\right) \smallsetminus \operatorname{pa}\left( v\right) = B \) . A vertex \( w \in \operatorname{nd}\left( v\right) \smallsetminus \operatorname{pa}\left( v\r... | Yes |
Theorem 15.5.1. The maximum likelihood estimator of \( \sum \) in the model (3) is given by\n\n(4)\n\n\[ \n{\widehat{\sigma }}_{{t}_{j}} = {s}_{{t}_{j}},\;i = j\\text{ or }\\left( {i, j}\\right) \\in E, \n\]\n\n(5)\n\n\[ \n{\lambda }_{ij} = 0,\;i \\neq j\\text{ and }\\left( {i, j}\\right) \\notin E, \n\]\n\nwhere \( \\... | This result follows from the general theory of exponential families. See Lauritzen (1996), Theorem 5.3 and Appendix D.1. | No |
Theorem 15.5.2. Let \( L \) and \( M \) be \( p \times p \) positive definite matrices. There exists a unique positive definite matrix \( \mathbf{K} \) such that\n\n(6)\n\n\[ {k}_{ij} = {l}_{ij},\;i = j\text{ or }\left( {i, j}\right) \in E, \]\n\n(7)\n\n\[ {k}^{\prime \prime } = {m}^{\prime \prime }, \]\n\n\[ i \neq j\... | The proof of Theorem 15.5.2 depends on several lemmas. In the maximum likelihood estimation \( L = S, M = I \) or any other diagonal matrix, and \( \mathbf{K} = \widehat{\mathbf{\sum }} \) . | No |
Lemma 15.5.1. Suppose \( \mathbf{P} \) and \( \mathbf{R} \) are positive definite. Then:\n\n(i) \( I\left( {\mathbf{P} \mid \mathbf{R}}\right) > 0,\mathbf{P} \neq \mathbf{R} \), and \( I\left( {\mathbf{P} \mid \mathbf{P}}\right) = 0 \) .\n\n(ii) If \( \left\{ {P}_{n}\right\} \) and \( \left\{ {R}_{n}\right\} \) are seq... | Proof. (i) Let the roots of \( \left| {\mathbf{P} - s\mathbf{R}}\right| = 0 \) be \( {s}_{1} \leq \cdots \leq {s}_{p} \) . Then\n\n(9)\n\n\[ \log \left| {\mathbf{P}{\mathbf{R}}^{-1}}\right| + \operatorname{tr}\left( {\mathbf{I} - \mathbf{P}{\mathbf{R}}^{-1}}\right) = \mathop{\sum }\limits_{{t = 1}}^{p}\left( {\log {s}_... | Yes |
Lemma 15.5.2. Let\n\n\[ \mathbf{P} = \left\lbrack \begin{array}{ll} {\mathbf{P}}_{11} & {\mathbf{P}}_{12} \\ {\mathbf{P}}_{21} & {\mathbf{P}}_{22} \end{array}\right\rbrack ,\;\mathbf{R} = \left\lbrack \begin{array}{ll} {\mathbf{R}}_{11} & {\mathbf{R}}_{12} \\ {\mathbf{R}}_{21} & {\mathbf{R}}_{22} \end{array}\right\rbra... | Proof. (i) Let\n\n\[ {Q}^{-1} = \left\lbrack \begin{matrix} S & {R}^{21} \\ {R}^{21} & {R}^{22} \end{matrix}\right\rbrack \]\n\nThen \( I = {Q}^{-1}Q \) can be solved for \( S = {P}_{11}^{-1} + {R}^{12}{\left( {R}^{22}\right) }^{-1}{R}^{21};Q = {\left( {Q}^{-1}\right) }^{-1} \) follows from Theorem A.3.3. Then (ii) fol... | Yes |
Theorem 1. Let \( X \) be a random vector with finite mean \( \mu \) and finite covariance \( \sum \) . Then\n\n\[ \mathbb{E}\left\lbrack {{X}^{\prime }{\Lambda X}}\right\rbrack = \operatorname{tr}\left( {\Lambda \sum }\right) + {\mu }^{\prime }{\Lambda \mu } \] | Proof. Note: we use properties of the trace of a matrix that will be discussed in the next lecture.\n\n\[ \mathbb{E}\left\lbrack {{X}^{\prime }{\Lambda X}}\right\rbrack = \operatorname{tr}\left( {\mathbb{E}\left\lbrack {{X}^{\prime }{\Lambda X}}\right\rbrack }\right) = \mathbb{E}\left\lbrack {\operatorname{tr}\left( {{... | No |
Theorem 2. (Expected Euclidean distance). Suppose \( X, Y \) are in-\n\ndependent, identically distributed random variables with mean \( \mu \) and covariance \( \sum \) . Then\n\n\[ \n\mathbb{E}\left\lbrack {\parallel X - Y{\parallel }_{2}^{2}}\right\rbrack = \mathop{\sum }\limits_{j}{\sigma }_{jj} \n\] | Proof.\n\n\( \mathbb{E}\left\lbrack {\left| \right| X - Y{\left| \right| }_{2}^{2}}\right\rbrack = \mathbb{E}\left\lbrack {{\left( X - Y\right) }^{\prime }I\left( {X - Y}\right) }\right\rbrack \)\n\n\[ \n= \operatorname{tr}\left( {I\sum }\right) + {\left( \mathbb{E}\left\lbrack X - Y\right\rbrack \right) }^{\prime }I\l... | Yes |
The defining contrast (optimal) for blocking factor \( \left( b\right) \) is | \[ b = {ABCD} \] | Yes |
Example 14.3. (a) \( {AB} \) interaction. (b) \( {AE} \) interaction | gap will be effective as long as \( B = \) gas flow is at the low level. However, if \( B \) is at the high level, then \( A \) must be at the low level to achieve low uniformity. Figure \( {14.9b} \) indicates that controlling \( \mathbf{E} = \mathrm{{RF}} \) power at the low level is effective in reducing uniformity,... | No |
The defining contrast (optimal) for blocking factor \( \left( b\right) \) is | \[ b = {ABCD} \] | Yes |
Proposition 4.1. \( \left| {\operatorname{eiq}\left( G\right) }\right| = {2}^{c\left( G\right) } \) . | Proof. Let \( \dot{\sigma } = \left\langle {{\sigma }_{1},{\sigma }_{2}}\right\rangle \in \dot{\operatorname{eq}}\left( G\right) \) . Each isolated node in \( G \) is labelled with both \( {\sigma }_{1} \) and \( {\sigma }_{2} \) . Let \( c \) be a cycle in \( G \) . If the label of the root in \( c \) is \( {\sigma }_... | Yes |
Consider a matrix given by\n\n\[ A = \left( \begin{array}{llll} 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{array}\right) \] | All its random paths are listed as follows.\n\n\[ 1\; \Rightarrow \;2\; \Rightarrow \;3\; \Rightarrow \;4\;{\sigma }_{1} = \{ 1,2,3,4\} \;{K}_{{\sigma }_{1}} = 4 \]\n\n\[ 1 \Rightarrow 3 \Rightarrow \times \;{\sigma }_{2} = \{ 1,3, \sim \} \;{K}_{{\sigma }_{2}} = 0 \]\n\n\[ \begin{matrix} 4 & \Rightarrow & 2 & \Rightar... | No |
Assume a matrix\n\n\[ \nA = \left( \begin{array}{lll} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}\right) \n\] | Hence all its random paths, together with their path values, are given as follows:\n\n\[ \n\begin{array}{lll} {i}_{1} & {i}_{2} & {i}_{3} \end{array} \n\]\n\n\[ \n1 \Rightarrow 3 \Rightarrow 2,\;{\sigma }_{1} = \{ 1,3,2\} ,\;{K}_{{\sigma }_{1}} = 3 \n\]\n\n\[ \n2 \Rightarrow 1 \Rightarrow \times ,\;{\sigma }_{2} = \{ 2... | No |
Definition 2.4. Suppose that the response variable \( Y \) and at least one predictor variable \( {x}_{i} \) are quantitative. Then the multiple linear regression (MLR) model is\n\n\[ \n{Y}_{i} = {x}_{i,1}{\beta }_{1} + {x}_{i,2}{\beta }_{2} + \cdots + {x}_{i, p}{\beta }_{p} + {e}_{i} = {\mathbf{x}}_{i}^{T}\mathbf{\bet... | In matrix notation, these \( n \) equations become\n\n\[ \n\mathbf{Y} = \mathbf{X}\mathbf{\beta } + \mathbf{e} \]\n\n(2.2)\n\nwhere \( \mathbf{Y} \) is an \( n \times 1 \) vector of dependent variables, \( \mathbf{X} \) is an \( n \times p \) matrix of predictors, \( \mathbf{\beta} \) is a \( p \times 1 \) vector of un... | Yes |
Proposition 2.1. Suppose that the regression estimator \( \mathbf{b} \) of \( \mathbf{\beta } \) is used to find the residuals \( {r}_{i} \equiv {r}_{i}\left( \mathbf{b}\right) \) and the fitted values \( {\widehat{Y}}_{i} \equiv {\widehat{Y}}_{i}\left( \mathbf{b}\right) = {\mathbf{x}}_{i}^{T}\mathbf{b} \) . Then in th... | Proof. The identity line in the response plot is \( Y = {\mathbf{x}}^{T}\mathbf{b} \) . Hence the vertical deviation is \( {Y}_{i} - {\mathbf{x}}_{i}^{T}\mathbf{b} = {r}_{i}\left( \mathbf{b}\right) \) . | Yes |
Proposition 2.2. Suppose that \( \mathbf{X} \) is an \( n \times p \) matrix of full rank \( p \) . Then\n\na) \( \mathbf{H} \) is symmetric: \( \mathbf{H} = {\mathbf{H}}^{T} \) .\n\nb) \( \mathbf{H} \) is idempotent: \( \mathbf{H}\mathbf{H} = \mathbf{H} \) .\n\nc) \( {\mathbf{X}}^{T}\mathbf{r} = \mathbf{0} \) so that ... | Proof. a) \( {\mathbf{X}}^{T}\mathbf{X} \) is symmetric since \( {\left( {\mathbf{X}}^{T}\mathbf{X}\right) }^{T} = {\mathbf{X}}^{T}{\left( {\mathbf{X}}^{T}\right) }^{T} = {\mathbf{X}}^{T}\mathbf{X} \) . Hence \( {\left( {\mathbf{X}}^{T}\mathbf{X}\right) }^{-1} \) is symmetric since the inverse of a symmetric matrix is ... | Yes |
Proposition 2.3. Assume that a constant is in the MLR model. Then \( {SSTO} = {SSE} + {SSR}. \) | Proof.\n\n\[ \n{SSTO} = \mathop{\sum }\limits_{{i = 1}}^{n}{\left( {Y}_{i} - {\widehat{Y}}_{i} + {\widehat{Y}}_{i} - \bar{Y}\right) }^{2} = {SSE} + {SSR} + 2\mathop{\sum }\limits_{{i = 1}}^{n}\left( {{Y}_{i} - {\widehat{Y}}_{i}}\right) \left( {{\widehat{Y}}_{i} - \bar{Y}}\right) . \n\] \n\nHence the result follows if \... | Yes |
For the Gladstone (1905) data, the response variable \( Y = \) brain weight, \( {x}_{1} \equiv 1,{x}_{2} = \) size of head, \( {x}_{3} = \operatorname{sex},{x}_{4} = \) breadth of head, \( {x}_{5} = \) circumference of head. Assume that the response and residual plots look good and test whether at least one of the nont... | Solution: i) Ho: \( {\beta }_{2} = \cdots = {\beta }_{5} = 0 \) Ha: not Ho\nii) \( {F}_{o} = {196.24} \) from output.\niii) The pval \( = {0.0} \) from output.\niv) The pval \( < \delta \) ( \( = {0.05} \) since \( \delta \) was not given). So reject Ho. Hence there is an MLR relationship between brain weight and the p... | Yes |
The experimenter expected the response to be independent of the predictors, and 19 cases were used. However, the ANOVA \( F \) test suggested that the predictors were important. The third case was an outlier and easily detected in the response and residual plots (not shown). After deleting the outlier, the response and... | The 4 step ANOVA \( F \) test is\n\ni) Ho: \( {\beta }_{2} = \cdots = {\beta }_{4} = 0 \) Ha: not Ho\n\nii) \( {F}_{o} = {0.10} \) .\n\niii) pval \( = {0.9585} \) .\n\niv) The pval \( > \delta \) ( \( = {0.05} \) since \( \delta \) was not given). So fail to reject Ho. Hence there is not an MLR relationship between fra... | No |
Theorem 2.7. Assume that the MLR model has a constant \( {\beta }_{1} \). | a)\n\n\[ \n{F}_{o} = \frac{MSR}{MSE} = \frac{{R}^{2}}{1 - {R}^{2}}\frac{n - p}{p - 1}.\n\]\n\nb) If the errors \( {e}_{i} \) are iid \( N\left( {0,{\sigma }^{2}}\right) \), and if Ho: \( {\beta }_{2} = \cdots = {\beta }_{p} = 0 \) is true, then \( {F}_{o} \) has an \( F \) distribution with \( p - 1 \) numerator and \(... | Yes |
For the Buxton (1920) data suppose that the response \( Y \) \( = \) height and the predictors were a constant, head length, nasal height, bigo-nal breadth, and cephalic index. Five outliers were deleted leaving 82 cases. Figure 2.3 shows a response plot of the fitted values versus the response \( Y \) with the identit... | The plot was made with the following \( R \) commands, using the lregpack function piplot.\n\n\[ \mathrm{x} < - \operatorname{buxx}\left\lbrack {-\mathrm{c}\left( {{61},{62},{63},{64},{65}}\right) ,}\right\rbrack \]\n\n\[ Y < - \operatorname{buxy}\left\lbrack {-c\left( {{61},{62},{63},{64},{65}}\right) }\right\rbrack \... | No |
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