Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Theorem 3.6.3. Suppose the m-component vector \( \mathbf{Z} \) has the density \( {\left| \Phi \right| }^{-\frac{1}{2}}h\left\lbrack {{\left( z - v\right) }^{\prime }{\Phi }^{-1}\left( {z - v}\right) }\right\rbrack \), where \( {w}^{\frac{1}{2}m}h\left( w\right) \) has a finite positive maximum at \( {w}_{h} \) and \( ... | Proof. Let \( \mathbf{\Psi } = {\left| \mathbf{\Phi }\right| }^{-1/m}\mathbf{\Phi } \) and\n\n(41)\n\n\[ d = {\left( z - v\right) }^{\prime }{\Phi }^{-1}\left( {z - v}\right) = \frac{{\left( z - v\right) }^{\prime }{\Psi }^{-1}\left( {z - v}\right) }{{\left| \Phi \right| }^{1/m}}.\]\n\nThen \( \left( {\nu ,\Phi }\right... | Yes |
Theorem 3.6.4. Let \( \mathbf{X}\left( {N \times p}\right) \) have the density (28), where \( {w}^{\frac{1}{2}{Np}}g\left( w\right) \) has a finite positive maximum at \( {w}_{g} \) . Then the maximum likelihood estimators of \( \mathbf{\mu } \) and \( \mathbf{A} \) are | (44)\n\n\[ \widehat{\mathbf{\mu }} = \bar{x},\;\widehat{\mathbf{\Lambda }} = \frac{Np}{{w}_{g}}\mathbf{A}, \]\n\nwhere \( A = \mathop{\sum }\limits_{{\alpha = 1}}^{N}\left( {{x}_{\alpha } - \bar{x}}\right) {\left( {x}_{\alpha } - \bar{x}\right) }^{\prime } \) . | Yes |
Corollary 3.6.1. Let \( X\left( {N \times p}\right) \) have the density (28). Then the maximum likelihood estimators of \( \mathbf{v},\left( {{\lambda }_{11},\ldots ,{\lambda }_{pp}}\right) \), and \( {\rho }_{ij}, i, j = 1,\ldots, p \), are \( \bar{x} \) , \( \left( {p/{w}_{g}}\right) \left( {{a}_{11},\ldots ,{a}_{pp}... | Proof. Corollary 3.6.1 follows from Theorem 3.6.3 and Corollary 3.2.1. | No |
Theorem 3.6.5. Let \( f\left( X\right) \) be a vector-valued function of \( X\left( {N \times p}\right) \) such that\n\n(45)\n\n\[ f\left( {X + {\varepsilon }_{N}{v}^{\prime }}\right) = f\left( X\right) \]\nfor all \( \mathbf{v} \) and\n\n(46)\n\n\[ f\left( {cX}\right) = f\left( X\right) \]\n\nfor all \( c \) . Then th... | Proof. Substitution of the representation (27) into \( f\left( X\right) \) gives\n\n(47)\n\n\[ f\left( X\right) = f\left( {Y{C}^{\prime } + {\varepsilon }_{N}{\mu }^{\prime }}\right) = f\left( {Y{C}^{\prime }}\right) \]\n\nby (45). Let \( f\left( X\right) = h\left( {\operatorname{vec}X}\right) \) . Then by (46), \( h\l... | Yes |
Lemma 4.2.1. If \( {Y}_{1},\ldots ,{Y}_{n} \) are independently distributed, if \( {Y}_{\alpha } = \left( {{Y}_{\alpha }^{\left( 1\right) },{Y}_{\alpha }^{\left( 2\right) }{}^{\prime }}\right) \) has the density \( f\left( {y}_{\alpha }\right) \), and if the conditional density of \( {Y}_{\alpha }^{\left( 2\right) } \)... | Proof. The marginal density of \( {Y}_{1}^{\left( 1\right) }\ldots ,{Y}_{n}^{\left( 1\right) } \) is \( \mathop{\prod }\limits_{{\alpha = 1}}^{n}{f}_{1}\left( {y}_{\alpha }^{\left( 1\right) }\right) \), where \( {f}_{1}\left( {y}_{\alpha }^{\left( 1\right) }\right) \) is the marginal density of \( {\mathbf{Y}}_{\alpha ... | Yes |
Theorem 4.2.1. Let \( {\mathbf{X}}_{1},\ldots ,{\mathbf{X}}_{N} \) be independent, each with distribution \( N\left( {\mathbf{\mu },\mathbf{\sum }}\right) \) . If \( {\rho }_{ij} = 0 \), the density of \( {r}_{ij} \) defined by (1) is (17). | From (17) we see that the density is symmetric about the origin. For \( N > 4 \), it has a mode at \( r = 0 \) and its order of contact with the \( r \) -axis at \( \pm 1 \) is \( \frac{1}{2}\left( {N - 5}\right) \) for \( N \) odd and \( \frac{1}{2}N - 3 \) for \( N \) even. Since the density is even, the odd moments ... | No |
Theorem 4.2.3. Let \( \{ U\left( n\right) \} \) be a sequence of \( m \) -component random vectors and \( \mathbf{b} \) a fixed vector such that \( \sqrt{n}\left\lbrack {U\left( n\right) - \mathbf{b}}\right\rbrack \) has the limiting distribution \( N\left( {\mathbf{0},\mathbf{T}}\right) \) as \( n \rightarrow \infty \... | Proof. See Serfling (1980), Section 3.3, or Rao (1973), Section 6a.2. A function \( g\left( u\right) \) is said to have a differential at \( b \) or to be totally differentiable at \( \mathbf{b} \) if the partial derivatives \( \partial g\left( u\right) /\partial u \), exist at \( u = \mathbf{b} \) and for every \( \ma... | No |
Theorem 4.2.5. Let \( z \) be defined by (65), where \( r \) is the correlation coefficient of a sample of \( N\left( { = n + 1}\right) \) from a bivariate normal distribution with correlation \( \rho \) ; let \( \zeta \) be defined by (66). Then \( \sqrt{n}\left( {z - \zeta }\right) \) has a limiting normal distributi... | It can be shown that to a closer approximation\n\n(67)\n\n\[ \mathcal{E}z \sim \zeta + \frac{\rho }{2n}. \]\n\n(68)\n\n\[ \mathcal{E}{\left( z - \zeta \right) }^{2} \sim \frac{1}{n - 2} \sim \mathcal{E}{\left( z - \zeta - \frac{\rho }{2n}\right) }^{2}. \]\n\nThe latter follows from\n\n(69)\n\n\[ \mathcal{E}{\left( z - ... | Yes |
Theorem 4.3.1. Let \( {x}_{1},\ldots ,{x}_{N} \) be a sample from \( N\left( {\mathbf{\mu },\sum }\right) \), where \( \mathbf{\mu } \) and \( \sum \) are partitioned as in (1). Define \( A \) by (4) and \( \left( {{\overrightarrow{x}}^{\left( 1\right) \prime }{\overrightarrow{x}}^{\left( 2\right) \prime }}\right) = \)... | (7)\n\n\[ \widehat{\mathbf{B}} = {A}_{12}{A}_{22}^{-1},\;{\widehat{\sum }}_{{11} \cdot 2} = \frac{1}{N}\left( {{A}_{11} - {A}_{12}{A}_{22}^{-1}{A}_{21}}\right) ,\n\]\nand \( {\widehat{\mathbf{\sum }}}_{22} = \left( {1/N}\right) {A}_{22} \), respectively. | Yes |
Theorem 4.3.2. Let \( {x}_{1},\ldots ,{x}_{N} \) be a sample of \( N \) from \( N\left( {\mu ,\sum }\right) \) . The maximum likelihood estimators of \( {\rho }_{{ij} \cdot q + 1,\ldots .p} \), the partial correlations of the first \( q \) components conditional on the last \( p - q \) components, are given by | \[ {\widehat{\rho }}_{{ij} \cdot q + 1\ldots \ldots p} = \frac{{a}_{{ij} \cdot q + 1\ldots ., p}}{\sqrt{{a}_{{ii} \cdot q + 1,\ldots, p}{a}_{{jj} \cdot q + 1,\ldots, p}}},\;i, j = 1,\ldots, q, \] where \[ \left( {a}_{1) \cdot q + 1.\;.p}\right) = {A}_{11} - {A}_{12}{A}_{22}^{-1}{A}_{21} = {A}_{{11} \cdot 2}. \] | Yes |
Theorem 4.3.3. Suppose \( {\mathbf{Y}}_{1},\ldots ,{\mathbf{Y}}_{m} \) are independent with \( {\mathbf{Y}}_{a} \) distributed according to \( N\left( {\Gamma {w}_{\alpha },\mathbf{\Phi }}\right) \), where \( {w}_{\alpha } \) is an r-component vector. Let \( \mathbf{H} = \mathop{\sum }\limits_{{\alpha = 1}}^{m}{\mathbf... | Proof. The rows of \( Y = \left( {{Y}_{1},\ldots ,{Y}_{m}}\right) \) are random vectors in an \( m \) -dimensional space, and the rows of \( W = \left( {{w}_{1},\ldots ,{w}_{m}}\right) \) are fixed vectors in that space. The idea of the proof is to rotate coordinate axes so that the last \( r \) axes are in the space s... | Yes |
Theorem 4.3.5. If the cdf of \( {r}_{ij} \) based on a sample of \( N \) from a normal distribution with correlation \( {\rho }_{ij} \) is denoted by \( F\left( {r \mid N,{\rho }_{ij}}\right) \), then the cdf of the sample partial correlation \( {r}_{{ijq} + 1,\ldots p} \) based on a sample of \( N \) from a normal dis... | This distribution was derived by Fisher (1924). | No |
Suppose that on the basis of a sample of size \( N \) we wish to obtain a confidence interval for \( {\rho }_{{ij} \cdot q + 1,\ldots, p} \) . The sample partial correlation is \( {r}_{{ij} \cdot q + 1,\ldots, p} \) . | The procedure is to use David’s charts for \( N - \left( {p - q}\right) \) . In the example at the end of Section 4.3.1, we might want to find a confidence interval for \( {\rho }_{{12} \cdot 3} \) with confidence coefficient 0.95 . The sample partial correlation is \( {r}_{{12} \cdot 3} = {0.759} \) . We use the chart... | Yes |
Theorem 4.4.1. The residuals \( {x}_{1\alpha }^{ * } \) are uncorrelated in the sample with the components of \( {\mathbf{x}}_{\alpha }^{\left( 2\right) },\alpha = 1,\ldots, N \) . For every vector \( \mathbf{a} \n\n\[ \text{(7)}\mathop{\sum }\limits_{{\alpha = 1}}^{N}{\left\lbrack {x}_{1\alpha } - {\bar{x}}_{1} - {\wi... | Proof. Since the sample mean of the residuals is 0 , the vector of sample covariances between \( {x}_{1\alpha }^{ * } \) and \( {x}_{\alpha }^{\left( 2\right) } \) is proportional to\n\n(8)\n\n\[ \mathop{\sum }\limits_{{\alpha = 1}}^{N}\left\lbrack {\left( {{x}_{1\alpha } - {\bar{x}}_{1}}\right) - {\widehat{\mathbf{\be... | Yes |
Theorem 4.4.4. Let \( R \) be the sample multiple correlation coefficient between \( {X}_{\left( 1\right) } \) and \( {\mathbf{X}}^{\left( 2\right) }{}^{\prime } = \left( {{X}_{2},\ldots ,{X}_{p}}\right) \) based on \( N \) observations \( \left( {{x}_{11},{x}_{1}^{\left( 2\right) }}\right) ,\ldots ,\left( {{x}_{1N},{x... | The conditional density (from Theorem 5.4.1) of \( F = \left\lbrack {{R}^{2}/\left( {1 - {R}^{2}}\right) }\right\rbrack \lbrack (N - \) \( p)/\left( {p - 1}\right) \rbrack \) is\n\n(37)\n\n\[ \frac{\left( {p - 1}\right) \exp \left\lbrack {-\frac{1}{2}{\beta }^{\prime }{A}_{22}\beta /{\sigma }_{{11} \cdot 2}}\right\rbra... | Yes |
Theorem 4.4.5. The density of the square of the multiple correlation coefficient, \( {R}^{2} \), between \( {X}_{1} \) and \( {X}_{2},\ldots ,{X}_{p} \) based on a sample of \( N = n + 1 \) is given by (42) or (43) [or (45) in the case of \( n - p + 1 \) even], where \( {\bar{R}}^{2} \) is the corresponding population ... | The moments of \( R \) are\n\n(46)\n\n\[ \mathcal{E}{R}^{h} = \frac{{\left( 1 - {\bar{R}}^{2}\right) }^{\frac{1}{2}n}}{\Gamma \left\lbrack {\frac{1}{2}\left( {n - p + 1}\right) }\right\rbrack \Gamma \left( {\frac{1}{2}n}\right) }\mathop{\sum }\limits_{{\mu = 0}}^{\infty }\frac{{\left( {\bar{R}}^{2}\right) }^{\mu }{\Gam... | Yes |
Theorem 4.4.7. On the basis of observations \( {x}_{1},\ldots ,{x}_{N} \) from \( N\left( {\mu ,\sum }\right) \), of all tests of \( \bar{R} = 0 \) at a given significance level with power depending only on \( \bar{R} \), the test with critical region given by \( R \) greater than a constant is uniformly most powerful. | Theorem 4.4.7 follows from Theorem 4.4.6 in the same way that Theorem 5.6.4 follows from Theorem 5.6.1. | No |
Theorem 4.5.1. Let \( f\left( s\right) \) be a vector-valued function such that each component of \( f\left( s\right) \) has a nonzero differential at \( s = \mathbf{\sigma } \) . Suppose \( S \) is the covariance of a sample from (1) such that \( \mathcal{E}{R}^{4} < \infty \) . Then | \[ \sqrt{N}\left\lbrack {f\left( s\right) - f\left( \mathbf{\sigma }\right) }\right\rbrack = \frac{\partial f\left( \mathbf{\sigma }\right) }{\partial {\mathbf{\sigma }}^{\prime }}\sqrt{N}\left( {s - \mathbf{\sigma }}\right) + {o}_{p}\left( 1\right) \] \[ \overset{d}{ \rightarrow }N\left\{ {\mathbf{0},\frac{\partial f\... | Yes |
Corollary 4.5.1. If\n\n(4)\n\n\[ f\left( {cs}\right) = f\left( s\right) \]\n\nfor all \( c > 0 \) and all positive definite \( S \) and the conditions of Theorem 4.5.1 hold, then\n\n(5)\n\n\[ \sqrt{N}\left\lbrack {f\left( s\right) - f\left( \mathbf{\sigma }\right) }\right\rbrack \overset{d}{ \rightarrow }N\left\lbrack ... | Proof. From (4) we deduce\n\n(6)\n\n\[ 0 = \frac{\partial f\left( {cs}\right) }{\partial c} = \frac{\partial f\left( {cs}\right) }{\partial {s}^{\prime }}\frac{\partial \left( {cs}\right) }{\partial c} = \frac{\partial f\left( {cs}\right) }{\partial {s}^{\prime }}s. \]\n\nThat is,\n\n(7)\n\n\[ \frac{\partial f\left( \m... | Yes |
Corollary 4.5.2. Under the conditions of Theorem 4.5.1,\n\n\[ \sqrt{\frac{N}{1 + \widehat{\kappa }}}\frac{\left( {r}_{ij} - {\rho }_{ij}\right) }{\sqrt{1 - {r}_{ij}^{2}}}\overset{d}{ \rightarrow }N\left( {0,1}\right) \] | As in the case of the observations normally distributed,\n\n\[ \sqrt{\frac{N}{1 + \widehat{\kappa }}}\left( {\frac{1}{2}\log \frac{1 + {r}_{ij}}{1 - {r}_{ij}} - \frac{1}{2}\log \frac{1 + {\rho }_{ij}}{1 - {\rho }_{ij}}}\right) \overset{d}{ \rightarrow }N\left( {0,1}\right) . \] | No |
Theorem 4.5.3. When \( X \) has the vector elliptical density (16), the distributions of \( {r}_{n},{r}_{{nj} + 1} \), and \( {R}^{2} \) are the distributions derived for normally distributed observations. | It follows from Theorem 4.5.3 that the asymptotic distributions of \( {r}_{ij} \) . \( {r}_{{ij}, q + 1,\ldots, p} \), and \( {R}^{2} \) are the same as for sampling from normal distributions. | No |
Lemma 4.5.1. Let \( V = \left( {{v}_{1},\ldots ,{v}_{p}}\right) \), where \( {v}_{i} \) is an \( N \) -component vector. \( i = 1,\ldots, p \) . Define recursively \( {w}_{1} = {v}_{1} \) ,\n\n\[{\mathbf{w}}_{i} = {\mathbf{v}}_{i} - \mathop{\sum }\limits_{{j = 1}}^{{i - 1}}\frac{{\mathbf{v}}_{i}^{\prime }{\mathbf{w}}_{... | The proof of the lemma is given in the first part of Section 7.2 and as the Gram-Schmidt orthogonalization in the Appendix (Section A.5.1). This lemma generalizes the construction in Section 3.2; see Figure 3.1. See also Figure 7.1. | No |
Theorem 4.5.5. Let \( f\left( X\right) \) be a vector-valued function of \( X\left( {N \times p}\right) \) such that\n\n(23)\n\n\[ f\left( {X + {\varepsilon }_{N}{v}^{\prime }}\right) = f\left( X\right) \]\n\nfor all \( v \) and\n\n(24)\n\n\[ f\left( {X{G}^{\prime }}\right) = f\left( X\right) \]\n\nfor all \( G\left( {... | Proof. From (23) we find that \( f\left( X\right) = f\left( {Y{C}^{\prime }}\right) \), and from (24) we find \( f\left( {Y{C}^{\prime }}\right) = f\left( {U{T}^{\prime }{C}^{\prime }}\right) = f\left( U\right) \), which is the same for arbitrary and normal densities (18). | Yes |
Corollary 4.5.5. Let \( f\left( X\right) \) be a vector-valued function of \( X\left( {N \times p}\right) \) with the density (18), where \( \mathbf{v} = \mathbf{0} \) . Suppose (24) holds for all \( G\left( {p \times p}\right) \) . Then the distribution of \( f\left( X\right) \) for an arbitrary density (18) is the sa... | The condition (24) of Corollary 4.5.5 is that \( f\left( X\right) \) is invariant with respect to linear transformations \( X \rightarrow {XG} \) . | No |
Lemma 5.2.1. For any \( p \times p \) nonsingular matrices \( C \) and \( H \) and any vector \( \mathbf{k} \) ,\n\n\[{\mathbf{k}}^{\prime }{\mathbf{H}}^{-1}\mathbf{k} = {\left( \mathbf{C}\mathbf{k}\right) }^{\prime }{\left( {\mathbf{{CHC}}}^{\prime }\right) }^{-1}\left( \mathbf{{Ck}}\right) .\] | Proof. The right-hand side of (15) is\n\n\[{\left( Ck\right) }^{\prime }{\left( CH{C}^{\prime }\right) }^{-1}\left( {Ck}\right) = {k}^{\prime }{C}^{\prime }{\left( {C}^{\prime }\right) }^{-1}{H}^{-1}{C}^{-1}{Ck}\]\n\n\[= {k}^{\prime }{H}^{-1}k.\] | Yes |
Corollary 5.2.1. Let \( {x}_{1},\ldots ,{x}_{N} \) be a sample from \( N\left( {\mathbf{\mu },\mathbf{\sum }}\right) \), and let \( {T}^{2} = \) \( N{\left( \bar{x} - {\mathbf{\mu }}_{0}\right) }^{\prime }{S}^{-1}\left( {\bar{x} - {\mathbf{\mu }}_{0}}\right) \) . The distribution of \( \left\lbrack {{T}^{2}/\left( {N -... | The above derivation of the \( {T}^{2} \) -distribution is due to Bowker (1960). The noncentral \( F \) -density and tables of the distribution are discussed in Section 5.4. | Yes |
Theorem 5.2.3. Let \( \\left\\{ {X}_{\\alpha }\\right\\} ,\\alpha = 1,2,\\ldots \\), be a sequence of independently identically distributed random vectors with mean vector \( \\mathbf{\\mu } \) and covariance matrix \( \\mathbf{\\sum } \) ; let \( {\\widetilde{\\mathbf{X}}}_{N} = \\left( {1/N}\\right) \\mathop{\\sum }\... | Proof. By the central limit theorem (Theorem 4.2.3) the limiting distribution of \( \\sqrt{N}\\left( {{\\overline{\\mathbf{X}}}_{N} - \\mathbf{\\mu }}\\right) \) is \( N\\left( {\\mathbf{0},\\mathbf{\\sum }}\\right) \). The sample covariance matrix converges stochastically to \( \\mathbf{\\sum } \). Then the limiting d... | Yes |
Lemma 5.3.1. If \( \mathbf{v} \) is a vector of \( p \) components and if \( \mathbf{B} \) is a nonsingular \( p \times p \) matrix, then \( {\mathbf{v}}^{\prime }{\mathbf{B}}^{-1}\mathbf{v} \) is the nonzero root of\n\n\[ \left| {\mathbf{v}{\mathbf{v}}^{\prime } - \lambda \mathbf{B}}\right| = 0 \] | Proof. The nonzero root, say \( {\lambda }_{1} \), of (4) is associated with a characteristic vector \( \beta \) satisfying\n\n\[ \nu {\nu }^{\prime }\beta = {\lambda }_{1}{B\beta }.\]\n\nSince \( {\lambda }_{1} \neq 0,{\mathbf{v}}^{\prime }\mathbf{B} \neq \mathbf{0} \) . Multiplying on the left by \( {\mathbf{v}}^{\pr... | Yes |
Lemma 5.3.2 (Generalized Cauchy-Schwarz Inequality). For a positive definite matrix \( S \) , \n\n\[ \n{\left( {\gamma }^{\prime }y\right) }^{2} \leq {\gamma }^{\prime }{S\gamma }{y}^{\prime }{S}^{-1}y \n\] | Proof. Let \( b = {\gamma }^{\prime }y/{\gamma }^{\prime }{S\gamma } \) . Then \n\n\[ \n0 \leq {\left( y - bS\gamma \right) }^{\prime }{S}^{-1}\left( {y - {bS\gamma }}\right) \n\] \n\n\[ \n= {y}^{\prime }{S}^{-1}y - b{\gamma }^{\prime }S{S}^{-1}y - {y}^{\prime }{S}^{-1}{S\gamma b} + {b}^{2}{\gamma }^{\prime }S{S}^{-1}{... | Yes |
Corollary 5.3.1. The estimator for \( p \geq 3 \)\n\n(39)\n\n\[{\left( 1 - \frac{a}{N{\left( \bar{x} - \mathbf{v}\right) }^{\prime }{A}^{-1}\left( {\bar{x} - \mathbf{v}}\right) }\right) }^{ + }\left( {\bar{x} - \mathbf{v}}\right) + \mathbf{v}\]\n\nhas smaller risk than (36) and is minimax for \( 0 < a < 2\left( {p - 2}... | Proof. This corollary follows from Theorem 5.3.1 and Lemma 3.5.2. | No |
Given the observations \( {x}_{1},\ldots ,{x}_{N} \) from \( N\left( {\mathbf{\mu },\sum }\right) \), of all tests of \( \mathbf{\mu } = \mathbf{0} \) based on \( \widetilde{x} \) and \( A = \sum \left( {{x}_{\alpha } - \bar{x}}\right) {\left( {x}_{\alpha } - \bar{x}\right) }^{\prime } \) that are invariant with respec... | Proof. First, as we have seen in Section 5.2.1, any test based on \( {T}^{2} \) is invariant. Second, this function is essentially the only invariant, for if \( f\left( {\widetilde{\mathbf{x}}, A}\right) \) is invariant, then \( f\left( {\widetilde{x}, A}\right) = f\left( {{\bar{x}}^{ * }, I}\right) \), where only the ... | Yes |
Theorem 5.6.2. On the basis of observations \( {x}_{1},\ldots ,{x}_{N} \) from \( N\left( {\mathbf{\mu },\mathbf{\sum }}\right) \), of all tests of \( \mathbf{\mu } = \mathbf{0} \) that are invariant with respect to transformations \( {x}_{\alpha }^{ * } = C{x}_{\alpha } \) (C nonsingular), the \( {T}^{2} \) -test is a... | Proof. Let \( \psi \left( {{x}_{1},\ldots ,{x}_{N}}\right) \) be the critical function of an invariant test. Then\n\n(2)\n\n\[ \mathcal{E}\left\lbrack {\psi \left( {{x}_{1},\ldots ,{x}_{N}}\right) }\right\rbrack = {\mathcal{E}}_{\bar{x}, A}\left\{ {\mathcal{E}\left\lbrack {\psi \left( {{x}_{1},\ldots ,{x}_{N}}\right) \... | Yes |
Theorem 5.6.3. Given observations \( {x}_{1},\ldots ,{x}_{N} \) from \( N\left( {\mathbf{\mu },\mathbf{\sum }}\right) \), of all tests of \( \mathbf{\mu } = \mathbf{0} \) based on \( \widetilde{x} \) and \( A = \sum \left( {{x}_{a} - \bar{x}}\right) {\left( {x}_{a} - \bar{x}\right) }^{\prime } \) with power depending o... | Proof. We wish to reduce this theorem to Theorem 5.6.1 by identifying the class of tests with power depending on \( N{\mathbf{\mu }}^{\prime }{\mathbf{\sum }}^{-1}\mathbf{\mu } \) with the class of invariant tests. We need the following definition:\n\nDefinition 5.6.2. A test \( \psi \left( {{x}_{1},\ldots ,{x}_{N}}\ri... | Yes |
Theorem 5.6.4. On the basis of observations \( {x}_{1},\ldots ,{x}_{N} \) from \( N\left( {\mathbf{\mu },\sum }\right) \) . of all tests of \( \mathbf{\mu } = \mathbf{0} \) with power depending only on \( N{\mathbf{\mu }}^{\prime }{\mathbf{\sum }}^{-1}\mathbf{\mu } \), the \( {T}^{2} \) -test is a uniformly most powerf... | Theorem 5.6.4 was first proved by Simaika (1941). The results and proofs given in this section follow Lehmann (1959). Hsu (1945) has proved an optimal property of the \( {T}^{2} \) -test that involves averaging the power over \( \mathbf{\mu } \) and \( \mathbf{\sum } \) . | Yes |
Theorem 5.6.5 (Stein). Let \( \left( {\mathcal{V},\mathcal{B}, m,\Omega, P}\right) \) be an exponential family and \( {\Omega }_{0} \) a nonempty proper subset of \( \Omega \) . (i) Let \( A \) be a subset of \( \mathcal{V} \) that is closed and convex. (ii) Suppose that for every vector \( \omega \in {\mathcal{Y}}^{\p... | Proof. The critical function of the test with acceptance region \( A \) is \( {\phi }_{A}\left( y\right) = 0.y \in A \), and \( {\phi }_{A}\left( y\right) = 1, y \notin A \) . Suppose \( \phi \left( y\right) \) is the critical function of a better fest, that is,\n\n(10)\n\n\[ \int \phi \left( y\right) d{P}_{\omega }\le... | Yes |
Corollary 5.6.2. If the conditions of Theorem 5.6.5 hold except that \( A \) is not necessarily closed, but the boundary of \( A \) has m-measure 0, then the conclusion of Theorem 5.6.5 holds. | Proof. The closure of \( A \) is convex (Problem 5.18), and the test with acceptance region equal to the closure of \( A \) differs from \( A \) by a set of probability 0 for all \( \omega \in \Omega \) . Furthermore,\n\n(15)\n\n\[ A \cap \left\{ {y \mid {\omega }^{\prime }y > c}\right\} = \varnothing \; \Rightarrow \;... | No |
Theorem 5.6.6. Based on observations \( {x}_{1},\ldots ,{x}_{N} \) from \( N\left( {\mathbf{\mu },\sum }\right) \) , Hotelling’s \( {T}^{2} \) -test is admissible for testing the hypothesis \( \mathbf{\mu } = \mathbf{0} \) . | Proof. To apply Theorem 5.6.5 we put the distribution of the observations into the form of an exponential family. By Theorems 3.3.1 and 3.3.2 we can transform \( {x}_{1},\ldots ,{x}_{N} \) to \( {z}_{\alpha } = \mathop{\sum }\limits_{{\beta = 1}}^{N}{c}_{\alpha \beta }{x}_{\beta } \), where \( \left( {c}_{\alpha \beta ... | Yes |
Lemma 5.6.1. Let \( B = A + N\bar{x}{\bar{x}}^{\prime } \) . Then\n\n\[ N{\bar{x}}^{\prime }{A}^{-1}\bar{x} = \frac{N{\bar{x}}^{\prime }{B}^{-1}\bar{x}}{1 - N{\bar{x}}^{\prime }{B}^{-1}\bar{x}}. \] | Proof of Lemma. If we let \( \mathbf{B} = \mathbf{A} + \sqrt{N}\widetilde{\mathbf{x}}\sqrt{N}{\widetilde{\mathbf{x}}}^{\prime } \) in (10) of Section 5.2, we obtain by Corollary A.3.1\n\n\[ \frac{1}{1 + {T}^{2}/\left( {N - 1}\right) } = {\lambda }^{2/N} = \frac{\left| B - \sqrt{N}\bar{x}\sqrt{N}{\bar{x}}^{\prime }\righ... | No |
Theorem 5.7.1. Let \( {x}_{1},\ldots ,{x}_{N} \) be a sample from (1). Assume \( \mathcal{E}{R}^{2} < \infty \) . Then \( {T}^{2}\overset{d}{ \rightarrow }{\chi }_{p}^{2} \) . | Proof. Theorem 3.6.2 implies that \( N{\left( \bar{x} - \mathbf{\mu }\right) }^{\prime }{\sum }^{-1}\left( {\bar{x} - \mathbf{\mu }}\right) \overset{d}{ \rightarrow }{\chi }_{p}^{2} \) and \( N(\bar{x} \) \( - \mathbf{\mu }{)}^{\prime }{\sum }^{-1}\left( {\bar{x} - \mathbf{\mu }}\right) - {T}^{2}\overset{p}{ \rightarro... | Yes |
Theorem 5.7.2. Suppose \( X \) has the density (3) with \( \mathbf{v} = \mathbf{0} \) and \( {T}^{2} = \) \( N{\bar{x}}^{\prime }{S}^{-1}\bar{x} \) . Then \( \left\lbrack {{T}^{2}/\left( {N - 1}\right) }\right\rbrack \left\lbrack {\left( {N - p}\right) /p}\right\rbrack \) has the distribution of \( {F}_{p, N - p} = \) ... | Thus the tests of hypotheses and construction of confidence regions at stated significance and confidence levels are valid for left spherical distributions.\n\nThe \( {T}^{2} \) -criterion for \( H : \mathbf{v} = \mathbf{0} \) is\n\n(4)\n\n\[ {T}^{2} = N{\bar{x}}^{\prime }{S}^{-1}\bar{x}\overset{d}{ = }N{\bar{u}}^{\pri... | No |
Theorem 6.3.1. If \( {q}_{1} \) and \( {q}_{2} \) are a priori probabilities of drawing an observation from population \( {\pi }_{1} \) with density \( {p}_{1}\left( x\right) \) and \( {\pi }_{2} \) with density \( {p}_{2}\left( x\right) \) , respectively, and if the cost of misclassifying an observation from \( {\pi }... | \[ \Pr \left\{ {\left. {\frac{{p}_{1}\left( x\right) }{{p}_{2}\left( x\right) } = \frac{{q}_{2}C\left( {1 \mid 2}\right) }{{q}_{1}C\left( {2 \mid 1}\right) }}\right| \;{\pi }_{t}}\right\} = 0, \] \( i = 1,2 \) , then the procedure is unique except for sets of probability zero. | No |
Theorem 6.3.3. If (14) holds, then every admissible procedure is a Bayes procedure. | The proof of Theorem 6.3.3 shows that the class of Bayes procedures is complete. For if \( R \) is any procedure outside the class, we construct a Bayes procedure \( {R}^{ * } \) so that \( P\left( {2 \mid 1, R}\right) = P\left( {2 \mid 1,{R}^{ * }}\right) \) . Then, since \( {R}^{ * } \) is admissible, \( P\left( {1 \... | Yes |
Theorem 6.3.4. If (14) holds, the class of Bayes procedures is minimal complete. | Finally, let us consider the minimax procedure. Let \( P\left( {i \mid j,{q}_{1}}\right) = P\left( {i \mid j, R}\right) \) , where \( R \) is the Bayes procedure corresponding to \( {q}_{1}.P\left( {i \mid j,{q}_{1}}\right) \) is a continuous function of \( {q}_{1}.P\left( {2 \mid 1,{q}_{1}}\right) \) varies from 1 to ... | Yes |
Theorem 6.4.1. If \( {\pi }_{i} \) has the density (1), \( i = 1,2 \), the best regions of classification are given by\n\n\[ \n{R}_{1} : {x}^{\prime }{\mathbf{\sum }}^{-1}\left( {{\mathbf{\mu }}^{\left( 1\right) } - {\mathbf{\mu }}^{\left( 2\right) }}\right) - \frac{1}{2}{\left( {\mathbf{\mu }}^{\left( 1\right) } + {\m... | If a priori probabilities \( {q}_{1} \) and \( {q}_{2} \) are known, then \( k \) is given by\n\n\[ \nk = \frac{{q}_{2}C\left( {1 \mid 2}\right) }{{q}_{1}C\left( {2 \mid 1}\right) }. \n\]\n\nIn the particular case of the two populations being equally likely and the costs being equal, \( k = 1 \) and \( \log k = 0 \) . ... | Yes |
Theorem 6.6.1. As \( {N}_{1} \rightarrow \infty ,{N}_{2} \rightarrow \infty \), and \( {N}_{1}/{N}_{2} \rightarrow \) a positive limit \( (n = \) \( \left. {{N}_{1} + {N}_{2} - 2}\right) \) | (1) \( \Pr \left\{ {\left. {\frac{W - \frac{1}{2}{\Delta }^{2}}{\Delta } \leq u}\right| \;{\pi }_{1}}\right\} \n\n\[ \n= \Phi \left( u\right) - \phi \left( u\right) \left\{ {\frac{1}{2{N}_{1}{\Delta }^{2}}\left\lbrack {{u}^{3} + \left( {p - 3}\right) u - {p\Delta }}\right\rbrack }\right. \n\] \n\n\[ \n+ \frac{1}{2{N}_{... | Yes |
[
\\Pr \\left\\{ {W \\leq 0 \\mid {\\pi }_{1},\\mathop{\\lim }\\limits_{{n \\rightarrow \\infty }}\\frac{{N}_{1}}{{N}_{2}} = 1}\\right\\}
] | [
= \\Phi \\left( {-\\frac{1}{2}\\Delta }\\right) + \\frac{1}{n}\\phi \\left( {\\frac{1}{2}\\Delta }\\right) \\left\\lbrack {\\frac{p - 1}{\\Delta } + \\frac{p}{4}\\Delta }\\right\\rbrack + o\\left( {n}^{-1}\\right)
]
[
= \\Pr \\left\\{ {W \\geq 0 \\mid {\\pi }_{2},\\mathop{\\lim }\\limi... | Yes |
Theorem 6.6.2. If \( {N}_{1}/{N}_{2} \rightarrow \) a positive limit as \( n \rightarrow \infty \) | \[ \Pr \left\{ {\left. {\frac{W - \frac{1}{2}{D}^{2}}{D} \leq u}\right| \;{\pi }_{1}}\right\} = \Phi \left( u\right) - \phi \left( u\right) \left\{ {\frac{1}{{N}_{\mathrm{I}}}\left( {\frac{u}{2} - \frac{p - 1}{\Delta }}\right) + \frac{1}{n}\left\lbrack {\frac{{u}^{3}}{4} + \left( {p - \frac{3}{4}}\right) u}\right\rbrac... | Yes |
Theorem 6.6.4. As \( {N}_{1} \rightarrow \infty ,{N}_{2} \rightarrow \infty \), and \( {N}_{1}/{N}_{2} \rightarrow \) a positive limit,\n\n(15)\n\n\[ \Pr \left\{ {\sqrt{n}\frac{P\left( {2 \mid 1, D{u}_{1} + \frac{1}{2}{D}^{2},{\bar{x}}^{\left( 1\right) },{\bar{x}}^{\left( 2\right) }, S}\right) - \Phi \left( {u}_{1}\rig... | \[ = \Phi \left\lbrack {x - \frac{\left( {p - 1}\right) n/{N}_{1} - \left( {p - \frac{3}{4} + n/{N}_{1}}\right) {u}_{1} - {u}_{1}^{3}/4}{\sqrt{n}{\left\lbrack \frac{1}{2}{u}_{1}^{2} + n/{N}_{1}\right\rbrack }^{\frac{1}{2}}}}\right\rbrack + O\left( {n}^{-2}\right) . \] | Yes |
Theorem 6.6.5. As \( {N}_{1} \rightarrow \infty ,{N}_{2} \rightarrow \infty \), and \( {N}_{1}/{N}_{2} \) approaches a positive limit,\n\n(16)\n\n\[ \Pr \left\{ {\left. {\frac{Z - \frac{1}{2}{\Delta }^{2}}{\Delta } \leq u}\right| \;{\pi }_{1}}\right\} \] | \[ = \Phi \left( u\right) - \phi \left( u\right) \left\{ {\frac{1}{2{N}_{1}{\Delta }^{2}}\left\lbrack {{u}^{3} + \Delta {u}^{2} + \left( {p - 3}\right) u - \Delta }\right\rbrack }\right. \] \[ + \frac{1}{2{N}_{2}{\Delta }^{2}}\left\lbrack {{u}^{3} + \Delta {u}^{2} + \left( {p - 3 - {\Delta }^{2}}\right) u - {\Delta }^{... | Yes |
\[ \Pr \left\{ {\left. {\frac{Z - \frac{1}{2}{D}^{2}}{D} \leq u}\right| \;{\pi }_{1}}\right\} \] | \[ = \Phi \left( u\right) - \phi \left( u\right) \left\{ {\frac{1}{2{N}_{1}\Delta }\left\lbrack {{u}^{2} + {\Delta u} - \left( {p - 1}\right) }\right\rbrack }\right. \]\n\[ - \frac{1}{2{N}_{2}\Delta }\left\lbrack {{u}^{2} + {2\Delta u} + p - 1 + {\Delta }^{2}}\right\rbrack \]\n\[ \left. {+\frac{1}{4n}\left\lbrack {{u}^... | Yes |
Theorem 6.6.8. As \( {N}_{1} \rightarrow \infty ,{N}_{2} \rightarrow \infty \), and \( {N}_{1}/{N}_{2} \rightarrow \) a positive limit, | \[ \Pr \left\{ {2\sqrt{\frac{{N}_{1}{N}_{2}}{{N}_{1} + {N}_{2}}}\left| {\;\frac{P\left( {2 \mid 1,0,{\bar{x}}^{\left( 1\right) },{\bar{x}}^{\left( 2\right) }, S}\right) - \Phi \left( {-\frac{1}{2}\Delta }\right) }{\phi \left( {\frac{1}{2}\Delta }\right) } \leq x}\right. }\right\} \]\n\[ = \Phi \left\lbrack {x - 2\sqrt{... | Yes |
Theorem 6.7.1. If \( q \), is the a priori probability of drawing an observation from population \( \pi \), with density \( {p}_{1}\left( x\right), i = 1,\ldots, m \), and if the cost of misclassifying an observation from \( {\pi }_{i} \) as from \( {\pi }_{j} \) is \( C\left( {j \mid i}\right) \), then the regions of ... | Proof. We now verify this result. Let\n\n(7)\n\n\[ {h}_{j}\left( x\right) = \mathop{\sum }\limits_{\substack{{i = 1} \\ {i \neq j} }}^{m}{q}_{i}{p}_{i}\left( x\right) C\left( {j \mid i}\right) .\n\nThen the expected loss of a procedure \( R \) is\n\n(8)\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{m}{\int }_{{R}_{j}}{h}_{j}\... | Yes |
Theorem 6.7.2. If \( {q}_{i} > 0, i = 1,\ldots, m \), then a Bayes procedure is admissible. | We shall now assume that \( C\left( {i \mid j}\right) = 1, i \neq j \), and \( \Pr \left\{ {{p}_{i}\left( x\right) = 0 \mid {\pi }_{j}}\right\} = 0 \) . The latter condition implies that all \( {p}_{i}\left( x\right) \) are positive on the same set (except fcr a set of measure 0). Suppose \( {q}_{i} = 0 \) for \( i = 1... | No |
Theorem 6.8.1. If \( {q}_{t} \) is the a priori probability of drawing an observation from \( {\pi }_{t} = N\left( {{\mathbf{\mu }}^{\left( i\right) },\mathbf{\sum }}\right), i = 1,\ldots, m \), and if the costs of misclassification are equal, then the regions of classification, \( {R}_{1},\ldots ,{R}_{m} \), that mini... | It should be noted that each \( {u}_{jk}\left( x\right) \) is the classification function related to the \( j \) th and \( k \) th populations, and \( {u}_{jk}\left( x\right) = - {u}_{kj}\left( x\right) \) . Since these are linear functions, the region \( {R}_{i} \) is bounded by hyperplanes. If the means span an \( \l... | Yes |
Theorem 6.8.2. If \( {\pi }_{t} \) is \( N\left( {{\mathbf{\mu }}^{\left( t\right) },\mathbf{\sum }}\right) \) and the costs of misclassification are equal. then the regions of classification, \( {R}_{1},\ldots ,{R}_{m} \), that minimize the maximum conditional expected loss are defined by (3), where \( {u}_{jk}\left( ... | As an example consider the case of \( m = 3 \) . There is no loss of generality in taking \( p = 2 \), for the density for higher \( p \) can be projected on the two-dimensional plane determined by the means of the three populations if they are not collinear (i.e., we can transform the vector \( x \) into \( {u}_{12},{... | Yes |
Lemma 6.10.1. If \( {\sum }_{1} \) and \( {\sum }_{2} \) are positive definite and \( {t}_{1} > 0,{t}_{2} > 0 \), then\n\n(18)\n\n\[ \n{\sum }_{2}{\left\lbrack {t}_{1}{\sum }_{1} + {t}_{2}{\sum }_{2}\right\rbrack }^{-1}{\sum }_{1}\n\]\n\nis positive definite. | Proof. The matrix (18) is\n\n(19)\n\n\[ \n{\sum }_{2}{\left\lbrack {\sum }_{1}\left( {t}_{1}{\sum }_{2}^{-1} + {t}_{2}{\sum }_{1}^{-1}\right) {\sum }_{2}\right\rbrack }^{-1}{\sum }_{1} = {\left( {t}_{1}{\sum }_{2}^{-1} + {t}_{2}{\sum }_{1}^{-1}\right) }^{-1}.\n\] | Yes |
Lemma 7.2.1. Conditional on \( {w}_{1},\ldots ,{w}_{t - 1} \) (or equivalently on \( {v}_{1},\ldots ,{v}_{t - 1} \) ), \( {t}_{i1},\ldots ,{t}_{i, t - 1} \) and \( {t}_{ti}^{2} \) are independently distributed; \( {t}_{ij} \) is distributed according to \( N\left( {0,1}\right), i > j \) ; and \( {t}_{ti}^{2} \) has the... | Proof. The coordinates of \( {v}_{1} \) referred to the new orthogonal coordinates with \( {v}_{1},\ldots ,{v}_{t - 1} \) defining the first coordinate axes are independently normally distributed with means 0 and variances 1 (Theorem 3.3.1). \( {t}_{u}^{2} \) is the sum of the coordinates squared omitting the first \( ... | No |
Corollary 7.2.1. Let \( {Z}_{1},\ldots ,{Z}_{n}\left( {n \geq p}\right) \) be independently distributed, each according to \( N\left( {\mathbf{0},\mathbf{I}}\right) \) ; let \( \mathbf{A} = \mathop{\sum }\limits_{{\alpha = 1}}^{n}{\mathbf{Z}}_{\alpha }{\mathbf{Z}}_{\alpha }^{\prime } = \mathbf{T}{\mathbf{T}}^{\prime } ... | Since \( {t}_{ti} \) has density \( {2}^{-\frac{1}{2}\left( {n - t - 1}\right) }{t}^{n - t}{e}^{-\frac{1}{2}{t}^{2}}/\Gamma \left\lbrack {\frac{1}{2}\left( {n + 1 - i}\right) }\right\rbrack \), the joint density of \( {t}_{ij}, j = 1,\ldots, i, i = 1,\ldots, p \), is\n\n(6)\n\n\[ \mathop{\prod }\limits_{{i = 1}}^{p}\fr... | Yes |
Theorem 7.2.1. Let \( {Z}_{1},\ldots ,{Z}_{n}\left( {n \geq p}\right) \) be independently distributed, each according to \( N\left( {\mathbf{0},\mathbf{\sum }}\right) \) ; let \( \mathbf{A} = \mathop{\sum }\limits_{{\alpha = 1}}^{n}{\mathbf{Z}}_{\alpha }{\mathbf{Z}}_{\alpha }^{\prime } \) ; and let \( \mathbf{A} = {\ma... | \[ \frac{\mathop{\prod }\limits_{{i = 1}}^{p}{t}_{ii}^{*n - i}{e}^{-\frac{1}{2}\operatorname{tr}{\sum }^{-1}{T}^{ * }{T}^{ * }}}{{2}^{\frac{1}{2}p\left( {n - 2}\right) }{\pi }^{p\left( {p - 1}\right) /4}{\left| \sum \right| }^{\frac{1}{2}n}\mathop{\prod }\limits_{{i = 1}}^{p}\Gamma \left\lbrack {\frac{1}{2}\left( {n + ... | Yes |
Theorem 7.2.2. Let \( {Z}_{1},\ldots ,{Z}_{n} \) be independently distributed, each according to \( N\left( {\mathbf{0},\mathbf{\sum }}\right) \). The density of \( A = \mathop{\sum }\limits_{{\alpha = 1}}^{n}{Z}_{\alpha }{Z}_{\alpha }^{\prime } \) is | \[ \frac{{\left| A\right| }^{\frac{1}{2}\left( {n - p - 1}\right) }{e}^{-\frac{1}{2}\operatorname{tr}{\sum }^{-1}A}}{{2}^{\frac{1}{2}{pn}}{\pi }^{p\left( {p - 1}\right) /4}{\left| \sum \right| }^{\frac{1}{2}n}\mathop{\prod }\limits_{{i = 1}}^{p}\Gamma \left\lbrack {\frac{1}{2}\left( {n + 1 - i}\right) }\right\rbrack } ... | Yes |
Corollary 7.2.2. Let \( {X}_{1},\ldots ,{X}_{N}\left( {N > p}\right) \) be independently distributed, each according to \( N\left( {\mathbf{\mu },\mathbf{\sum }}\right) \). Then the density of \( A = \mathop{\sum }\limits_{{\alpha = 1}}^{N}\left( {{X}_{\alpha } - \bar{X}}\right) {\left( {X}_{\alpha } - \bar{X}\right) }... | The density (14) will be denoted by \( w\left( {A \mid \sum, n}\right) \), and the associated distribution will be termed \( W\left( {\mathbf{\sum }, n}\right) \). If \( n < p \), then \( A \) does not have a density, but its distribution is nevertheless defined, and we shall refer to it as \( W\left( {\mathbf{\sum }, ... | Yes |
Corollary 7.2.3. Let \( {\mathbf{X}}_{1},\ldots ,{\mathbf{X}}_{N}\left( {N > p}\right) \) be independently distributed, each according to \( N\left( {\mathbf{\mu },\mathbf{\sum }}\right) \) . The distribution of \( S = \left( {1/n}\right) \mathop{\sum }\limits_{{\alpha = 1}}^{N}\left( {{X}_{\alpha } - \bar{X}}\right) {... | Proof. \( S \) has the distribution of \( \mathop{\sum }\limits_{{\alpha = 1}}^{n}\left\lbrack {\left( {1/\sqrt{n}}\right) {Z}_{\alpha }}\right\rbrack {\left\lbrack \left( 1/\sqrt{n}\right) {Z}_{\alpha }\right\rbrack }^{\prime } \), where \( \left( {1/\sqrt{n}}\right) {Z}_{1},\ldots ,\left( {1/\sqrt{n}}\right) {Z}_{N} ... | Yes |
\[ {\int }_{B > 0}{\left| B\right| }^{t - \frac{1}{2}\left( {p + 1}\right) }{e}^{-\ln B}{dB} = {\pi }^{p\left( {p - 1}\right) /4}\mathop{\prod }\limits_{{t = 1}}^{p}\Gamma \left\lbrack {t - \frac{1}{2}\left( {i - 1}\right) }\right\rbrack . \] | Proof. Here \( B > 0 \) denotes \( B \) positive definite. Since (14) is a density, its integral for \( \mathbf{A} > \mathbf{0} \) is 1 . Let \( \mathbf{\sum } = \mathbf{I},\mathbf{A} = 2\mathbf{B}\left( {d\mathbf{A} = {2d}\mathbf{B}}\right) \), and \( n = {2t} \) . Then the fact that the integral is 1 is identical to ... | No |
Theorem 7.3.4. Let \( A \) and \( \sum \) be partitioned into \( q \) and \( p - q \) rows and columns,\n\n\[ \mathbf{A} = \left( \begin{array}{ll} {\mathbf{A}}_{11} & {\mathbf{A}}_{12} \\ {\mathbf{A}}_{21} & {\mathbf{A}}_{22} \end{array}\right) ,\;\mathbf{\sum } = \left( \begin{array}{ll} {\mathbf{\sum }}_{11} & {\mat... | Proof. \( A \) is distributed as \( \mathop{\sum }\limits_{{\alpha = 1}}^{n}{Z}_{\alpha }{Z}_{\alpha }^{\prime } \), where the \( {Z}_{\alpha } \) are independent, each with the distribution \( N\left( {\mathbf{0},\mathbf{\sum }}\right) \) . Partition \( {Z}_{\alpha } \) into subvectors of \( q \) and \( p - q \) compo... | Yes |
Theorem 7.3.5. Let \( A \) and \( \sum \) be partitioned into \( {p}_{1},{p}_{2},\ldots ,{p}_{q} \) rows and columns \( \left( {{p}_{1} + \cdots + {p}_{q} = p}\right) \), \[ \mathbf{A} = \left( \begin{matrix} {\mathbf{A}}_{11} & \cdots & {\mathbf{A}}_{1q} \\ \vdots & & \vdots \\ {\mathbf{A}}_{q1} & \cdots & {\mathbf{A}... | Proof. \( A \) is distributed as \( \mathop{\sum }\limits_{{\alpha = 1}}^{n}{Z}_{\alpha }{Z}_{\alpha }^{\prime } \), where \( {Z}_{1},\ldots ,{Z}_{n} \) are independently distributed, each according to \( N\left( {\mathbf{0},\mathbf{\sum }}\right) \). Let \( {\mathbf{Z}}_{\alpha } \) be partitioned \[ {Z}_{\alpha } = \... | Yes |
Lemma 7.4.1. If the \( N \times N \) symmetric matrix \( {C}_{i} \) has rank \( {r}_{i}, i = 1,\ldots, m \) , and\n\n(1)\n\n\[ \mathop{\sum }\limits_{{l = 1}}^{m}{C}_{l} = {I}_{N} \]\n\nthen\n\n(2)\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{m}{r}_{i} = N \]\n\nis a necessary and sufficient condition for there to exist an \... | Proof. The necessity follows from the fact that (1) implies that the sum of (3) over \( i = 1,\ldots, m \) is \( {I}_{N} \) . Now let us prove the sufficiency; we assume (2).\n\nThere exists an orthogonal matrix \( {P}_{1} \) such that \( {P}_{i}{C}_{i}{P}_{1}^{\prime } \) is diagonal with diagonal elements the charact... | Yes |
Theorem 7.4.1. Suppose \( {Y}_{1},\ldots ,{Y}_{N} \) are independently distributed, each according to \( N\left( {\mathbf{0},\sum }\right) \) . Suppose the matrix \( \left( {c}_{\alpha \beta }^{1}\right) = C \), used in forming\n\n(8)\n\n\[ \n{Q}_{1} = \mathop{\sum }\limits_{{\alpha ,\beta = 1}}^{N}{c}_{\alpha \beta }^... | It follows from (3) that \( {C}_{i} \) is idempotent. See Section A. 2 of the Appendix.\n\nThis theorem is useful in generalizing results from the univariate analysis of variance. (See Chapter 8.) As an example of the use of this theorem, let us prove that the mean of a sample of size \( N \) times its transpose and a ... | Yes |
Theorem 7.5.1. If \( V = \left( {{v}_{1},\ldots ,{v}_{p}}\right) \), then the square of the p-dimensional volume of the parallelotope with \( {\mathbf{v}}_{1},\ldots ,{\mathbf{v}}_{\rho } \) as principal edges is \( \left| {{\mathbf{V}}^{\prime }\mathbf{V}}\right| \) . | Proof. If \( p = 1 \), then \( \left| {{\mathbf{V}}^{\prime }\mathbf{V}}\right| = {\mathbf{v}}_{1}^{\prime }{\mathbf{v}}_{1} = {\begin{Vmatrix}{\mathbf{v}}_{1}\end{Vmatrix}}^{2} \), which is the square of the one-dimensional volume of \( {\mathbf{v}}_{1} \) . If two \( k \) -dimensional parallelotopes have bases consis... | Yes |
Corollary 7.5.1. The square of the p-dimensional volume of the parallelo-topc with the rows of (2) as principal edges is \( \left| \mathbf{A}\right| \), where \( \mathbf{A} \) is given by (3). | We see that\n\n(6)\n\n\[ \left| A\right| = \left| \begin{matrix} \mathop{\sum }\limits_{\alpha }{y}_{1\alpha }^{2} & \cdots & \mathop{\sum }\limits_{\alpha }{y}_{1\alpha }{y}_{p - 1,\alpha } & \mathop{\sum }\limits_{\beta }{y}_{1\beta }{y}_{p\beta } \\ \vdots & & \vdots & \vdots \\ \mathop{\sum }\limits_{\alpha }{y}_{p... | Yes |
Theorem 7.5.2. Let \( \left| S\right| \) be defined by (1), where \( {x}_{1},\ldots ,{x}_{N} \) are the \( N \) vectors of a sample. Then \( \left| S\right| \) is proportional to the sum of squares of the volumes of all the different parallelotopes formed by using as principal edges \( p \) vectors with \( p \) of \( {... | The distribution of \( \left| S\right| \) is the same as the distribution of \( \left| \mathbf{A}\right| /{\left( N - 1\right) }^{p} \), where \( \mathbf{A} = \mathop{\sum }\limits_{{\alpha = 1}}^{n}{\mathbf{Z}}_{\alpha }{\mathbf{Z}}_{\alpha }^{\prime } \) and \( {\mathbf{Z}}_{1},\ldots ,{\mathbf{Z}}_{n} \) are distrib... | Yes |
Theorem 7.5.3. The distribution of the generalized variance \( \left| S\right| \) of a sample \( {X}_{1},\ldots ,{X}_{N} \) from \( N\left( {\mathbf{\mu },\mathbf{\sum }}\right) \) is the same as the distribution of \( \left| \mathbf{\sum }\right| /{\left( N - 1\right) }^{p} \) times the product of \( p \) independent ... | If \( p = 1,\left| S\right| \) has the distribution of \( \left| \sum \right| \cdot {\chi }_{N - 1}^{2}/\left( {N - 1}\right) \) . If \( p = 2,\left| S\right| \) has the distribution of \( \left| \sum \right| {\chi }_{N - 1}^{2} \cdot {\chi }_{N - 2}^{2}/{\left( N - 1\right) }^{2} \) . It follows from Problem 7.15 or 7... | Yes |
Theorem 7.7.1. If \( A \) has the distribution \( W\left( {\sum, m}\right) \), then \( B = {A}^{-1} \) has the density\n\n(1)\n\n\[ \frac{{\left| \Psi \right| }^{\frac{1}{2}m}{\left| B\right| }^{-\frac{1}{2}\left( {m + p + 1}\right) }{e}^{-\frac{1}{2}\operatorname{tr}\Psi {B}^{-1}}}{{2}^{\frac{1}{2}m}{\Gamma }_{p}\left... | Proof. By Theorem A.4.6 of the Appendix, the Jacobian of the transformation \( A = {B}^{-1} \) is \( {\left| B\right| }^{-\left( {p + 1}\right) } \) . Substitution of \( {B}^{-1} \) for \( A \) in (16) of Section 7.2 and multiplication by \( {\left| B\right| }^{-\left( {p + 1}\right) } \) yields (1). | Yes |
Theorem 7.7.2. If \( A \) has the distribution \( W\left( {\sum, n}\right) \) and \( \sum \) has the a priori distribution \( {W}^{-1}\left( {\mathbf{\Psi }, m}\right) \), then the conditional distribution of \( \sum \) is \( {W}^{-1}(\mathbf{A} + \) \( \Psi, n + m) \) . | Proof. The joint density of \( \mathbf{A} \) and \( \mathbf{\sum } \) is\n\n(2)\n\n\[ \frac{{\left| \mathbf{\Psi }\right| }^{\frac{1}{2}m}{\left| \mathbf{\sum }\right| }^{-\frac{1}{2}\left( {n + m + p + 1}\right) }{\left| A\right| }^{\frac{1}{2}\left( {n - p - 1}\right) }{e}^{-\frac{1}{2}\operatorname{tr}\left( {A + \P... | Yes |
Corollary 7.7.2. If \( {nS} \) has the distribution \( W\left( {\sum, n}\right) ,\sum \) has the a priori distribution \( {W}^{-1}\left( {\mathbf{\Psi }, m}\right) \), and the loss function is \( \operatorname{tr}\left( {D - \sum }\right) G\left( {D - \sum }\right) H \), where G and \( \mathbf{H} \) are positive defini... | Proof. It follows from Section 3.4.2 that the Bayes estimator for \( \sum \) is \( \mathcal{E}\left( {\sum \mid S}\right) \) . From Theorem 7.7.2 we see that \( {\sum }^{-1} \) has the a posteriori distribution \( W\left\lbrack {{\left( nS + \mathbf{\Psi }\right) }^{-1}, n + m}\right\rbrack \) . The theorem results fro... | Yes |
Lemma 7.7.1. If \( A \) has the distribution \( W\left( {\sum, n}\right) \), then\n\n\[ \mathcal{E}{A}^{-1} = \frac{1}{n - p - 1}{\sum }^{-1} \] | Proof. If \( C \) is a nonsingular matrix such that \( \sum = C{C}^{\prime } \), then \( A \) has the distribution of \( {CB}{C}^{\prime } \), where \( B \) has the distribution \( W\left( {I, n}\right) \), and \( \mathcal{E}{A}^{-1} = \) \( {\left( {C}^{\prime }\right) }^{-1}\left( {\mathcal{E}{B}^{-1}}\right) {C}^{-1... | Yes |
Theorem 7.7.3. Let \( {\mathbf{x}}_{1},\ldots ,{\mathbf{x}}_{N} \) be observations from \( N\left( {\mathbf{\mu },\mathbf{\sum }}\right) \) . Suppose \( \mathbf{\mu } \) and \( \sum \) have the a priori density \( n\left( {\mathbf{\mu } \mid \mathbf{v},\left( {1/K}\right) \sum }\right) \times {w}^{-1}\left( {\sum \mid ... | Proof. Since \( \widetilde{\mathbf{x}} \) and \( n\mathbf{S} = \mathbf{A} \) are a sufficient set of statistics, we can consider the joint density of \( \overline{\mathbf{x}},\mathbf{A},\mathbf{\mu } \), and \( \mathbf{\sum } \), which is\n\n(8)\n\n\[ \frac{{K}^{\frac{1}{2}p}{N}^{\frac{1}{2}p}{\left| \Psi \right| }^{\f... | Yes |
Corollary 7.7.3. If \( {x}_{1},\ldots ,{x}_{N} \) are observations from \( N\left( {\mathbf{\mu },\sum }\right) \), if \( \mathbf{\mu } \) and \( \sum \) have the a priori density \( n\left\lbrack {\mathbf{\mu } \mid \mathbf{v},\left( {1/K}\right) \sum }\right\rbrack \times {w}^{-1}\left( {\sum \mid \mathbf{\Psi }, m}\... | (13)\n\n\[ \frac{1}{N + K}\left( {N\bar{x} + {K\nu }}\right) \]\n\nand\n\n(14)\n\n\[ \frac{1}{N + m - p - 1}\left\lbrack {{nS} + \mathbf{\Psi } + \frac{NK}{N + \widetilde{K}}\left( {\widetilde{x} - \mathbf{v}}\right) {\left( \widetilde{x} - \mathbf{v}\right) }^{\prime }}\right\rbrack , \] | Yes |
Theorem 7.7.4. If \( {x}_{1},\ldots ,{x}_{N} \) are observations from \( N\left( {\mathbf{\mu },\sum }\right) \) and if \( \mathbf{\mu } \) and \( \sum \) have the a priori density \( n\left\lbrack {\mathbf{\mu } \mid \mathbf{v},\left( {1/K}\right) \sum }\right\rbrack \times {w}^{-1}\left( {\sum \mid \mathbf{\Psi }, m}... | Proof. The exponent in (12) is \( - \frac{1}{2} \) times\n\n(16)\n\n\[ \operatorname{tr}\left\lbrack {B + \left( {N + K}\right) \left( {\mu - {\mu }^{ * }}\right) {\left( \mu - {\mu }^{ * }\right) }^{\prime }}\right\rbrack {\sum }^{-1}. \]\n\nThen the integral of (12) with respect to \( \sum \) is\n\n(17)\n\n\[ \frac{{... | Yes |
Theorem 7.8.1. The quadratic risk of \( {aA} \) is minimized at \( a = 1/\left( {n + p + 1}\right) \) . and its value is \( p\left( {p + 1}\right) /\left( {n + p + 1}\right) \) . | Proof. By the invariance of the loss function\n\n(3)\n\n\[ \n{\mathcal{E}}_{\sum }{L}_{q}\left( {\sum ,{aA}}\right) = {\mathcal{E}}_{I}{L}_{q}\left( {I, a{A}^{ * }}\right) \n\]\n\n\[ \n= {\mathcal{E}}_{I}\operatorname{tr}{\left( a{A}^{ * } - I\right) }^{2} \n\]\n\n\[ \n= {\mathcal{E}}_{I}\left( {{a}^{2}\mathop{\sum }\l... | Yes |
Theorem 7.8.2. With respect to the quadratic loss function the best estimator invariant with respect to linear transformations \( \sum \rightarrow {H\sum }{H}^{\prime }, A \rightarrow {HA}{H}^{\prime } \), where \( H \) is lower triangular, is \( G\left( A\right) = {TD}{T}^{\prime } \), where \( D \) is the diagonal ma... | Since \( d = {\mathbf{F}}^{-1}f \) is not proportional to \( \mathbf{\varepsilon } = {\left( 1,\ldots ,1\right) }^{\prime } \), that is, \( \mathbf{F}\mathbf{\varepsilon } \) is not proportional to \( f \) (see Problem 7.28), this estimator has a smaller (quadratic) loss than any estimator of the form \( {aA} \) (which... | Yes |
Theorem 7.8.3. The estimator \( G\left( A\right) \) defined in Theorem 7.8.2 is minimax with respect to the quadratic loss function. | In the case of \( p = 2 \)\n\n(12)\n\n\[ \n{d}_{1} = \frac{{\left( n + 1\right) }^{2} - \left( {n - 1}\right) }{{\left( n + 1\right) }^{2}\left( {n + 3}\right) - \left( {n - 1}\right) },\;{d}_{2} = \frac{\left( {n + 1}\right) \left( {n + 2}\right) }{{\left( n + 1\right) }^{2}\left( {n + 3}\right) - \left( {n - 1}\right... | Yes |
Theorem 7.8.4. With respect to the likelihood loss function, the best estimator invariant with respect to linear transformations \( \sum \rightarrow {H\sum }{H}^{\prime }, A \rightarrow {HA}{H}^{\prime } \) , where \( \mathbf{H} \) is lower triangular, is \( \mathbf{G}\left( A\right) = \mathbf{{TD}}{\mathbf{T}}^{\prime... | \[ {\mathcal{E}}_{\sum }L\left\lbrack {\sum, G\left( A\right) }\right\rbrack = \mathop{\sum }\limits_{{j = 1}}^{p}\log \left( {n + p - {2j} + 1}\right) - \mathop{\sum }\limits_{{j = 1}}^{p}{c}^{n}\log {\chi }_{n + 1 - j}^{2}. \] | Yes |
Theorem 7.9.1. If \( \sum = {I}_{p} \), the limiting distribution of \( \sqrt{N}\left( {\widetilde{T} - {I}_{p}}\right) \) is normal with mean 0 . The variance of a diagonal element is \( \left( {{3\kappa } + 2}\right) /4 \) ; the covariance of two diagonal elements is \( \kappa /4 \) ; the variance of an off-diagonal ... | Let \( \mathbf{X} = \mathbf{v} + \mathbf{C}\mathbf{Y} \), where \( \mathbf{Y} \) has the density \( g\left( {{y}^{\prime }y}\right) ,\mathbf{\Lambda } = \mathbf{C}{\mathbf{C}}^{\prime } \), and \( \sum = \mathcal{E}(\mathbf{X} \) \( - \mathbf{v}){\left( X - \mathbf{v}\right) }^{\prime } = \left( {\mathcal{E}{R}^{2}/p}\... | No |
Theorem 7.9.2. Define \( T = \left( {t}_{ij}\right) \) by \( {Y}^{\prime }Y = T{T}^{i},{t}_{ij} = 0, i < j \), and \( {t}_{ii} \geq 0 \) . If the density of \( \mathbf{Y} \) is \( g\left( {{\mathbf{Y}}^{\prime }\mathbf{Y}}\right) \), then the density of \( \mathbf{T} \) is\n\n(5)\n\n\( \mathop{\prod }\limits_{{i = 1}}^... | Proof. Let \( Y = \left( {{v}_{1},\ldots ,{v}_{p}}\right) \) . Define \( {w}_{i} \) and \( {w}_{i} \) recursively by \( {w}_{1} = {v}_{1},{u}_{1} = \) \( {\mathbf{w}}_{1}/\begin{Vmatrix}{\mathbf{w}}_{1}\end{Vmatrix} \n\n(6)\n\n\[ \n{\mathbf{w}}_{1} = {\mathbf{v}}_{1} - \mathop{\sum }\limits_{{j = 1}}^{{i - 1}}{\mathbf{... | Yes |
Theorem 7.9.3. If \( X\left( {N \times p}\right) \) has the density\n\n(12)\n\n\[ \n{\left| C\right| }^{-N}g\left\lbrack {{C}^{-1}{X}^{\prime }X{\left( {C}^{\prime }\right) }^{-1}}\right\rbrack \n\]\n\nthen the lower triangular matrix \( {\mathbf{T}}^{ * } \) satisfying \( {\mathbf{X}}^{\prime }\mathbf{X} = {\mathbf{T}... | Let \( A = {X}^{\prime }X = {T}^{ * }{T}^{*\prime } \). | No |
Theorem 7.9.4. If \( X \) has the density (12), then \( A = {X}^{\prime }X \) has the density | \[ \frac{{\pi }^{\frac{1}{2}p\left\lbrack {N - \frac{1}{2}\left( {p - 1}\right) }\right\rbrack }}{{\Gamma }_{p}\left( {\frac{1}{2}N}\right) {\left| \mathbf{A}\right| }^{\frac{1}{2}N}}{\left| A\right| }^{\frac{1}{2}\left( {N - p + 1}\right) }g\left\lbrack {{C}^{-1}A{\left( {C}^{\prime }\right) }^{-1}}\right\rbrack . \] | Yes |
Lemma 8.2.1. Let\n\n\[ B = \mathop{\sum }\limits_{{\alpha = 1}}^{N}{x}_{\alpha }{z}_{\alpha }^{\prime }{\left( \mathop{\sum }\limits_{{\alpha = 1}}^{N}{z}_{\alpha }{z}_{\alpha }^{\prime }\right) }^{-1}. \]\n\nThen for any \( p \times q \) matrix \( \mathbf{F} \)\n\n\[ \mathop{\sum }\limits_{{\alpha = 1}}^{N}\left( {{x}... | Proof. The left-hand side of (4) is\n\n\[ \mathop{\sum }\limits_{{\alpha = 1}}^{N}\left\lbrack {\left( {{x}_{\alpha } - B{z}_{\alpha }}\right) + \left( {B - F}\right) {z}_{\alpha }}\right\rbrack {\left\lbrack \left( {x}_{\alpha } - B{z}_{\alpha }\right) + \left( B - F\right) {z}_{\alpha }\right\rbrack }^{\prime }, \]\n... | Yes |
Lemma 8.2.2. If \( A \) and \( G \) are positive definite, tr \( {FA}{F}^{\prime }G > 0 \) for \( F \neq \mathbf{0} \) . | Proof. Let \( A = H{H}^{\prime }, G = K{K}^{\prime } \) . Then\n\n(9)\n\[ \operatorname{tr}{FA}{F}^{\prime }G = \operatorname{tr}{FH}{H}^{\prime }{F}^{\prime }K{K}^{\prime } = \operatorname{tr}{K}^{\prime }{FH}{H}^{\prime }{F}^{\prime }K \]\n\n\[ = \operatorname{tr}\left( {{K}^{\prime }{FH}}\right) {\left( {K}^{\prime ... | Yes |
Theorem 8.2.1. If \( {x}_{\alpha } \) is an observation from \( N\left( {\mathbf{B}{z}_{\alpha },\sum }\right) ,\alpha = 1,\ldots, N \), with \( \left( {{z}_{1}\ldots \ldots {z}_{N}}\right) \) of rank \( q \), the maximum likelihood estimator of \( \mathbf{B} \) is given by (10), where \( C = \mathop{\sum }\limits_{\al... | A useful algebraic result follows from (12) and (4) with \( \mathbf{F} = \mathbf{0} \) :\n\n(13)\n\n\[ N\widehat{\sum } = \mathop{\sum }\limits_{{\alpha = 1}}^{N}{x}_{\alpha }{x}_{\alpha }^{\prime } - \widehat{\mathbf{B}}A{\widehat{\mathbf{B}}}^{\prime } = \mathop{\sum }\limits_{{\alpha = 1}}^{N}{x}_{\alpha }{x}_{\alph... | Yes |
Theorem 8.2.2. The maximum likelihood estimator \( \widehat{\mathbf{B}} \) based on a set of \( N \) observations, the \( \alpha \) th from \( N\left( {\mathbf{B}{z}_{\alpha },\mathbf{\sum }}\right) \), is normally distributed with mean \( \mathbf{B} \), and the covariance matrix of the \( i \) th and \( j \) th rows o... | The density then can be written [by virtue of (4)]\n\n(17)\n\n\[ \frac{1}{{\left( 2\pi \right) }^{\frac{1}{2}{pN}}{\left| \mathbf{\sum }\right| }^{\frac{1}{2}N}}\exp \left( {-\frac{1}{2}\operatorname{tr}\left\langle {{\mathbf{\sum }}^{-1}\left\lbrack {\left( {\widehat{\mathbf{B}} - \mathbf{B}}\right) {.1}{\left( \wideh... | Yes |
Theorem 8.2.4. The least squares estimator is the best linear unbiased estimator of \( {\beta }_{rg} \) . | Proof. Let \( {\widetilde{\beta }}_{tg} = \mathop{\sum }\limits_{{\alpha = 1}}^{N}\mathop{\sum }\limits_{{j = 1}}^{p}{f}_{j\alpha }{x}_{j\alpha } \) be an arbitrary unbiased estimator of \( {\beta }_{tg} \), and let \( {\widehat{\beta }}_{tg} = \mathop{\sum }\limits_{{\alpha = 1}}^{N}\mathop{\sum }\limits_{{h = 1}}^{q}... | Yes |
Theorem 8.3.1. The likelihood ratio criterion (11) for testing the null hypothesis \( {\mathbf{B}}_{1} = \mathbf{0} \) is invariant with respect to transformations \( {x}_{\alpha }^{ * } = D{x}_{\alpha },\alpha = \) \( 1,\ldots, N \), for nonsingular \( D \) . | Proof. The estimators in terms of \( {x}_{\alpha }^{ * } \) are\n\n(12) \( {\widehat{\mathbf{B}}}^{ * } = {DC}{A}^{-1} = D\widehat{\mathbf{B}} \) ,\n\n(13) \( {\widehat{\mathbf{\sum }}}_{\Omega }^{ * } = \frac{1}{N}\mathop{\sum }\limits_{{\alpha = 1}}^{N}\left( {D{x}_{\alpha } - D\widehat{\mathbf{B}}{z}_{\alpha }}\righ... | Yes |
\[ {\widehat{\mathbf{B}}}_{2\omega } - {\widehat{\mathbf{B}}}_{2\Omega } = \left( {{\widehat{\mathbf{B}}}_{1\Omega } - {\mathbf{B}}_{1}^{ * }}\right) {A}_{12}{A}_{22}^{-1}. \] | Proof. The normal equation \( {\widehat{\mathbf{\beta }}}_{\Omega }A = C \) is written in partitioned form \[ \left( {{\widehat{\mathbf{B}}}_{1\Omega }{A}_{11} + {\widehat{\mathbf{B}}}_{2\Omega }{A}_{21},{\widehat{\mathbf{B}}}_{1\Omega }{A}_{12} + {\widehat{\mathbf{B}}}_{2\Omega }{A}_{22}}\right) = \left( {{C}_{1},{C}_... | Yes |
Lemma 8.4.2. The criterion \( U \) has the distribution of \[ U = \frac{\left| G\right| }{\left| G + H\right| } \] where \( G \) is distributed according to \( W\left( {\mathbf{\sum }, n}\right), H \) is distributed according to \( W\left( {\mathbf{\sum }, m}\right) \), where \( m = {q}_{1} \), and \( G \) and \( H \) ... | Let \[ G = N{\widehat{\mathbf{\sum }}}_{\Omega } = X{X}^{\prime } - X{Z}^{\prime }{\left( Z{Z}^{\prime }\right) }^{-1}Z{X}^{\prime }, \] \[ G + H = N{\widehat{\mathbf{\sum }}}_{\Omega } + \left( {{\widehat{\mathbf{B}}}_{1\Omega } - {\mathbf{B}}_{1}^{ * }}\right) {A}_{{11} \cdot 2}{\left( {\widehat{\mathbf{B}}}_{1\Omega... | Yes |
Lemma 8.4.3. Let \( y \) be an \( N \) -component row vector and \( U \) an \( r \times N \) matrix. Then the sum of squares of the residuals of \( y \) from its regression on \( U \) is\n\n(9)\n\n\[ \frac{\left| \begin{array}{ll} y{y}^{\prime } & y{U}^{\prime } \\ U{y}^{\prime } & U{U}^{\prime } \end{array}\right| }{\... | Proof. By Corollary A.3.1 of the Appendix,(9) is \( y{y}^{\prime } - y{U}^{\prime }{\left( U{U}^{\prime }\right) }^{-1}U{y}^{\prime } \) , which is the sum of squares of residuals as indicated in (13) of Section 8.2. | Yes |
Lemma 8.4.4. \( {V}_{t} \) defined by (8) is the ratio of the sum of squares of the residuals of \( {y}_{11},\ldots ,{y}_{1N} \) from their regression on \( {\mathbf{y}}_{1}^{\left( 1 - 1\right) },\ldots ,{\mathbf{y}}_{N}^{\left( i - 1\right) } \) and \( \mathbf{Z} \) to the sum of squares of residuals of \( {y}_{t1},\... | Proof. The numerator of \( {V}_{1} \) can be written [from (13) of Section 8.2]\n\n(10)\n\n\[ \frac{\left| {G}_{1}\right| }{\left| {G}_{t - 1}\right| } = \frac{\left| {Y}_{t}{Y}_{t}^{\prime } - {Y}_{t}{Z}^{\prime }{\left( Z{Z}^{\prime }\right) }^{-1}Z{Y}_{t}^{\prime }\right| }{\left| {Y}_{t - 1}{Y}_{t - 1}^{\prime } - ... | Yes |
Theorem 8.4.1. The distribution of \( U \) defined by (3) is the distribution of the product \( \mathop{\prod }\limits_{{i = 1}}^{p}{V}_{i} \), where \( {V}_{1},\ldots ,{V}_{p} \) are independent and \( {V}_{i} \) has the density (11). | The cdf of \( U \) can be found by integrating the joint density of \( {V}_{1},\ldots ,{V}_{p} \) over the range\n\n(12)\n\n\[ \mathop{\prod }\limits_{{i = 1}}^{p}{V}_{i} \leq u \] | No |
Theorem 8.4.3. The hih moment of \( U\left\lbrack {\text{if }h > - \frac{1}{2}\left( {n + 1 - p}\right) }\right\rbrack \) is\n\n(21)\n\n\[ \mathcal{E}{U}^{h} = \mathop{\prod }\limits_{{i = 1}}^{p}\frac{\Gamma \left\lbrack {\frac{1}{2}\left( {n + 1 - i}\right) + h}\right\rbrack \Gamma \left\lbrack {\frac{1}{2}\left( {n ... | \[ = \mathop{\prod }\limits_{{i = 1}}^{p}\frac{\Gamma \left\lbrack {\frac{1}{2}\left( {N - {q}_{1} - {q}_{2} + 1 - i}\right) + h}\right\rbrack \Gamma \left\lbrack {\frac{1}{2}\left( {N - {q}_{2} + 1 - i}\right) }\right\rbrack }{\Gamma \left\lbrack {\frac{1}{2}\left( {N - {q}_{1} - {q}_{2} + 1 - i}\right) }\right\rbrack... | Yes |
Theorem 8.4.7. If \( p \) is even or if \( m \) is even, the density of \( {U}_{p, m, n} \) can be expressed as a linear combination of terms \( {\left( -\log u\right) }^{k}{u}^{l} \), where \( k \) is an integer and \( l \) is a half integer. | From (35) we see that the cumulative distribution function of \( - \log U \) is a linear combination of terms \( {w}^{k}{e}^{-{lw}} \) and hence the cumulative distribution function of \( U \) is a linear combination of terms \( {\left( -\log u\right) }^{k}{u}^{l} \) . The values of \( k \) and \( l \) and the coeffici... | No |
Theorem 8.6.1. Let \( {x}_{\alpha } \) be an observation from \( N\left( {{\mathbf{\beta }}_{1}{z}_{\alpha }^{*\left( 1\right) } + {\mathbf{\beta }}_{2}^{ * }{z}_{\alpha }^{\left( 2\right) },\sum }\right) \) , where \( {\sum }_{\alpha }{z}_{\alpha }^{*\left( 1\right) }{z}_{\alpha }^{\left( 2\right) } = \mathbf{0} \) an... | The likelihood ratio criterion is a function of\n\n(5)\n\n\[ \nU = \frac{\left| G\right| }{\left| G + H\right| } = \frac{\left| KG{K}^{\prime }\right| }{\left| KG{K}^{\prime } + KH{K}^{\prime }\right| } = \frac{\left| I\right| }{\left| I + L\right| } \n\] \n\n\[ \n= \mathop{\prod }\limits_{{i = 1}}^{p}{\left( 1 + {l}_{... | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.