STATISTICS 512 TECHNIQUES OF MATHEMATICS FOR ...

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42 • Positive definite matrices. If a symmetric matrix M is such that x 0 Mx ≥ 0forallx, wesay that M is positive semi-definite (p.s.d.) or nonnegative definite (n.n.d.). We write M ≥ 0. (The text reserves the term p.s.d. for the case in which equality is attained for at least one non-zero x; this convention is somewhat unusual and won’t be followed here.) The preceding discussion shows (how?) that M is p.s.d. iff all eigenvalues are non-negative. If x 0 Mx 0forallx 6= 0, we say thatM is positive definite (p.d.). We write M 0. Equivalently, all eigenvalues are positive. — Geometric interpretation: If M 0 then |M| 0 (why?) and the set n x | x 0 M −1 x = 2o is transformed, via the (orthogonal) transformation y = V 0 x,intotheset ⎧ ⎨ ⎩ y | X =1 2 = 2 ⎫ ⎬ ⎭
43 This is the ellipsoid in R with semi-axes of lengths √ q along the coordinate axes (and volume ∝ |M|). Thus (why?) the original set, obtained from the second via the transformation x = Vy, is an ellipsoid as well, whose semi-axes have the same lengths but are now in the directions of the eigenvectors of M. The following three results illustrate the adage that “a symmetric matrix is almost diagonal”. • Matrix square roots. Can we define a notion of the square root of a (n.n.d.) matrix? Start by thinking of a diagonal matrix, in which case the method is obvious: If D is a diagonal matrix with non-negative diagonal elements, then we can define the square root D 12 to be the diagonal matrix with the roots of these elements on its diagonal. Now extend to the general case. If M ≥ 0 we write M = VD λ V 0 ,whereV is orthogonal
Page 1 and 2: STATISTICS 512 TECHNIQUES OF MATHEM
Page 3 and 4: II LIMITS, CONTINUITY, DIFFEREN- TI
Page 5 and 6: 19 Numerical optimization: Steepest
Page 7 and 8: 7 1. Introduction; matrix manipulat
Page 9 and 10: - is of rather limited usefulness.
Page 11 and 12: 11 — Block matrices ... a particu
Page 13 and 14: 13 for a random matrix X. You shoul
Page 15 and 16: 15 Here the observations (rows) hav
Page 17 and 18: 17 3. Identity element: There is 0
Page 19 and 20: 19 — Fact 1: Every vector space h
Page 21 and 22: 21 3) Used often: (A 0 A)=(A). Proo
Page 23 and 24: 23 areallsolvable. Wewrite[b 1 ··
Page 25 and 26: 25 • Angle between nonzero vecto
Page 27 and 28: 27 (why?). Geometrically, an orthog
Page 29 and 30: 29 orthogonal unit vectors q 1 q h
Page 31 and 32: 31 4. LSEs; Spectral theory • Rec
Page 33 and 34: 33 (ii) In terms of QR-decompositio
Page 35 and 36: 35 and = Ã P ⎛ ⎜ ⎝ R S Q !
Page 37 and 38: 37 • Now suppose that M is symmet
Page 39 and 40: 39 • Spectral Decomposition Theor
Page 41: 41 5. Examples & applications Conse
Page 45 and 46: 45 • If H is idempotent then (i)
Page 47 and 48: 47 and similarly define 2 2 as the
Page 49 and 50: 49 Application 2. By the Cauchy-Sch
Page 51 and 52: 51 6. Limits; continuity; probabili
Page 53 and 54: 53 — ⊂ R is closed if it cont
Page 55 and 56: 55 bounded, q say =(0). It can be
Page 57 and 58: 57 7. Random variables; distributio
Page 59 and 60: 59 • Since the set = (−∞] is
Page 61 and 62: 61 • Jensen’s Inequality: If :
Page 63 and 64: 63 • If → and the function
Page 65 and 66: 65 • Linearity, product, quotient
Page 67 and 68: 67 • Taylor’s Theorem:“Suffic
Page 69 and 70: 69 We want to show that (), which
Page 71 and 72: 71 9. Applications: transformations
Page 73 and 74: 73 monotonic. Example: suppose ∼
Page 75 and 76: 75 Now suppose that → and (ac
Page 77 and 78: 77 This can make it problematic to
Page 79 and 80: 79 10. Sequences and series • Con
Page 81 and 82: 81 Then for each , sup | () − ()
Page 83 and 84: 83 Thus a convergent sequence (i.e.
Page 85 and 86: 85 Then for 0 we have 0 = + X
Page 87 and 88: 87 11. Power series; moment and pro
Page 89 and 90: 89 • If P ∞ =0 converges fo
Page 91 and 92: 91 • By (ii), we can repeat the p
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93 • The moment generating functi
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95 12. Branching processes • Impo
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97 Considering the probabilities of
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99 Since = ( −1 )and is continu
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101 P(N
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103 Then clearly () ≤ () ≤
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105 • Monotonic, bounded function
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107 • Now define () = Z ()
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109 • Improper Riemann integrals,
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111 14. Riemann and Riemann-Stieltj
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113 • A generalization of the Rie
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115 • Improper R-S integrals defi
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117 • Cauchy-Schwarz inequality:
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119 constant’) ∈ [0 1) as →
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15. Moment generating functions; Ch
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123 • Suppose ∼ (0 1), with p.
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125 • Chebyshev’s Inequality fu
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127 Let be fixed but arbitrary. Ex
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129 • Slutsky’s Theorem: If
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131 Part IV MULTIDIMENSIONAL CALCUL
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133 • Derivatives. Put e = (00 1
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135 — If = 1 then the Jacobian m
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137 1. If the partial derivatives o
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139 quantile function. Differentiat
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141 17. Implicit Function Theorem;
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143 • The Inverse Function Theore
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145 • Example. Write the characte
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147 • Often we seek extrema of mu
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149 under which the satisfaction of
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151 Since x 2 is a stationary point
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153 If sup or equivalently if ()
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155 Here and elsewhere the assumpti
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157 • Example: Let be independe
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159 order that 2 () integrate to 1
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161 Note that x ¯ ¯ = ¯¯¯¯¯
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163 with = 0 to minimize (by tria
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165 —Example:solve () =log − 1=
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167 regression, so I’ll illustrat
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169 Assuming convergence, the limit
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171 • For i.i.d. observations wit
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173 • The MLE is generally obtain
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175 so that ³ covθ h˙(θ) i =´
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177 We have (by the WLLN) that X Ã
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with gradient ˙(θ) = Ã − +
Page 181 and 182:
181 • Method of moments: Define p
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• The limit of the NR-process is
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185 • Now let (X) be any unbiased
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187 22. Minimax M-estimation I •
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189 • The asymptotic variance doe
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191 • We will show that such a pa
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193 • By the Lemma, () in(22.3)is
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195 • By comparison with (22.4) w
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We note that and () = − 0 = −
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199 • A solution (there are three
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• The asymptotic normality result
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203 where the infimum is over all s
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205 ∪ =1 ;inthiscasedefine R =
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207 • If is Lebesgue measure the
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209 • The expected value of a r.v
Page 211:
211 • It was stated above that if
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