STATISTICS 512 TECHNIQUES OF MATHEMATICS FOR ...

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— Optimization - find numbers or functions minimizing certain objectives, e.g. designing experiments for maximum information/minimum variance etc.; associated numerical methods. 8 • In Statistics, matrices are convenient ways to store and refer to data. As well, in Regression for instance, there are important structural features that come from examining the algebraic properties of the vector space formed from all linear combinations of the columns of a matrix. • As for the first of these - matrices as data storage - you should learn various ways to manipulate matrices. In particular, the formula for the product of two matrices - that the ( ) element of the product AB of an × matrix A and a × matrix B is given by [AB] = X =1 A B
- is of rather limited usefulness. Rather, one should treat either the rows or the columns of matrices as the basic elements. Some examples: — Define (column) vector in R ;sum,transpose, scalar product, outer product. — Matrix as a column of rows, or row of columns: A × = ⎛ ⎜ ⎝ a 0 1 a 0 2 . a 0 ⎞ ⎟ ⎠ =(α 1.α 2 . ···.α ) — If X × has rows n x 0 o (note: vectors are =1 columns, rows are transposed vectors), and θ is a × 1 vector, then Xθ = ⎛ ⎜ ⎝ x 0 1 . x 0 . x 0 ⎞ ⎟ ⎠ θ = ⎛ ⎜ ⎝ x 0 1 θ . x 0 θ . x 0 θ — If X × has columns n o z ,andθ is a ×1 =1 ⎞ ⎟ ⎠ 9
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: 7 1. Introduction; matrix manipulat
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 and 42: 41 5. Examples & applications Conse
Page 43 and 44: 43 This is the ellipsoid in R with
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
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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
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67 • Taylor’s Theorem:“Suffic
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69 We want to show that (), which
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71 9. Applications: transformations
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73 monotonic. Example: suppose ∼
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75 Now suppose that → and (ac
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77 This can make it problematic to
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79 10. Sequences and series • Con
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81 Then for each , sup | () − ()
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83 Thus a convergent sequence (i.e.
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85 Then for 0 we have 0 = + X
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87 11. Power series; moment and pro
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89 • If P ∞ =0 converges fo
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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
Page 107 and 108:
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
Page 117 and 118:
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
Page 185 and 186:
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
Page 193 and 194:
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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