Asymptotic Methods in Statistical Inference - Statistics Centre

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190 22. Efficiency; multiparameter estimation; method of moments • Efficiency. Recall that if ˆ is the MLE (or a one-step approximation) then √ ³ ³ˆ − ´ → 0 −1 ()´ and −1 () ≤ [ √ ( − )] for any unbiased . Consider now estimates of , not necessarily unbiased in finite samples but satisfying √ ( − ) → (0()) with () continuous and 0 () ∞. Under (C1)-(C7) it can be shown that, if () is also continuous, then () ≥ −1 (). An estimator attaining this lower bound is called efficient. In particular, the MLE is efficient if () iscontinuous. By the delta method, if is efficient for then ( )isefficient for () at all points where 0 () 6= 0,since √ ( ( ) − ()) → ⎛ ⎝0 £ 0 () ¤ 2 () = −1 ( ()) ⎞ ⎠
191 • The MLE is not the only efficient estimator, for instance ‘+ ³ −12´’ isalsoefficient. A less trivial example is if 1 ∼ ³ = 2´ for a known (= 1). To obtain the MLE ˆ it is convenient to first obtain ˆ, where =1. We define 2 = P 2 and calculate ( () ∝ − exp − 1 X ( − ) 2 ) 2 2 ½ = exp − 1 X ( − ) 2¾ 2 () = + log − 1 2 2 2 + ¯ 0 () = ( 1 − 2 + ¯ and then ) ˆ = r ³ ¯ + ¯´2 +42 2 2 ˆ = 1ˆ = s µ 2 ¯ 2 + 2 − 2 ¯
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STATISTICS 665 ASYMPTOTIC METHODS I
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II LARGE-SAMPLE INFERENCE 42 5 Intr
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V LIKELIHOOD METHODS 169 19 Maximum
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Part I PROBABILISTIC PRELIMINARIES
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The ‘rate’ of convergence refer
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11 imagine ‘large’ but remain
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13 • For the ( ) distribution,
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15 —Corollary1: → If h (
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17 2. Slutsky’s Theorem; conseque
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— By the CLT, − q (1 − ) 1
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21 • Corollary 1: If → the
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23 • First an application. Let 2
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25 with d.f. (), then there is a
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27 next lecture), = 1− " ( )+
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29 • CLT via the Edgeworth expans
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31 (why?) with ‘cumulant generati
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Thus in (3.2), () () isthec.f.of
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35 • Delta method. Suppose that
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37 • Uniformity—read§2.6 • C
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39 suppose that → 0. Then X =1
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41 • Under this condition ˆ is (
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43 5. Introduction to asymptotic te
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45 • We often work instead with t
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47 • Two-sample problems. Suppose
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49 • Example 2: 1 ∼ P(),
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51 term is Ã √ ˆ ( ( 0 )
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53 • More useful is to study the
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55 • Example: t-test of a mean.
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57 • Efficacy. Test : = 0 vs.
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59 Put = − and test for tre
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61 7. Relative efficiency • Relat
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• The constraint (1) ∼ (2)
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65 =1. Theefficacy is = 0 q (0)
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67 • Example 3. In this example
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69 Proof of ‘ 1’: From 1 − (
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71 • We will consider only ‘lev
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73 • Example 1. Suppose we (mista
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75 To see this take 0 = 0 for simp
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77 The numerator of is normally d
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79 9. Confidence intervals • X =(
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81 • Example. 1 ∼ () ( 0);
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83 • A strong CI on a population
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85 • A more involved example of a
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87 We exhibit 1 () 2 () → 1−
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10. Point estimation; Asymptotic re
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91 Recall that if 0 () 6= 0 (assum
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93 where (with being a bound on |
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• Asymptotic Relative Efficiency.
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97 0.0 0.1 0.2 0.3 0.4 0.5 0.6 -3 -
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99 that then √ ³ ¯ − ´
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101 — This is also the ARE of the
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103 We minimize Z h 2 ()+2 2 ()
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105 In a broad class of cases, comp
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107 The ARE of the estimates consid
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109 We have lim →∞ { ∗ () −
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111 13. Random vectors; multivariat
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113 • X () () → c (constant)
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115 • Multivariate normality. We
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117 • If X ∼ (μ Σ) then an
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119 14. Multivariate applications
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121 One can eliminate the last elem
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123 have = (|)+ ¯ (| ¯) =
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125 at = 1. In this latter case Ã
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127 Proof: Write Then +1 = − X
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129 15. Expectation functionals; U-
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131 • A closely related estimate,
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1≤≤ 133 — e.g. = 2; let =
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135 • Rao-Blackwell Theorem: We c
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137 • Here we get the variance of
Page 139 and 140: 139 16. Asymptotic normality of U-s
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Page 143 and 144: 143 • Proofofclaim1: [( − ) |
Page 145 and 146: 145 • V-statistics. Recall that i
Page 147 and 148: 147 17. Influence function analysis
Page 149 and 150: 149 By using the fact that | 2 −
Page 151 and 152: 151 — e.g. If ( )= [], then ³
Page 153 and 154: 153 By the CLT and Slutsky’s theo
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Page 157 and 158: 157 18. Bootstrapping • Suppose t
Page 159 and 160: 159 • In each of the examples abo
Page 161 and 162: 161 • Example 2. ( )=bias. Writ
Page 163 and 164: 163 • For the measure (18.1) we t
Page 165 and 166: 165 (True for functionals (·) con
Page 167 and 168: 167 • Indeed, since ( ) has plu
Page 169 and 170: 169 Part V LIKELIHOOD METHODS
Page 171 and 172: 171 By this, for large samples and
Page 173 and 174: 173 • Assume: (C1) Identifiabilit
Page 175 and 176: 175 • Under stronger conditions o
Page 177 and 178: 177 Then (here we use only twice; t
Page 179 and 180: 179 • Information Inequality. (Cr
Page 181 and 182: 181 • Example: 1 the indicator
Page 183 and 184: 183 Theorem: Under (C1)-(C7) and th
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Page 187 and 188: 187 Claim: (i) (ii) (iii) 0 ( 0 )
Page 189: 189 Define the ‘scores’ ( −
Page 193 and 194: 193 The variance-minimizing choice
Page 195 and 196: 195 • Multiparameter likelihood e
Page 197 and 198: 197 • Efficiency. Under appropria
Page 199 and 200: 199 asymptotic normality of the sam
Page 201 and 202: 201 • Likelihood ratio test. Put
Page 203 and 204: 203 and 000 ( ∗∗ ) is (1)
Page 205 and 206: 205 • Example. 1 ∼ Logistic,
Page 207 and 208: 207 • Multiparameter inferences.
Page 209 and 210: 209 For testing, I( 0 ) is replaced
Page 211 and 212: 211 24. Examples Example: 1 ∼
Page 213 and 214: 213 with and I (θ 0 )= 1 I 11 (θ
Page 215 and 216: 215 • The likelihood ratio statis
Page 217 and 218: 217 • If instead of testing =
Page 219 and 220: 219 The parameter vector is θ = ³
Page 221 and 222: 221 25. Higher order asymptotics
Page 223 and 224: 223 As in Lecture 10, cov h i 2 h
Page 225 and 226: 225 • The normal approximation, a
Page 227 and 228: 227 By uniqueness of m.g.f.s, (;
Page 229 and 230: 229 • Example 2. Let = P 1 w
Page 231: 231 • The error of Saddlepoint
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