Asymptotic Methods in Statistical Inference - Statistics Centre

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134 An optimality property of U-statistics. Recall that, in any family F of distributions, the vector X () of order statistics is a sufficient statistic for ∈ F. Given X () = x () , the data must have been some re-arrangement of x () : ³ X = x|X() = ³ 1 (1) ()´´ = ! for each permutation x = ³ ³ 1 ´ of (1) ()´; this does not depend on the ‘parameter’ . Note that the U-statistic corresponding to ( 1 )is afunctionofX () : h ( 1 ) |X () i = 1 ³ ´ X ( 1 ) ³ 1 ´ = since, given X () , all such a-tuples are equally likely. Example: =3, =2. GivenX (3) = h1 5 6i the sample must have been some permutation of {1 5 6}. For instance 1 =1, 2 =6, 3 =5. Allthreevalues (1 5) = (5 1), (1 6) = (6 1), (6 5) = (5 6) are then equally likely, with h i (1 5) + (1 6) + (6 5) ( 1 2 ) |X (3) = 3
135 • Rao-Blackwell Theorem: We can always reduce the variance of an unbiased estimate by conditioning on X () . Proof: Let = ( 1 ) be any unbiased estimator of and define ˜ = h |X () i Then (Double Expectation Theorem) [ ˜] = X() ½ |X() h |X() i ¾ = [] = Furthermore the decomposition [] = h ³ h ³ |X ()í + |X()í shows that [] ≥ h ³ |X ()í = h ˜ i This inequality is strict unless 0 = h ³ |X ()í ½ ∙ ³ h i´2 = X() |X() − |X() |X()¸¾ = ½ ∙ ³ − ˜´2 |X()¸¾ which holds iff ³ = ˜´ =1. ∙ ³ = − ˜´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);
Page 83 and 84: 83 • A strong CI on a population
Page 85 and 86: 85 • A more involved example of a
Page 87 and 88: 87 We exhibit 1 () 2 () → 1−
Page 89 and 90: 10. Point estimation; Asymptotic re
Page 91 and 92: 91 Recall that if 0 () 6= 0 (assum
Page 93 and 94: 93 where (with being a bound on |
Page 95 and 96: • Asymptotic Relative Efficiency.
Page 97 and 98: 97 0.0 0.1 0.2 0.3 0.4 0.5 0.6 -3 -
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Page 101 and 102: 101 — This is also the ARE of the
Page 103 and 104: 103 We minimize Z h 2 ()+2 2 ()
Page 105 and 106: 105 In a broad class of cases, comp
Page 107 and 108: 107 The ARE of the estimates consid
Page 109 and 110: 109 We have lim →∞ { ∗ () −
Page 111 and 112: 111 13. Random vectors; multivariat
Page 113 and 114: 113 • X () () → c (constant)
Page 115 and 116: 115 • Multivariate normality. We
Page 117 and 118: 117 • If X ∼ (μ Σ) then an
Page 119 and 120: 119 14. Multivariate applications
Page 121 and 122: 121 One can eliminate the last elem
Page 123 and 124: 123 have = (|)+ ¯ (| ¯) =
Page 125 and 126: 125 at = 1. In this latter case Ã
Page 127 and 128: 127 Proof: Write Then +1 = − X
Page 129 and 130: 129 15. Expectation functionals; U-
Page 131 and 132: 131 • A closely related estimate,
Page 133: 1≤≤ 133 — e.g. = 2; let =
Page 137 and 138: 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
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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
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Page 179 and 180: 179 • Information Inequality. (Cr
Page 181 and 182: 181 • Example: 1 the indicator
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185 has a limit distribution, so th
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187 Claim: (i) (ii) (iii) 0 ( 0 )
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189 Define the ‘scores’ ( −
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191 • The MLE is not the only eff
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193 The variance-minimizing choice
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195 • Multiparameter likelihood e
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197 • Efficiency. Under appropria
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199 asymptotic normality of the sam
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201 • Likelihood ratio test. Put
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203 and 000 ( ∗∗ ) is (1)
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205 • Example. 1 ∼ Logistic,
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207 • Multiparameter inferences.
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209 For testing, I( 0 ) is replaced
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211 24. Examples Example: 1 ∼
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213 with and I (θ 0 )= 1 I 11 (θ
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215 • The likelihood ratio statis
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217 • If instead of testing =
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219 The parameter vector is θ = ³
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221 25. Higher order asymptotics
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223 As in Lecture 10, cov h i 2 h
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225 • The normal approximation, a
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227 By uniqueness of m.g.f.s, (;
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229 • Example 2. Let = P 1 w
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231 • The error of Saddlepoint
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