Asymptotic Methods in Statistical Inference - Statistics Centre

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152 • Asymptotic normality. By Taylor’s Theorem, expanding ( )around =0gives ( ) = ( 0 )+˙( 0 ; 1 ) + (), whence ( 1 ) = ( 0 )+˙( 0 ; 1 )+Remainder. Typically ˙( 0 ; 1 )= 1 [()] (17.1) for some function . With 1 = we obtain () =(). Then with 1 = 0 we obtain Thus 0 [()] = 0 ( 1 )= ( 0 )+ 1 [()] + Remainder and then, with 1 = ˆ , we have (a ‘Mean Value Theorem’) √ ³ ³ ˆ ´ − (0 )´ = 1 √ X =1 ( )+ √ where the ( ) are i.i.d. r.v.s with mean 0 and variance 2 ( 0 )= 0 h 2 () i
153 By the CLT and Slutsky’s theorem, if √ → 0 (17.2) where = ³ ˆ ´ − (0 ) − 1 X =1 ( ) we have √ ³ ³ ˆ ´ − (0 )´ → ³ 0 2 ( 0 )´ (17.3) Sometimes we just use (17.3) as a guide to what to expect, and then prove it using another technique. In other cases we can verify (17.1) and (17.2) so as to infer (17.3). The latter approach is taken in this example. Put ( )= h ( () − ()) 2 i ,with
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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 ¯ − ´
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 and 134: 1≤≤ 133 — e.g. = 2; let =
Page 135 and 136: 135 • Rao-Blackwell Theorem: We c
Page 137 and 138: 137 • Here we get the variance of
Page 139 and 140: 139 16. Asymptotic normality of U-s
Page 141 and 142: 141 Under , ( + ) = () ∼ (0 1)
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: 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
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 and 190: 189 Define the ‘scores’ ( −
Page 191 and 192: 191 • The MLE is not the only eff
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
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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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