Asymptotic Methods in Statistical Inference - Statistics Centre

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86 Thus, for any 0, ( ∈ ) ³ √ = || ≤ 2 ´ ( ³ √ || ≤ 2 || − 1| ´ · = (| − 1| ) ( ³ √ || ≤ 2 || − 1| ≥ ´ · + (| − 1| ≥ ) so that = ≤ ≤ ¯ ¯ ( ∈ ) − (1 − ) ¯ ¯ ≤ ³ √ || ≤ 2 || − 1| ´ · (| − 1| ) − (1 − ) ¯ + (| − 1| ≥ ) ¯ = ¯ · − (1 − ) ¯ + ³ 1 − ´ ³ ³ ¯ ¯ − (1 − ) − 1 − ´¯¯¯ + 1 − ´ ¯ ¯ ¯ − (1 − ) ¯ + ³ 1+ ´³ 1 − ´ ¯ ¯ ¯ − (1 − ) ¯ +2 ³ 1 − ´ (9.2) ¯ ) )
87 We exhibit 1 () 2 () → 1− and () →0 such that → 1 →∞ ³ √ 1): 1 () ≤ || ≤ 2 || − 1| ´ ≤ 2 () 2): (| − 1| ) ≥ () Then (9.2) is, uniformly in , ≤ () = max{| 1 () − (1 − )| | 2 () − (1 − )|} +2 (1 − ()) Now () → max {| 1 () − (1 − )| | 2 () − (1 − )|} as →∞, and this may be made arbitrarily small. For 1) note that ³ || ≤ 2 √ || − 1| ´ is maximized by choosing aslargeaspossible(= 1+), yielding 2 () =2Φ ³ 2 √ 1+´ − 1, and similarly 1 () =2Φ ³ √ 2 1 − ´ − 1. Here we use the independence of and . Both these bounds → 1 − as → 0.
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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
Page 35 and 36: 35 • Delta method. Suppose that
Page 37 and 38: 37 • Uniformity—read§2.6 • C
Page 39 and 40: 39 suppose that → 0. Then X =1
Page 41 and 42: 41 • Under this condition ˆ is (
Page 43 and 44: 43 5. Introduction to asymptotic te
Page 45 and 46: 45 • We often work instead with t
Page 47 and 48: 47 • Two-sample problems. Suppose
Page 49 and 50: 49 • Example 2: 1 ∼ P(),
Page 51 and 52: 51 term is Ã √ ˆ ( ( 0 )
Page 53 and 54: 53 • More useful is to study the
Page 55 and 56: 55 • Example: t-test of a mean.
Page 57 and 58: 57 • Efficacy. Test : = 0 vs.
Page 59 and 60: 59 Put = − and test for tre
Page 61 and 62: 61 7. Relative efficiency • Relat
Page 63 and 64: • The constraint (1) ∼ (2)
Page 65 and 66: 65 =1. Theefficacy is = 0 q (0)
Page 67 and 68: 67 • Example 3. In this example
Page 69 and 70: 69 Proof of ‘ 1’: From 1 − (
Page 71 and 72: 71 • We will consider only ‘lev
Page 73 and 74: 73 • Example 1. Suppose we (mista
Page 75 and 76: 75 To see this take 0 = 0 for simp
Page 77 and 78: 77 The numerator of is normally d
Page 79 and 80: 79 9. Confidence intervals • X =(
Page 81 and 82: 81 • Example. 1 ∼ () ( 0);
Page 83 and 84: 83 • A strong CI on a population
Page 85: 85 • A more involved example of a
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 -
Page 99 and 100: 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
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137 • Here we get the variance of
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139 16. Asymptotic normality of U-s
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141 Under , ( + ) = () ∼ (0 1)
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143 • Proofofclaim1: [( − ) |
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145 • V-statistics. Recall that i
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147 17. Influence function analysis
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149 By using the fact that | 2 −
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151 — e.g. If ( )= [], then ³
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153 By the CLT and Slutsky’s theo
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But −1 X =1 155 ( ≥ ) =
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157 18. Bootstrapping • Suppose t
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159 • In each of the examples abo
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161 • Example 2. ( )=bias. Writ
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163 • For the measure (18.1) we t
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165 (True for functionals (·) con
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167 • Indeed, since ( ) has plu
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169 Part V LIKELIHOOD METHODS
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171 By this, for large samples and
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173 • Assume: (C1) Identifiabilit
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175 • Under stronger conditions o
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177 Then (here we use only twice; t
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179 • Information Inequality. (Cr
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181 • Example: 1 the indicator
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183 Theorem: Under (C1)-(C7) and th
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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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