Asymptotic Methods in Statistical Inference - Statistics Centre

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8 1. Convergence concepts • Large sample procedures are typically used to give tractable, approximate solutions to otherwise intractable problems. The example at the end of this lecture illustrates the fact that the ‘improvement’ realized by an exact solution, over an approximate one, is typically of a smaller order (asymptotically) than the underlying sampling variation. • Example: We commonly base inferences about a population mean on the ‘t-statistic’ √ (¯ − ) = If the — independent, identically distributed (‘i.i.d.’) — observations are non-normal then the exact distribution of is, in general, very intractable. But ( ≤ ) → Φ() as →∞
The ‘rate’ of convergence refers to the fact that the error in the approximation is typically of order 1 √ , and a correction of the form ( ≤ ) =Φ()+ 1 √ () + smaller order terms (1.1) is available (Edgeworth expansion). Question — what if the observations are not even independent? 9 • Sequences { } { } — They are asymptotically equivalent ( ∼ ) as →∞if → 1 — is of a smaller order than ( = ( )) if → 0 — is of a smaller, or the same, order as ( = ( )) if | | is bounded — and are of the same order ( ³ ) if = ( )and = ( )
Page 1 and 2: STATISTICS 665 ASYMPTOTIC METHODS I
Page 3 and 4: II LARGE-SAMPLE INFERENCE 42 5 Intr
Page 5 and 6: V LIKELIHOOD METHODS 169 19 Maximum
Page 7: Part I PROBABILISTIC PRELIMINARIES
Page 11 and 12: 11 imagine ‘large’ but remain
Page 13 and 14: 13 • For the ( ) distribution,
Page 15 and 16: 15 —Corollary1: → If h (
Page 17 and 18: 17 2. Slutsky’s Theorem; conseque
Page 19 and 20: — By the CLT, − q (1 − ) 1
Page 21 and 22: 21 • Corollary 1: If → the
Page 23 and 24: 23 • First an application. Let 2
Page 25 and 26: 25 with d.f. (), then there is a
Page 27 and 28: 27 next lecture), = 1− " ( )+
Page 29 and 30: 29 • CLT via the Edgeworth expans
Page 31 and 32: 31 (why?) with ‘cumulant generati
Page 33 and 34: 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.
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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
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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, (;
Page 229 and 230:
229 • Example 2. Let = P 1 w
Page 231:
231 • The error of Saddlepoint
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Asymptotic Methods in Statistical Inference - Statistics Centre

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