Asymptotic Methods in Statistical Inference - Statistics Centre

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18 • Order in probability. — If → 0thenwesaythat = ( ). — = ( )if is ‘bounded in probability’: For any there is an = () and an = () such that implies (| | ≤ ) ≥ 1 − . Equivalently, lim →∞ (| | ≤ ) = 1. Note that lim →∞ lim →∞ (| | ≤ ) = 1 is not enough — = ∼ ( 1) satisfies this but is not (1). Why not? • In particular, if ∼ ( ), then: — = −1 P =1 ,where = (‘ experiment is a success’) ∼ (1), and so ˆ = → [ 1 ]= by the WLLN. (Bernoulli’s Law of Large Numbers). Thus ˆ − =( − ) → 0, hence is (1).
— By the CLT, − q (1 − ) 19 has a limit distribution, hence is (1) (proven below). • You should show: → ⇒ = (1). • Proof that convergence in law implies bounded in probability: Suppose → .Wearetofind such that (| | ≤ ) ≥ 1− for all sufficiently large . Let be the d.f. of . Suppose we can choose numbers 0 such that: 1. ± are continuity points of ; 2. 0 ⇒ | (±) − (±)| 4(possible since (±) → (±)); 3. () − (−) ≥ 1 − 2 .
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 and 8: Part I PROBABILISTIC PRELIMINARIES
Page 9 and 10: The ‘rate’ of convergence refer
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: 17 2. Slutsky’s Theorem; conseque
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.
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 − (
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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
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77 The numerator of is normally d
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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 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.
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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
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109 We have lim →∞ { ∗ () −
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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
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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-
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 and 152:
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 ( ≥ ) =
Page 157 and 158:
157 18. Bootstrapping • Suppose t
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159 • In each of the examples abo
Page 161 and 162:
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
Page 167 and 168:
167 • Indeed, since ( ) has plu
Page 169 and 170:
169 Part V LIKELIHOOD METHODS
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171 By this, for large samples and
Page 173 and 174:
173 • Assume: (C1) Identifiabilit
Page 175 and 176:
175 • Under stronger conditions o
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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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185 has a limit distribution, so th
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
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197 • Efficiency. Under appropria
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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
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217 • If instead of testing =
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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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