Asymptotic Methods in Statistical Inference - Statistics Centre

More documents

Recommendations

Info

184 it suffices (Slutsky!) to establish: (i) 0 ( 0 ) √ → (0( 0 )) (ii) − 00 ( 0 ) → ( 0 ) 0 Proofof(i): 0 ( 0 ) √ = 1 √ P log ( ) |=0 , a normalized sum of i.i.d. r.v.s with mean 0 and variance ( 0 ). Proofof(ii): − 00 ( 0 ) = 1 X Ã ! − 2 2 log ( ) " → 0 − 2 2 log () # | 0 = ( 0 ) • There may be multiple roots of the likelihood equation, in which case the theory so far does not tell us which one to choose. This can be dealt with as follows. Assume that we have some estimator ˜ which is √ -consistent, i.e. √ ³˜ − 0´ is (1). This holds in particular if √ ³˜ − 0´
185 has a limit distribution, so that for instance a plug-in estimator could be used. Now perform one step of the Newton-Raphson method to solve 0 () = 0, starting with ˜ : ³˜ ´ = ˜ − 0 (21.1) ³˜ ´ We will show that, under these conditions (and the assumptions of the previous theorem), √ ( − 0 ) → ³ 0 −1 ( 0 )´ (21.2) Note that by this, is itself √ -consistent and so can be used as the starting point for one more iteration. A common prescription is the ‘3-step’ estimate — do three steps of Newton-Raphson, starting with ˜ . (Butwhynot‘ThreestepsifI like what I see, a few more otherwise’?) 00
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 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.
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 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 -
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
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
Page 153 and 154: 153 By the CLT and Slutsky’s theo
Page 155 and 156: But −1 X =1 155 ( ≥ ) =
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: 183 Theorem: Under (C1)-(C7) and 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
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
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
Page 217 and 218: 217 • If instead of testing =
Page 219 and 220: 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
show all

Asymptotic Methods in Statistical Inference - Statistics Centre

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?