Asymptotic Methods in Statistical Inference - Statistics Centre

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104 12. Biased estimation; Pitman closeness • Biased estimation. In general, to compare estimators 1 2 of () with variances 2 () and biases () welookattheARE 2 1 = lim ( 1) ( 2 ) = lim ( 1)+ 2 ( 1 ) ( 2 )+ 2 ( 2 ) In the examples studied previously 2 was of order ( −2 ) and the ARE reduced to a comparison of the limiting, or asymptotic, variances. We look below at an example in which and 2 are of the same order, and in which 2 plays a significant role, asymptotically. First we look at another measure of performance of an estimator. • A related measure of accuracy is ‘Pitman closeness’ (| − | ≤ ) for specified ; onewants this probability to be large. For an asymptotic treatment we would typically have to normalize, and consider something like lim ( √ | − | ≤ ).
105 In a broad class of cases, comparisons based on Pitman closeness of two estimators reduce to comparing the (exact or asymptotic) biases. Suppose that is the d.f. of ( − []) , andthat is symmetric about 0 with a unimodal density (i.e. is a decreasing function of ||). Then with (bias) = []− we have that the Pitman closeness is (| − | ≤ ) = µ ∙ µ ¸ − + − 1 − = () say. This is an even function of whose derivative is µ µ ¸ () = ∙ −1 + − − This (odd) function of is 0if0 (since then | + | | − |); thus under these conditions the Pitman closeness is a decreasing function of || for any 0. Thus in comparing two estimators, for each of which ( − []) ∼ exactly or asymptotically (same , same ) the estimate with smaller (absolute) bias || — hence smaller mse — is Pitman closer.
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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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Page 55 and 56: 55 • Example: t-test of a mean.
Page 57 and 58: 57 • Efficacy. Test : = 0 vs.
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Page 61 and 62: 61 7. Relative efficiency • Relat
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Page 67 and 68: 67 • Example 3. In this example
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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);
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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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Page 101 and 102: 101 — This is also the ARE of the
Page 103: 103 We minimize Z h 2 ()+2 2 ()
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,
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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
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Page 145 and 146: 145 • V-statistics. Recall that i
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Page 149 and 150: 149 By using the fact that | 2 −
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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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