Lecture 34 Bootstrap confidence intervals


 Αθος Αλεξόπουλος
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1 Lecture 34 Bootstrap confidence intervals
2 Confidence Intervals θ: an unknown parameter of interest We want to find limits θ and θ such that Gt = P nˆθ θ t If G 1 1 α is known, then P θ θ = P θ θ = 1 α θ = ˆθ G 1 1 α/ n is an exact 1001 α% lower confidence limit for θ Traditional asymptotic approach: Gt Φt/σ, n Φ: the standard normal cdf σ: an unknown scale parameter ˆσ: a consistent estimator of σ, Gˆσt Φt Normal approximation NA 1001 α% confidence limits are θ N = ˆθ ˆσz 1 α / n, θ N = ˆθ + ˆσz 1 α / n z a = Φ 1 a
3 Bootstrap Confidence Intervals 1 The hybrid bootstrap HB A bootstrap estimator of Gt = P nˆθ θ t is Ĝt = P nˆθ ˆθ t G 1 1 α can be estimated by Ĝ 1 1 α HB lower and upper confidence limits: θ HB = ˆθ Ĝ 1 1 α/ n θ HB = ˆθ Ĝ 1 α/ n If Gt is nearly symmetric, then Ĝ 1 α can be replaced by Ĝ 1 1 α Hybrid: Use bootstrap estimate in the construction of confidence interval based on the sampling distribution of ˆθ θ.
4 2 Bootstrapt BT Gt = P nˆθ θ t Φt/σ ˆσ: a consistent estimator of σ Studentized statistic: nˆθ θ/ˆσ Ht = P nˆθ θ/ˆσ t Φt ˆσ : a bootstrap analog of ˆσ A bootstrap estimator of Ht: Ĥt = P nˆθ ˆθ/ˆσ t BT lower and upper confidence limits: θ BT = ˆθ ˆσĤ 1 1 α/ n θ BT = ˆθ + ˆσĤ 1 1 α/ n This is also a hybrid method It is called BT because it is based on the studentized variable nˆθ θ/ˆσ
5 3 The bootstrap percentile BP Bootstrap distribution histogram Kt = P ˆθ t 1 B 1001 α% BP lower confidence limit # of times ˆθ b t θ BP = K 1 α = inf{t : Kt α} ˆθ B θ BP αbth ordered value of ˆθ 1,..., 1001 α% BP upper confidence limit θ BP = K 1 1 α = inf{t : Kt 1 α} θ BP 1 αbth ordered value of ˆθ 1,..., There are two improvements over the BP ˆθ B
6 Accuracy: Justification Properties of the bootstrap percentile BP Lower confidence bound level 1 α θ BP = K 1 α, Kt = P ˆθ t Assumption A: There is a monotone transformation φ such that P ˆφ φ t = Ψt for all t and all P φ = φθ, ˆφ = φˆθ Ψ: a continuous, increasing cdf symmetric about 0 Result 1. If φ is known, then θ E = φ 1 ˆφ + z α, z α = Ψ 1 α, is an exact 1001 α% lower confidence bound, i.e., P θ E θ = 1 α
7 Proof P θ E θ = P φθ E φθ = P φφ 1 ˆφ + z α φ = P ˆφ + zα φ = P ˆφ φ zα = Ψ z α = 1 Ψz α = 1 α
8 Result 2. Under Assumption A, θ BP = θ E. This means that the BP interval is exactly correct under Assumption A To use the BP, we don t need to know φ The derivation of φ is replaced by computation If Assumption A holds approximately when n is large, then the BP intervals are approximately correct
9 Proof Let w = φθ BP ˆφ and ˆφ = φˆθ Since Assumption A holds for all P including P, Ψw = P ˆφ ˆφ w = P ˆφ ˆφ φθ BP ˆφ = P φˆθ φθ BP = P ˆθ θ BP = Kθ BP = α By the uniqueness of the quantile, w = Ψ 1 α = z α Then Hence φθ BP = ˆφ + z α θ BP = φ 1 ˆφ + z α = θ E
10 The bootstrap biascorrected percentile BC Is Assumption A too strong? Assumption B: There is a monotone transformation φ and a constant z 0 such that P ˆφ φ + z 0 t = Ψt for all t and all P Ψ: a continuous, increasing cdf symmetric about 0 If z 0 = 0, then Assumption B reduces to Assumption A. z 0 is a bias correction. Since Assumption B holds for all P including P, i.e., z 0 = Ψ 1 Kˆθ Kˆθ = P ˆθ ˆθ = P ˆφ ˆφ = P ˆφ ˆφ + z 0 z 0 = Ψz 0
11 Result 3. θ E = φ 1 ˆφ + z α + z 0 is an exact 1001 α% lower confidence limit for θ under Assumption B Proof: Similar to the proof of result 1. We need to know φ and Ψ in order to use θ E The BC confidence limits are θ BC = K 1 Ψz α + 2z 0 = K 1 Ψz α + 2Ψ 1 Kˆθ θ BC = K 1 Ψz 1 α + 2Ψ 1 Kˆθ When bootstrap Monte Carlo of size B is applied, θ BC BΨz α + 2z 0 th ordered value of ˆθ 1,..., ˆθ B θ BC BΨz 1 α + 2z 0 th ordered value of ˆθ 1,..., ˆθ B BC = BP if Kˆθ = 0.5, since Ψ = 0 i.e., ˆθ is the median of Kt bootstrap histogram Thus, BC is a biascorrected BP It shifts the BP bound towards left or right according to z 0 BC does not need the explicit form of φ But BC requires the form of Ψ typically, Ψ = Φ
12 Result 4. Under Assumption B, θ BC = θ E BC is exactly correct if Assumption B holds. If Assumption B holds approximately, then BC is approximately correct Proof: For any α, 1 α = Ψz 1 α = P ˆφ ˆφ + z 0 z 1 α = P ˆφ ˆφ + Ψ 1 1 α z 0 = P ˆθ φ 1 ˆφ + Ψ 1 1 α z 0 = Kφ 1 ˆφ + Ψ 1 1 α z 0 Hence, for all t, K 1 t = φ 1 ˆφ + Ψ 1 t z 0 Let t = Ψz α + 2z 0, K 1 Ψz α + 2z 0 = φ 1 ˆφ + z α + 2z 0 z 0 = φ 1 ˆφ + z α + z 0 = θ E
13 The bootstrap accelerated biascorrected percentile BC a Assumption B is weaker than Assumption A Assumption B is still too strong in some cases Weaker assumptions? Assumption C: There is a monotone transformation φ and constants z 0 and a acceleration constant such that ˆφ φ P 1 + aφ + z 0 t = Ψt for all t and all P Ψ: a continuous, increasing cdf symmetric about 0 If a = 0, Assumption C reduces to Assumption B. a corrects skewness
14 Result 5. Under Assumption C, θ E = φ 1 ˆφ + z α + z a ˆφ 1 az α + z 0 is an exact 1001 α% lower confidence limit for θ Proof: 1 α = Ψ z α ˆφ φ = P 1 + aφ + z 0 z α = P ˆφ φ zα + z aφ ˆφ + zα + z 0 = P 1 az α + z 0 φ = P ˆφ + z α + z a ˆφ φ 1 az α + z 0 = P θ E θ
15 We need to know a, Ψ, and φ in order to use θ E Under Assumption C, z 0 = Ψ 1 Kˆθ First, assume a is known The BC a lower confidence bound is θ BCa = K Ψ 1 z α + z 0 z az α + z 0 The BC a upper confidence bound is θ BCa = K Ψ 1 z 1 α + z 0 z az 1 α + z 0 When bootstrap Monte Carlo of size B is applied, θ BCa BΨ z 0 + zα+z 0 1 az α+z 0 th ordered value of ˆθ 1,..., ˆθ B θ BCa BΨ z 0 + z 1 α+z 0 1 az 1 α +z 0 th ordered value of ˆθ 1,..., ˆθ B We don t need the explicit form of φ But we need to know a and Ψ If a = 0, BC a reduces to BC
16 Result 6. If a is known, then θ BCa = θ E. Hence, BC a is exactly correct if Assumption C holds. If Assumption C holds approximately, then BC a is approximately correct Proof. 1 α = Ψz 1 α ˆφ = P ˆφ 1 + a ˆφ + z 0 z 1 α = P ˆφ ˆφ + z 1 α z a ˆφ ˆθ = P φ 1 ˆφ + z 1 α z a ˆφ = K φ 1 ˆφ + z 1 α z a ˆφ = K φ 1 ˆφ + Ψ 1 1 α z a ˆφ This shows that K 1 t = φ 1 ˆφ + Ψ 1 t z a ˆφ
17 Let t = Ψ z 0 + zα+z 0 1 az α+z 0, θ E = φ 1 ˆφ + z α + z a ˆφ 1 az α + z 0 = K Ψ 1 z 0 + = θ BCa z α + z 0 1 az α + z 0 The constant a is unknown and has to be estimated â: an estimator of a θ BCâ and θ BCâ are BCâ confidence limits They are also called ABC confidence limits
18 Asymptotic Accuracy A confidence set C is first order accurate if and second order accurate if P θ C = 1 α + On 1/2 P θ C = 1 α + On 1 For the case of ˆθ is a smooth function of sample means, we have shown the following summary: 1. The BT and bootstrap BC a onesided confidence intervals are second order accurate. 2. The the BP, BC, HB, and NA onesided confidence intervals are in general first order accurate. 3. The equaltail twosided confidence intervals produced by all five bootstrap methods and the normal approximation are second order accurate errors cancel each other.
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