fujisawa@ism.ac.jp 1
Contents 1. 2. 3. 4. γ- 2
1. 3
10 5.6, 5.7, 5.4, 5.5, 5.8, 5.5, 5.3, 5.6, 5.4, 5.2. 5.5 5.6 +5.7 +5.4 +5.5 +5.8 +5.5 +5.3 +5.6 +5.4 +5.2 =5.5. 10 outlier 5 5.6, 5.7, 5.4, 5.5, 5.8, 5.5, 5.3, 5.6, 5.4, 55.2. 10.5 5.6 +5.7 +5.4 +5.5 +5.8 +5.5 +5.3 +5.6 +5.4 +55.2 10 =10.5. 4
5.6, 5.7, 5.4, 5.5, 5.8, 5.5, 5.3, 5.6, 5.4, 5.2. 5.6, 5.7, 5.4, 5.5, 5.8, 5.5, 5.3, 5.6, 5.4, 55.2. 5.2 55.2. 5.5 10.5. x = x 1 + x 2 + + x n 1 + x n n as x n 5
10.5 (robust estimation) (robust inference) 6
ˆρ =0.44 ˆρ =0.91 ˆρ rob =0.85 7
Hampel et al. (1986) 8
2. 9
μ 5.6, 5.7, 5.4, 5.5, 5.8, 5.5, 5.3, 5.6, 5.4, 55.2. μ (median) x 1,...,x n x (1) x (2) x (n) Med({x i }) = x (k) n =2k 1 = (x (k 1) + x (k) )/2 n =2k 10
5.6, 5.7, 5.4, 5.5, 5.8, 5.5, 5.3, 5.6, 5.4, 55.2. 5.3, 5.4, 5.4, 5.5, 5.5, 5.6, 5.6, 5.7, 5.8, 55.2. Med({x i })=(5.5 +5.6)/2 =5.55 μ 11
0.0 0.1 0.2 0.3 0.4 0.5 dlnorm (x) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 4 2 0 2 4 x 0 2 4 6 8 x 12
σ 5.6, 5.7, 5.4, 5.5, 5.8, 5.5, 5.3, 5.6, 5.4, 5.2. 5.6, 5.7, 5.4, 5.5, 5.8, 5.5, 5.3, 5.6, 5.4, 55.2. S 2 = 1 n 1 S 2 =0.185 n (x i x) 2 S 2 =15.7 ˆσ rob (= MADN) =0.222 13
μ (trimmed mean) x (1) x [nα] x n [nα] x (n) 100α% 100α% ˆμ = 1 n 2[nα] n [nα] i=[nα]+1 x (i) [a] a 14
N(μ, σ 2 ) σ μ n (x i μ) =0 (μ = x) 15
μ φ(x; μ, σ 2 ) { } φ(x; μ, σ 2 1 )= 2πσ 2 exp (x μ)2 2σ 2 x φ(x ; μ, σ 2 ) 16
μ ˆμ n φ(x i ; μ, σ 2 )(x i μ) =0. σ 0.158 ˆμ =5.517 17
σ 2 f(x; θ) 18
3. x f θ (x) =f(x; θ) Basu et al. (1998, Biometrika) Eguchi and Kano (1998, unpublished) Jones et al. (2001, Biometrika) 19
μ n φ(x i ; μ, σ 2 )(x i μ) =0 n s(x i ; θ) =0 s(x; θ) = d dθ log f(x; θ) n f(x i ; θ) β s(x i ; θ) =0 β>0 20
n f(x i ; θ) β s(x i ; θ) =0. ] E fθ [f(x; θ) β s(x; θ) =0 μ ] ψ(x; θ) =f(x; θ) β s(x; θ) E fθ [f(x; θ) β s(x; θ) E fθ [ψ(x; θ)] = 0. 21
M 0= 1 n n 0 = 1 n = 1 n n n ψ(x i ; θ) [ ]} {f(x i ; θ) β s(x i ; θ) E fθ f(x; θ) β s(x; θ) f(x i ; θ) β s(x i ; θ) f(x; θ) 1+β s(x; θ)dx 22
KL ˆθ = arg max θ 1 n n log f(x i ; θ) 0= 1 n n s(x i ; θ) 1 n n log f(x i ; θ) 23
d KL (g, f θ )= E g [log f(x; θ)] = g(x) log f(x; θ)dx KL [ D KL (g, f) = E g log g(x) ] = E g [log g(x)] E g [log f(x)] f(x) = d KL (g, g) +d KL (g, f) ( ) D KL (g, f) 0. D KL (g, f) =0 g = f. 24
β- 0= 1 n f(x i ; θ) β s(x i ; θ) f(x; θ) 1+β s(x; θ)dx n KL β- d β (g, f θ )= 1 β g(x)f (x; θ) β dx + 1 1+β f(x; θ) 1+β dx 25
β- (β-divergence, density power divergence) D β (g, f) = d β (g, g) +d β (g, f) 1 = g(x) 1+β dx 1 β(1 + β) 1+β + 1 f(x) 1+β dx 1+β g(x)f (x) β dx KL lim D β(g, f) =D KL (g, f). β 0 26
γ- 0= 1 n f(x i ; θ) γ s(x i ; θ) E fθ [f(x; θ) γ s(x; θ)] n 1 0= 1 n / 1 n f(x i ; θ) γ s(x i ; θ) f(x i ; θ) γ n n / E fθ [f(x; θ) γ s(x; θ)] E fθ [f(x; θ) γ ] γ- d γ (g, f θ )= 1 γ log g(x)f (x; θ) γ dx + 1 1+γ log f(x; θ) 1+γ dx 27
γ- D γ (g, f) = d γ (g, g) +d γ (g, f) 1 = γ(1 + γ) log g(x) 1+γ dx 1 1+γ log + 1 1+γ log f(x) 1+γ dx g(x)f (x) γ dx KL lim D γ(g, f) =D KL (g, f). γ 0 28
β- d β (g, f) = 1 β g(x)f (x) β dx + 1 1+β f(x) 1+β dx γ- d γ (g, f) = 1 γ log g(x)f (x) γ dx + 1 1+γ log f(x) 1+γ dx log 29
4. γ- Fujisawa and Eguchi (2008). Robust parameter estimation with a small bias against heavy contamination. Journal of Multivariate Analysis, Vol.99, 2053-2081. 30
g(x) =(1 ε)f (x)+εδ(x) f: (= f θ ) δ: ε: ( { } 1/γ0 ν f = f(x)δ(x)dx 0 f(x) γ 0 δ(x)dx 0 γ0 >γ>0) 31
{ 1/γ0 ν f = f(x) γ 0 δ(x)dx} 0 x δ(x) =δ x (x) ν f = f(x ) 0 32
Review: δ a (x) f(x)δ a (x)dx = f(a) ( ) ḡ(x) = 1 n δ xi (x) g(x) n E g [h(x)] = g(x)h(x)dx ḡ(x)h(x)dx = 1 n n h(x i ) 33
γ- d γ (g, f) = 1 γ log g(x)f (x) γ dx + 1 1+γ log f(x) 1+γ dx γ d γ (ḡ, f) = 1 γ log ḡ(x)f (x) γ dx + 1 1+γ log = 1 γ log 1 n f(x i ) γ + 1 n 1+γ log f(x) 1+γ dx f(x) 1+γ dx 34
(γ-) ˆθ γ = arg min d γ (ḡ, f θ ) ( θ { = arg min dγ (ḡ, ḡ) +d γ (ḡ, f θ ) } ) = arg min D γ (ḡ, f θ ) θ θ θγ = arg min d γ(g, f θ ) θ θ γ θ ( 0) = O (εν γ ) 35
D γ (g, f) = d γ (g, g)+d γ (g, f) D γ (g, f θ ) = D γ (g, f) + D γ (f, f θ ) + O(εν γ ) h D fl (g; h) D fl (f; h) f D fl (g; f) g =(1 ")f + "ffi 36
(ε =0.05) g =(1 ε)n (0, 1) + εn(6, 1) f θ = N(μ, σ 2 ) 0.3 0.25 0.2 0.15 0.1 0.05 μ 0.2 0.4 0.6 0.8 1 σ 0.6 θγ : d γ(g, f) 0.5 0.4 θ (m) β : d β(g, f) 0.3 0.2 0.1 0.2 0.4 0.6 0.8 1 x : y : γ or β 37
(ε =0.2) g =(1 ε)n (0, 1) + εn(6, 1) f θ = N(μ, σ 2 ) 1.2 1 0.8 0.6 0.4 0.2 μ 0.2 0.4 0.6 0.8 1 σ 1.5 θγ : d γ(g, f) 1 θ (m) β : d β(g, f) 1.25 0.75 0.5 0.25 0.2 0.4 0.6 0.8 1 x : y : γ or β 38
N(μ, σ 2 ) θ =(μ, σ 2 ) w (a) i = f(x i ; θ (a) ) γ/ n μ (a+1) = (σ 2 ) (a+1) = n w (a) i x i n w (a) f(x i ; θ (a) ) γ i x 2 i (μ(a+1) ) 2 (1 + γ) d γ (ḡ, f θ (a)) d γ (ḡ, f θ (a+1)) d γ (ḡ, fˆθγ ) 39
n (ˆθγ θ γ) d N (0, Σ g (θ γ ) ) [ ] d J g (θ) =E g dθ ξ(x; θ) Σ g (θ) =J g (θ) 1 I g (θ)j g (θ) 1 I g (θ) =E g [ ξ(x; θ)ξ(x; θ) ] Σ g (θ γ )= 1 1 ε Σ f(θ )+O(εν γ ) 40
d(g, f) d(g, f) =φ(d γ (g, f)). φ(u) 41
THANK YOU Hironori Fujisawa Institute of Statistical Mathematics 42