kato-kuriki-2012-jjas-41-1.pdf

Size: px

Start display at page:

Download "kato-kuriki-2012-jjas-41-1.pdf"

ねんたろうおえづか
4 years ago
Views:

1 Vol. 41, No. 1 (2012), / JST CREST T 2 T 2 2 K K K K 2,,,,,. 1. t i y i 2 2 y i = f (t i ; c) + ε i, f (t; c) = c h t h = c ψ(t), i = 1,...,N (1) h=0 c = (c 0, c 1, c 2 ), ψ(t) = (1, t, t 2 ) 3 1 i ε i 0 t i T R m T = [a j, b j ] (2) a 1 b 1 < a 2 b m 1 < a m b m a 1 = [a 1, b 1 ] (, b 1 ] b m = [a m, b m ] [a m, ) c 2 ĉ 2 f (t; c) t T t T f (t; c) 0 (3) / 1

2 4.2 2 T (2) T = [a, b] T = m {a j} T = m {a j} (3) t = a j ( j = 1,...,m) (3) c { } K = K[T] = c f (t; c) 0 t T (Barvinok [1]) K (3) c K 3 H 0 : c = 0, H 1 : c K, H 2 : c R 3. (4) H 0 H 1 H 1 H 2 (3) 2 μ μ = 0 μ K 2 (Shapiro [8]) ( ) ( [4], Kuriki and Takemura [5]) K K K, 1 1 Karlin and Studden [2] T K K T m [a j, b j ] K K c ĉ ML ĉ K 2 c (4) ( ) 3 K K 4 Potthoff and Roy [6] Kato and Kuriki [3] 4 T 2

3 Vol. 41, No. 1 (2012) Q x,y Q = x Qy, x Q = x, x Q x A Π Q (x A) = argmin x c Q c A A Q x,y, x, Π(x A) (1) σ 2 ĉ ĉ ĉ N 3 (c, Σ) Σ=σ 2 Σ 0, Σ 0 = ( N i=1 ψ(t i)ψ(t i ) ) 1 σ 2 (ĉ, σ 2 ) σ 2 σ 2 ĉ ν = N 4 2 σ 2 χ 2 ν/ν ĉ N 3 (c, Σ) H 1 : c K ĉ K, Σ 1 H 0, H 2 0, ĉ 1 σ 2 H 0 vs. H 1 H 1 vs. H 2 λ 01 = ĉ K 2 Σ, λ 1 12 = ĉ ĉ K 2 Σ. (5) 1 ĉ K =Π Σ 1(ĉ K) ĉ K σ 2 β 01 = Σ = σ 2 Σ 0, ĉ K =Π Σ 1 (ĉ K) (5), (6) ĉ K 2 Σ 1 ĉ 2 Σ + ν, β ĉ ĉk 2 Σ 12 = 1 ĉ ĉ K 2 Σ + ν (6) 1 1 ĉ ĉ K 2 Σ = 1 ĉ 2 Σ ĉ 1 K 2 Σ 1 H 1 vs. H 2 Robertson, et al. [7] H 1 vs. H 2 σ 2 H 1 : c K c = 0 ( ) Kato and Kuriki [3] 3

4 λ 01 ĉ K (Robertson, et al. [7]) H 0 : c = 0 (5) λ 01, λ 12 P H0 (λ 01 a, λ 12 b) = 3 w i Ḡ i (a)ḡ 3 i (b). (7) i=0 Ḡ i i 2 Ḡ 0 (a) = 1(a 0), 0 (a < 0) (6) β 01, β 12 P H0 (β 01 a, β 12 b) = 3 w i B i i=0 2, 3 i+ν 2 (a) B 3 i 2, ν (b). (8) 2 B a,b (a, b) (8) {w i } (7) i w i i w i = 1 (λ 01,λ 12 ) {w i } 2 2 (β 01,β 12 ) 2 a = b = λ 01, λ 12, β 01, β 12 λ 01 λ 12 2 ( χ 2 ) (Shapiro [8], Takemura and Kuriki [10]) K {w i } (Wynn [13]) K K {w i } K 2 ( ) (Takemura and Kuriki [10, 11]) K 2 w 3 = Vol 2(K S 2 ) Vol 2 (S 2 ) w 1 = Vol 1 ( K (S 2 ) ) 2Vol 1 ((S 1 ) ) K K, w 2 = Vol 1( K S 2 ), 2Vol 1 (S 1 ), w 0 = Vol 2 (K (S 2 ) ) Vol. (9) 2 (S2 ) M[T] = co ({ ψ(t) t T }) K = K[T] = M[T] (Barvinok [1]) co(a) A K, K K, K S 2 = {x R 3 x Σ 1 = 1}, ( S n ) = {x R 3 x Σ = 1} Vol d,vol d, Σ 1,, Σ d 4

5 Vol. 41, No. 1 (2012) ( Vol d 1 S d 1 ) = Vol (( d 1 S d 1 ) ) 2π d/2 = Γ(d/2) (9) w 3, w 0 2 w 2, w 1 1 w i = w i = 1 (10) 2 i:even i:odd (Takemura and Kuriki [11]) w 3 = 1/2 w 1, w 0 = 1/2 w T 1. K ( ) K ( ) ν = 0 β 01 M = K S 2 K M θ ( ) Tube(M,θ) = { x S 2 min y M dist(x,y) θ}, dist(x,y) = cos 1 x,y. β 01 = Π(ĉ K) 2 ĉ 2 cos θ ĉ ĉ Tube(M,θ) H 0 ĉ/ ĉ Vol 2 (Tube(M,θ)) Vol 2 (S 2 ) = P H0 (β 01 cos θ) = 3 w i B i i=0 2, 3 i 2 (cos θ) 5

6 [4], Kuriki and Takemura [5] n w n+1,w n,w 1,w 0 (9) (10) n 4 (9) {w i } 0 i n+1 Kato and Kuriki [3] K K {w i } 0 i 3 w 1, w 2 (9) w 0, w 3 (10) w 2 w 1 K K (1 1 ) 4 (i) a 1 >, b m <, T = m [a j, b j ]. (ii) a 1 >, b m =, T = m 1 [a j, b j ] [a m, ). (iii) a 1 =, b m <, T = m j=2 [a j, b j ] (, b 1 ]. (iv) a 1 =, b m =, T = m 1 j=2 [a j, b j ] (, b 1 ] [a m, ). K f (t; c) 1 f (t; c) T (c K) c K (a) (b) (a) f (t; c) T (b) a 1 = b m = f (t; c) 1 ( ) Karlin and Studden [2] T = [0, ) (a) f (t; c) t 0 T ε>0 f (t 0 ; c ε(1, 0, 0)) = f (t 0 ; c) ε f (t 0 ;(1, 0, 0)) = ε <0. c ε(1, 0, 0) K (b) f (t; c ε(0, 0, 1)) = f (t; c) ε f (t;(0, 0, 1)) = εt 2 + o(t 2 ) t c ε(0, 0, 1) K c K (a), (b) a 1, b m t T c R 3 6

7 Vol. 41, No. 1 (2012) f (t; c + εc ) = f (t; c) + ε f (t; c ). c int K f (t; c) > 0 ε a 1 = b m = t ± ε f (t; c + εc ) > 0. c + εc K c int K 1 (b) 1 T f (t; c) a 1, b m a 1, b m a 1 = a m = 1 2 T 1 α(t a 1 ) + β(b m t) (a 1 >, b m < ), α(t a 1 ) + β (a 1 >, b m = ), f (t; c) = α + β(b m t) (a 1 =, b m < ), α (a 1 =, b m = ). α, β 0 1, 2 K ( ) 3 f (t; c), c K α, β 0 α, β > 0 α(t γ) 2, γ T (γ 2 ), α(t b j ) 2 + β(t b j )(t a j+1 ) (b j 1 ), j = 1,...,m 1, α(t a j+1 ) 2 + β(t b j )(t a j+1 ) (a j+1 1 ), j = 1,...,m 1. (i) a 1 >, b m < α(t b m ) 2 + β(b m t)(t a 1 ) α(b m t)(t a 1 ) + β(t a 1 ) 2 (b m 1 ), (a 1 1 ). (ii) a 1 >, b m = α + β(t a 1 ) (b m = 1 ), α(t a 1 ) + β(t a 1 ) 2 (a 1 1 ). (iii) a 1 =, b m < α(b m t) + β(b m t) 2 (b m 1 ), α + β(b m t) (a 1 = 1 ). (iv) a 1 =, b m = α (a 1 =, b m = 2 ). 7

8 ( ) 1 f (t; c), c K 1 (a) (b) (a) f (t; c) T f (t; c) γ int T ε>0 f (γ ε; c), f (γ + ε; c) 0 γ 2 f (t; c) = α(t γ) 2 γ = a j+1, b j ( j = 1,...,m) γ = a 1, b m γ 2 γ int T γ 1 γ = b j 1 f (t; c) 1 f (a j ; c) < 0 f (a j+1 ; c) < 0 f (t; c) 2 f (a j+1 ; c) 0 f (t; c) b j γ (b j, a j+1 ] 2 ( (t b j )(t γ) = (t b j ) t α α + β b j β ) α + β a j+1 α(t b j ) 2 + β(t b j )(t a j+1 ) γ = a j+1 1 f (t; c) a j+1 γ [b j, a j+1 ) 2 ( (t a j+1 )(t γ) = (t a j+1 ) t α α + β b j β ) α + β a j+1 α(t b j )(t a j+1 ) + β(t a j+1 ) 2 γ = a 1 1 a b m < b m = (T 1 ) (t a 1 ) = {α(t a 1 ) + β(b m t)} (t a 1 ) = α(t a 1 ) 2 + β(b m t)(t a 1 ), (T 1 ) (t a 1 ) = {α(t a 1 ) + β} (t a 1 ) = α(t a 1 ) 2 + β(t a 1 ) γ = b m 1 b m 1 2 a 1 > (T 1 ) (b m t) = {α(t a 1 ) + β(b m t)} (b m t) = α(b m t) 2 + β(b m t)(t a 1 ), a 1 = (T 1 ) (b m t) = {α + β(b m t)} (b m t) = α(b m t) + β(b m t) 2 8

9 Vol. 41, No. 1 (2012) f (t; c) T 1 (b) a 1 = b m = f (t; c) 2 α(t a 1 ) + β (a 1 >, b m = ), α(b m t) + β (a 1 =, b m < ), α (a 1 =, b m = ) f (t; c) α, β > 0 3 K K 3 ϕ(t) = (t 2, 2t, 1), ϕ j = (b j a j+1, b j a j+1, 1), j = 1,...,m 1, (t t 0 ) 2 = ϕ(t 0 ) ψ(t), (t b j )(t a j+1 ) = ϕ j ψ(t) ( b m a 1, b m + a 1, 1) (a 1 >, b m < ), ( a 1, 1, 0) (a 1 >, b m = ), ϕ m = (b m, 1, 0) (a 1 =, b m < ), (1, 0, 0) (a 1 =, b m = ) (t a 1 )(b m t),α(t a 1 ), α(b m t), α ϕ mψ(t) 3 K 4 α, β K = m ({ αϕ(t) t [a j, b j ] } { αϕ j + βϕ(b j ) } { αϕ j + βϕ(a j+1 ) }). (11) a m+1 = a 1 t = ± ϕ(t) = (1, 0, 0) (11) α, β > 0 {} (11) α, β > 0 K (2 ) 0 (11) K 4 K ( ) K {ψ(t) t T} {αψ(t) t T, α 0} {(x,y,z) x + z = 1} (x,y) t {αψ(t) α 0} ψ(t)/(1 + t 2 ) (x,y) 9

10 ( t, t ) = 1 ( 1 t t t, 2t ) ( 1 ) t 2 2, 0 1/2 {αψ(t) t T, α 0} K (extreme ray) ψ(b j ) ψ(a j+1 )() K ( 1( ) ) 5 α, β m ({ K = αψ(t) t [a j, b j ] } { αψ(b j ) + βψ(a j+1 ) }). (12) a m+1 = a 1 t = ± ψ(t) = (0, 0, 1) (12) α, β > 0 {} (12) α, β > 0 K (2 ) 0 (12) K 3.2. {w i } 4, 5 K K w i (9) w 1, w 2 w 2 = Vol 1( K S 2 ) 4π, w 1 = Vol 1 ( K (S 2 ) ). 4π (10) w 2 w 1 4 (11) w 2 ξ(t) = ϕ(t)/ ϕ(t) Σ 1 { αϕ(t) t [a j, b j ],α>0 } S 2 = { ξ(t) t [a j, b j ] } b j dξ(t) dt a j Σ 1 dt { αϕ j + βϕ(b j ) α, β > 0 } S 2 ϕ(b j )/ ϕ(b j ) Σ 1 ϕ j / ϕ j Σ 1 cos 1 ϕ(b j ),ϕ j Σ 1 ϕ(b j ) Σ 1 ϕ j Σ 1 { αϕ j + βϕ(a j+1 ) α, β > 0 } S 2 10

11 Vol. 41, No. 1 (2012) cos 1 ϕ j,ϕ(a j+1 ) Σ 1 ϕ j Σ 1 ϕ(a j+1 ) Σ 1 w 2 = 1 m { b j dξ(t) 4π dt a j Σ 1 dt + cos 1 ϕ(b j ),ϕ j Σ 1 ϕ(b j ) Σ 1 ϕ j Σ 1 + cos 1 ϕ j,ϕ(a j+1 ) } Σ 1 ϕ j Σ 1 ϕ(a j+1 ) Σ 1 (13) 5 (12) w 1 η(t) = ψ(t)/ ψ(t) Σ w 1 = 1 m { b j dη(t) 4π dt dt + cos 1 ψ(b j ),ψ(a j+1 ) } Σ (14) Σ ψ(b j ) Σ ψ(a j+1 ) Σ 6 a j ϕ(t) ξ(t) = ϕ(t) Σ 1, η(t) = ψ(t) ψ(t) Σ w 2, w 1 (13), (14) w 3 = 1 2 w 1, w 0 = 1 2 w ĉ K (5) (6) λ 01, λ 12, β 01, β 12 H 1 c ĉ K ĉ K 3 c K f (t; c) 1 K (extreme ray) (t γ) 2, γ T, (t b j )(t a j+1 ), j = 1,...,m 1. (i) a 1 >, b m < (b m t)(t a 1 ), (ii) a 1 >, b m = t a 1,1, (iii) a 1 =, b m < b m t,1, (iv) a 1 =, b m = 1. K 1 ψ 1 (t) = (1, t) Q 2 2 (Q 0) (t γ) 2, γ T, 1 ψ 1 (t) Qψ 1 (t) T f (t; c) () 11

12 7 T = m [a j, b j ] f (t; c) m 1 f (t; c) =ψ 1 (t) Qψ 1 (t) + u j (t b j )(t a j+1 ) u m (b m t)(t a 1 ) + u m (t a 1 ) u m (b m t) (a 1 >, b m < ), (a 1 >, b m = ), (a 1 =, b m < ). (15) ψ 1 (t) = (1, t) Q 2 2 u 1,...,u m 0. ( ) 1 (15) (15) T ĉ K (symmetric cone programming) (cylindrical algebraic decomposition) (15) f (t; c) e(q, u) ψ(t) e(q, u) 3 Q u = (u 1,...,u m ) 1 ĉ K d d ĉ c 2 Σ 1 (2 ), c = e(q, u) Q 0 u 1,...,u m 0 ( ), (), ( ) SeDuMi (Sturm [9]) u i = v 2 i, Q 2 2 R = (r ij) Q = R R ĉ e(q, u) 2 v Σ 1 i, r ij v i r ij Mathematica ([12] ) f (t; ĉ 1 ), f (t; ĉ 2 ) 2 H 1 t T f (t; c 1 ) f (t; c 2 ) = f (t; c 1 c 2 ) 0. 2 Potthoff and Roy [6] 8, 10, 12, 14 12

13 Vol. 41, No. 1 (2012) ĉ Σ 11 t i t + 11 x it 2 x it = f (t; c) + ε it, f (t; c) = c 0 + c 1 t + c 2 t 2. ε it i 0 Cov(ε it,ε it ) 2 f (t; c) f (t) = ĉ 0 + ĉ 1 t + ĉ 2 t 2, ĉ = (1.968, 0.310, 0.053). ĉ Σ = Σ Σ (3) 2 T T = [ 3, 3], [ 3, 1] [1, 3], { 3, 1, 1, 3} 3 T f (t; ĉ) ĉ K = ĉ λ 01 = ĉ K 2 Σ = ĉ 2 1 Σ = (w 0,w 1,w 2,w 3 ) = (0.030, 0.200, 0.470, 0.300) (T = [ 3, 3]), = (0.029, 0.198, 0.471, 0.302) (T = [ 3, 1] [1, 3]), = (0.027, 0.198, 0.473, 0.302) (T = { 3, 1, 1, 3}) p T = [ 3, 3] T = [ 3, 1] [1, 3] T = { 3, 1, 1, 3} H 0 : c = 0 vs. H 2 : c R 3 p p p [1] Barvinok, A. (2002). A Course in Convexity. AMS, Providence, Rhode Island. [2] Karlin, S. and Studden, W. (1966). Tchebycheff Systems: With Applications in Analysis and Statistics. Interscience Publishers, Wiley, New York. 13

14 [3] Kato, N. and Kuriki, N. (2011). Likelihood ratio test for positivity in polynomial regressions. arxiv: [4], (2008).., 60 (2), [5] Kuriki, S. and Takemura, A. (2009). Volume of tubes and the distribution of the maximum of a Gaussian random field. Selected Papers on Probability and Statistics, AMS Translations Series 2, 227, No. 2, [6] Potthoff, R. and Roy, S. (1964). A generalized multivariate analysis of variance model useful especially for growth curve problems. Biometrika, 51 (3-4), [7] Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference. Wiley, Chichester. [8] Shapiro, A. (1988). Towards a unified theory of inequality constrained testing in multivariate analysis. Internat. Statist. Rev., 56 (1), [9] Sturm, J. F. (1999). Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones. Optimization Methods and Software, 11 (1), [10] Takemura, A. and Kuriki, S. (1997). Weights of chi-bar-square distribution for smooth or piecewise smooth cone alternatives. Ann. Statist., 25 (6), [11] Takemura, A. and Kuriki, S. (2002). On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains. Ann. Appl. Probab., 12 (2), [12] Wolfram MathWorld. Cylindrical Algebraic Decomposition. [13] Wynn, H. P. (1975). Integrals for one-sided confidence bounds: a general result. Biometrika, 62, nkato@ism.ac.jp 14

15 Japanese J. Appl. Statist. 41 (1) (2012), Test for positivity in quadratic polynomial regression over intervals Naohiro Kato and Satoshi Kuriki The Graduate University for Advanced Studies and The Institute of Statistical Mathematics / JST, CREST Abstract A polynomial that is nonnegative over a given domain T is called a positive polynomial. We consider the likelihood ratio test for the hypothesis of positivity that the estimand quadratic polynomial regression curve is a positive polynomial when T is the union of intervals. We define hierarchical hypotheses including the hypothesis of positivity, and derive their null distributions as mixtures of chi-square distributions. According to the volume-of-tube method, the mixing probabilities are obtained through the evaluation of the volumes of boundaries of the closed convex cone K consisting of quadratic positive polynomials and its dual K. We introduce the parameterizations of the boundaries of K and K, and then provide expressions for the mixing probabilities. We demonstrate that the symmetric cone programming is useful for obtaining numerically the test statistics. Key words: chi-bar square distribution, moment cone, positive polynomial cone, repeated measurements, symmetric cone programming, Tchebycheff system. Corresponding author address: nkato@ism.ac.jp Received June 15, 2011; Received in final form October 11, 2011; Accepted October 15,

ohpmain.dvi

ohpmain.dvi 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,