2 1,2, , 2 ( ) (1) (2) (3) (4) Cameron and Trivedi(1998) , (1987) (1982) Agresti(2003)
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2 2 1,2, , 2 ( ) (1) (2) (3) (4) Cameron and Trivedi(1998) , (1987) (1982) Agresti(2003)
3 3 (1)-(4) AAA, AA+,A (1) (2) (3) (4) (5) (1)-(5) (DI) % (2006)
4 (50% (median ) 50% (quartile 25% 50% 75% ) quantile:20% 40% 60% 80% % (mode) 9 > > 6 (1991 pp.21-22) Quantile Regression Koenker(2005) 9
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6 6 r xy = (xi x)(y i y) (xi x) 2 (yi y) 2 (1) x y 1 r xy 1 x y x y %-70% pp.54-55) 14
7 7 id 4 X f(x) X P (a X b) = b f(x)dx (2) a x f(x) 0 + f(x)dx = 1 f(x) X X x X F (x) = P (X x) = x f(u)du (3) F (x) = f(x) 3 (1)F x 1 < x 2 F (x 1 ) < F (x 2 ) (2) lim F (x) = 0 x lim F (x) = 1 (3)F lim F (x) = F (a). 3 x x a+0 F (x) X E(X) E(X) = xf(x)dx (4) V (X) E(X) = µ
8 8 V (X) = (X µ)2 f(x)dx = E(X 2 ) 2µE(X) + µ 2 = E(X 2 ) (E(X)) 2 (5) D(X) D(X) = V (X) skewness SK SK = E(X µ) 3 /σ 3 (6) SK > 0 SK 0 kurtosis KT KT = E(X µ) 4 /σ 4 (7) KT = 3 (EK=excess kurtosis) EK = KT 3 (8) EK > 0 EK 0 A p A X X f(x) f(x) = n C x p x (1 p) n x, x = 0, 1, 2,...n (9) E(X) = np V (X) = np(1 p) np λ n, p 0 f(x) e λ λ x /x!, x = 0, 1, 2.. f(x) = e λ λ x /x!, x = 0, 1, 2.. (10) P o(λ) E(X) = λ V (X) = λ
9 9 15 f(x) f(x) = 1 ] (x µ)2 exp [ 2πσ 2σ 2, < x < (11) E(X) = µ V (X) = σ 2 N(µ, σ 2 ) 16 1/ 2πσ f(x)dx = 1 ] [ exp (x µ)2 2σ 2 dx = 2πσ (12) 17 k f(x 1, x 2,...x k ) 0... f(x 1, x 2,...x k )dx 1 dx 2...dx k = 1 (13) S S A P ((x 1, x 2,...x k ) A) =... f(x 1, x 2,...x k )dx 1 dx 2...dx k (14) A X i F (x i ) X i X j FIFA X (X µ)/σ µ = 0 σ = 1 N(0, 1) 17 (2003) Silverman (1986)
10 10 F (x 1, x 2,...x k ) = F 1 (x 1 )F 2 (x 2 )F 3 (x 3 )...F k (x k ) (15) X i X j X i x i X j g(x j x i ) = f(x j, x i ) h(x i ) (16) x j g(x j x i ) = f(x j, x i ) h(x i ) = h(x i )/h(x i ) = 1 (17) x j x j E(X j x i ) = x x jg(x j x i ) = µ xj x i (18) V (X j x i ) = x (x j µ xj x i ) 2 g(x j x i )dx j (19) 5
11 11 18 y = (y 1, y n ) θ = (θ 1, θ 2, θ p ) L(θ) 19 L(θ) θ θ θ L(θ) logl(θ) θ y i = βx i + ε i i = 1,, n (20) ε i N(0, σ 2 ) { log L(β) = log [(2πσ 2 ) n2 exp (y }] βx) (y βx) 2σ 2 = n 2 log(2πσ2 ) 1 2σ 2 (y βx) (y βx) β (21) β = ˆβ = x y/x x (22) log L(β) 2 log L(β) β 2 = x x/σ 2 (23) β 2 log L(β) β I(θ) y y f θ (y) { 2 } log L(θ) I(θ) = E θ 2 { 2 } log f θ (y) = E θ 2 (24) I(θ)
12 12 V {t(y)} 1 I(θ) (25) θ Z H 0 : θ = θ 0 n( θ θ 0 ) N(0, 1/I 1 (θ 0 )) θ 1 > θ 0 ni1 (θ 0 )( θ θ 0 ) > Z α (26) I 1 (θ) = I(θ)/n Z α α Z H 0 : θ = θ 0 H 1 : θ θ 0 H 0 θ 0 χ 2 (1) 2 log L( θ) L(θ 0 ) > χ2 α(1) (= Z 2 α/2 ) (27) χ 2 α(1) = Z 2 α/ ( Empirical Likelihood Mittelhammer, Judge and Miller (2000) Owen (2001)
13 13 [1] (1987) [2] (2006) [3] (2005) [4] (1991) [5] (1992) [6] (1982) [7] (1974) [8] (2003) [9] Cameron, A.C.and Trivedi, P.K.(1998) Regression Analysis of Count Data, Cambridge University Press. [10] Cameron, A.C. and Trivedi, P.K.(2005) Microeconometrics: Methods and Applications, Cambridge University Press. [11] Davidson, Russell and MacKinnon, James G.(2004) Econometric Theory and Methods, Oxford University Press. [12] Koenker, Roger. (2005) Quantile Regression, Cambridge University Press. [13] Mitterlhammer, Ron C.,Judge, Gerorge G. and Miller, Douglas, J.(2000) Econometric Foundations, Cambridge University Press. [14] Owen, Art B.(2001) Empirical Likelihood, Chapman & Hall
14 14 [15] Silverman, B.W.(1986) Density Estimation for Statistics and Data Analysis, Chipman & Hall. [16] Winklemann, Rainer and Boes, Stefan.(2005) Analysis of Microdata, Springer. [17] Wooldridge, Jeffrey. M.(2003) Econometric Analysis of Cross Section and Panel Data, The MIT Press
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() 10 9 30 1 Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [], [13]) Poincaré e m Poincaré e m Kähler-like Kähler-like Kähler M g M X, Y, Z (.1) Xg(Y, Z) = g( X Y, Z) + g(y, XZ)
1: 3.3 1/8000 1/ m m/s v = 2kT/m = 2RT/M k R 8.31 J/(K mole) M 18 g 1 5 a v t πa 2 vt kg (
1905 1 1.1 0.05 mm 1 µm 2 1 1 2004 21 2004 7 21 2005 web 2 [1, 2] 1 1: 3.3 1/8000 1/30 3 10 10 m 3 500 m/s 4 1 10 19 5 6 7 1.2 3 4 v = 2kT/m = 2RT/M k R 8.31 J/(K mole) M 18 g 1 5 a v t πa 2 vt 6 6 10
No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
(2004 ) 2 (A) (B) (C) 3 (1987) (1988) Shimono and Tachibanaki(1985) (2008) , % 2 (1999) (2005) 3 (2005) (2006) (2008)
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春期講座 ~ 極限 1 1, 1 2, 1 3, 1 4,, 1 n, n n {a n } n a n α {a n } α {a n } α lim n an = α n a n α α {a n } {a n } {a n } 1. a n = 2 n {a n } 2, 4, 8, 16,
春期講座 ~ 極限 1 1, 1 2, 1 3, 1 4,, 1 n, n n {a n } n a n α {a n } α {a n } α lim an = α n a n α α {a n } {a n } {a n } 1. a n = 2 n {a n } 2, 4, 8, 16, 32, n a n {a n } {a n } 2. a n = 10n + 1 {a n } lim an
X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
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() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.
![() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.](/thumbs/89/98384840.jpg "() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.")
() 6 f(x) [, b] 6. Riemnn [, b] f(x) S f(x) [, b] (Riemnn) = x 0 < x < x < < x n = b. I = [, b] = {x,, x n } mx(x i x i ) =. i [x i, x i ] ξ i n (f) = f(ξ i )(x i x i ) i=. (ξ i ) (f) 0( ), ξ i, S, ε >
1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1
sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω ω α 3 3 2 2V 3 33+.6T m T 5 34m Hz. 34 3.4m 2 36km 5Hz. 36km m 34 m 5 34 + m 5 33 5 =.66m 34m 34 x =.66 55Hz, 35 5 =.7 485.7Hz 2 V 5Hz.5V.5V V
わが国企業による資金調達方法の選択問題
* [email protected] ** [email protected] *** [email protected] No.05-J-3 2005 3 103-8660 30 No.05-J-3 2005 3 1990 * [email protected] ** [email protected]
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No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y
No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
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dy + P (x)y = Q(x) (1) dx dy dx = P (x)y + Q(x) P (x), Q(x) dy y dx Q(x) 0 homogeneous dy dx = P (x)y 1 y dy = P (x) dx log y = P (x) dx + C y = C exp
+ P (x)y = Q(x) (1) = P (x)y + Q(x) P (x), Q(x) y Q(x) 0 homogeneous = P (x)y 1 y = P (x) log y = P (x) + C y = C exp{ P (x) } = C e R P (x) 5.1 + P (x)y = 0 (2) y = C exp{ P (x) } = Ce R P (x) (3) αy
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1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =
1 8, : 8.1 1, z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = a ii x i + i
( z = x 3 y + y ( z = cos(x y ( 8 ( s8.7 y = xe x ( 8 ( s83.8 ( ( + xdx ( cos 3 xdx t = sin x ( 8 ( s84 ( 8 ( s85. C : y = x + 4, l : y = x + a,
[ ] 8 IC. y d y dx = ( dy dx ( p = dy p y dx ( ( ( 8 ( s8. 3 A A = ( A ( A (3 A P A P AP.3 π y(x = { ( 8 ( s8 x ( π < x x ( < x π y(x π π O π x ( 8 ( s83.4 f (x, y, z grad(f ( ( ( f f f grad(f = i + j