20 5 8..................................................2.....................................3 L.....................................4................................. 2 2. 3 2. (N )......................................... 3 2.2 (LN3 )..................................... 4 2.3 III (P3 )................................... 6 2.4 III (LP3 )................................ 7 2.5............................................ 8 2.6 (GEV ).................................... 9 2.7 (SQRT-ET )............................ 0 2.8 (3 )....................................... 2 3. 4 4. 4 5. 5 No.540 998 5 () No.540 998 5 (2) No.54 998 6 L-moments B2( ) Vol.B2-65 No. 2009 pp6-65 Derek A. Roff( ) ( ) 20 3 0
.. ( )T p T = p p = T () T x T N P (X < x T ) N = ( T ) N (2) ) N ( P (X x T ) N = T (3) T N P T 5 0 30 00 200 500 000 5000 N 5 5 5 0 20 50 00 500 P 0.328 0.590 0.844 0.904 0.905 0.905 0.905 0.905.2 x = x j (4) N S 2 = N C s = N (x j x) 2 (5) ( ) 3 xj x (6) S ˆσ 2 = N N S2 (7) N(N ) ˆγ = C s (8) N 2.3 L (PWM : Probbility Weighted Moments) L (L Moments) L (PWM) β r = 0 xf r df (r = 0,, 2,... ) (9) PWM L λ = β 0 (0) λ 2 = 2β β 0 () λ 3 = 6β 2 6β + β 0 (2) PWM L
b 0 = N b = b 2 = x (j) (3) N(N ) (j )x (j) (4) N(N )(N 2) (j )(j 2)x (j) (5) x (j) N j.4 F [x (i) ] = i α N + 2α (6) N i x (i) F [x (i) ] α i ( ) Weibull Blom Cunnne Gringorten Hzen α 0 0.375 0.40 0.44 0.5 2
2. 2. (N ) x () f(x) = [ exp 2πσx 2 ( ) ] 2 x µx σ x (7) (2) ( ) x µx F (x) = Φ σ x Φ(z) = 2π z exp ( 2 ) t2 dt (8) (3) p x p z = x µ x σ x x = µ x + σ x z (9) x p = µ x + σ x z p z p p = Φ(z) z (20) (4) (L ) b 0 = N x (j) b = N(N ) (j )x (j) b 2 = N(N )(N 2) (j )(j 2)x (j) (2) x (j) N j b i = β i λ i λ = β 0 λ 2 = 2β β 0 λ 3 = 6β 2 6β + β 0 (22) L { µ x = λ σ x = πλ 2 (23) 3
2.2 (LN3 ) x 3 () f(x) = { (x ) exp 2πσ y 2 [ ] } 2 ln(x ) µy σ y y = ln(x ) (24) (2) ( ) ln(x ) µy F (x) = Φ σ y Φ(z) = 2π z exp ( 2 ) t2 dt (25) (3) p x p z = ln(x ) µ y σ y x = + exp(µ y + σ y z) (26) x p = + exp(µ y + σ y z p ) z p p = Φ(z) z (27) (4) ( ) = x 2 () x (N) x m x () + x (N) 2x m > 0 x () + x (N) 2x m µ y = N N ln(x j ) (28) σ 2 y = N N [ln(x j ) µ y ] 2 x () x (N) x m ( ) x j x (j) (5) ( ) ( ) 3 xj x (29) x = N x j S 2 x = N (x j x) 2 C sx = N S x N(N ) µ x = x σ x = [N/(N )] /2 S x γ x = C sx (30) N 2 γ x Bobee Robitille ( ) ( ) B C sx 3 γ x = C sx (A + B C 3 sx ) (3) A =.0 + 7.0/N + 4.66/N 2 B =.69/N + 74.66/N 2 (32) µ x = + ɛ φ σ x = ɛ φ(φ ) γ x = (φ + 2) φ (33) ɛ = exp(µ y ) φ = exp ( 2 σ ) y (34) 4
[ φ = β + ] /3 [ β 2 + β ] /3 β 2 (β = + γ x 2 ) 2 (35) σ x ɛ = φ(φ ) (36) σ y = ln φ µ y = ln ɛ = µ x ɛ φ (37) 5
2.3 III (P3 ) x III Person type 3 distribution ( ) () f(x) = Γ(b) ( x c ) b ( exp x c ) > 0 : c x < (38) (2) ( ) x c F (x) = G G(w) = Γ(b) w 0 t b exp( t)dt ( > 0) (39) (3) p x p w = x c x = c + w (40) x p = c + w p w p p = G(w) w (4) (4) ( ) ( ) 3 xj x (42) x = N x j S 2 x = N (x j x) 2 C sx = N S x N(N ) µ x = x σ x = [N/(N )] /2 S x γ x = C sx (43) N 2 γ x Bobee Robitille ( III ) ( ) B C sx 2 γ x = C sx (A + B C 2 sx ) (44) A = + 6.5/N + 20.2/N 2 B =.48/N + 6.77/N 2 (45) µ x = c + b σ x 2 = 2 b γ x = 2 b 2 b = 4/γ x (b > 0) = σ x / b (γ x < 0 = σ x / b < 0) c = µ x b (46) (47) γ x < 0 < 0 w p p γ x III 6
2.4 III (LP3 ) x y = ln x III () f(x) = Γ(b) x ( ln x c ) b ( exp ln x c ) > 0 : exp(c) < x < (48) (2) ( ) ln x c F (x) = G G(w) = Γ(b) w 0 t b exp( t)dt ( > 0) (49) (3) p x p w = ln x c x = exp(c + w) (50) x p = exp(c + w p ) w p p = G(w) w (5) (4) ( ) ( ) 3 yj ȳ (52) y j = ln x j ȳ = N y j S 2 y = N (y j ȳ) 2 C sy = N S y N(N ) µ y = ȳ σ y = [N/(N )] /2 S y γ y = C sy (53) N 2 γ x Bobee Robitille ( III ) ( ) B C sy 2 γ y = C sy (A + B C 2 sy ) (54) A = + 6.5/N + 20.2/N 2 B =.48/N + 6.77/N 2 (55) µ y = c + b σ y 2 = 2 b γ y = 2 b 2 b = 4/γ y (b > 0) = σ y / b (γ y < 0 = σ y / b < 0) c = µ y b (56) (57) γ y < 0 < 0 w p p γ y b b Wilson-Hilferty p x p x p = exp(µ y + σ y K p ) K p = 2 ( + γ yz p γ y 2 ) 2 (58) γ y 6 36 γ y z p N(0, ) Wilson-Hilferty b < 0, 000 γ y < 0 z p p K p γ y γ y p γ y 7
2.5 x (Gumbel distribution) () f(x) = [ exp x c ( exp x c )] < x < (59) (2) [ ( F (x) = exp exp x c )] (60) (3) p x p [ ( p = exp exp x c )] x = c ln[ ln(p)] (6) x p = c ln[ ln(p)] (62) (4) (L ) b 0 = N x (j) b = N(N ) (j )x (j) b 2 = N(N )(N 2) (j )(j 2)x (j) (63) x (j) N j b i = β i λ i λ = β 0 λ 2 = 2β β 0 λ 3 = 6β 2 6β + β 0 (64) L { = λ 2 / ln 2 c = λ 0.5772 (65) 8
2.6 (GEV ) x (Generlized Extreme Vlue distribution) () f(x) = ( k x c ) [ /k ( exp k x c ) ] /k (k 0) (66) (2) F (x) = exp [ ( k x c ) ] /k (k 0) (67) (3) p x p p = exp [ ( k x c ) ] /k x = c + k { [ ln(p)] k} (68) x p = c + k { [ ln(p)] k} (69) (4) (L ) b 0 = N x (j) b = N(N ) (j )x (j) b 2 = N(N )(N 2) (j )(j 2)x (j) (70) x (j) N j b i = β i λ i λ = β 0 λ 2 = 2β β 0 λ 3 = 6β 2 6β + β 0 (7) L k = 7.8590d + 2.9554d 2 d = 2λ 2 ln(2) λ 3 + 3λ 2 ln(3) kλ 2 = ( 2 k ) Γ( + k) c = λ [ Γ( + k)] k (72) 9
2.7 (SQRT-ET ) x (SQRT exponentil-type distribution of mximum) () f(x) = b [ 2 exp ( bx + ) ( bx exp )] bx (x 0) (73) (2) [ ( F (x) = exp + ) ( bx exp )] bx (x 0) (74) (3) p x p [ ( p = exp + ) ( bx exp )] bx = exp [ ( + t p ) exp( t p )] (t p = bx) (75) x = t p 2 ln( + t p ) t p = ln [ ] b ln(p) (76) x p = t p 2 b ln( + t p ) t p = ln [ ] ln(p) (77) ( ) t p g(t p ) = ln( + t p ) t p ln [ ] ln(p) (78) g (t p ) = + t p (79) g(t p ) g(0) > 0 Newton- Rphson g(t p ) = 0 t p t p(n+) = t p(n) g(t p(n)) g (t p(n) ) (n) (80) t p p x p x t p = b x mx t p (4) ( ) b L L(, b) = ln f(x j ) = N ln + N ln b N ln 2 bxj exp ( ) bx j + bxj exp ( ) bx j (8) 0
L b 0 b L b = 0 = N bxj 2N N (bx j) exp ( ) = (82) bx j L 0 b 2 L = 0 = N N exp ( ) N bx j + bxj exp ( ) = 2 (83) bx j L = 2 h(b) = (b) 2 (b) = 0 b 2 > 0 > 0 b > 0 b > ( N ) 2 2N (84) xj b b (C ) /* Bisection method */ b=bb; /* >0 b ( ) */ b2=b+0.5; /* b+0.5 b2 */ bb=0.5*(b+b2); /* bb */ f=fsqr(nd,dtx,b,&,&2); /* h(b) */ f2=fsqr(nd,dtx,b2,&,&2); /* h(b2) */ ff=fsqr(nd,dtx,bb,&,&2); /* h(bb) */ do{ /* */ if(f*ff<0.0)b2=bb; if(ff*f2<0.0)b=bb; if(ff==0.0)brek; if(0.0<f*ff&&0.0<ff*f2){b=b2;b2=b+0.5;} /* */ bb=0.5*(b+b2); /* 0.5 */ f=fsqr(nd,dtx,b,&,&2); f2=fsqr(nd,dtx,b2,&,&2); ff=fsqr(nd,dtx,bb,&,&2); }while(0.00<fbs(-2)); /* h(b) <0.00 */
2.8 (3 ) x 3 (Weibull distribution) () f(x) = k ( ) [ k ( ) ] k x c x c exp (k = 0) (85) (2) [ ( ) ] k x c F (x) = exp (k 0) (86) (3) p x p [ ( ) ] k x c p = exp x = c + [ ln ( p)] /k (87) x p = x = c + [ ln ( p)] /k (88) (4) (L *) ) b 0 = N x (j) b = N(N ) (j )x (j) b 2 = N(N )(N 2) (j )(j 2)x (j) (89) x (j) N j b i = β i λ i λ = β 0 λ 2 = 2β β 0 λ 3 = 6β 2 6β + β 0 (90) L k = 285.3τ 6 658.6τ 5 + 622.8τ 4 37.2τ 3 + 98.52τ 2 2.256τ + 3.560 τ = λ 3 /λ 2 λ 2 = ( 2 /k ) Γ( + /k) c = λ Γ( + /k) (9) *) L-moments B2( ) Vol.B2-65 No. 2009 pp6-65 -2 k λ 3 τ 3 = λ3/λ 2 A k (5) ( ) 3 c k (I) c 2
x F (x) x [ ( ) ] k x c F (x) = exp (92) 2 ln{ ln[ F (x)]} = k ln(x c) k ln Y = A X + B (93) Y i = ln{ ln[ F (x i )]} X i = ln(x i c) k = A = exp( B/A) c c c 3 k c c k k Newton-Rphson (II) k c t = x c f(t) = k ( ) [ k ( ) ] k t t exp (94) L = N i= ln f(t i) k Newton-Rphson k c L k = 0 k + N i= ln t i N N i= [(ln t i) t i k ] N i= t i k = 0 (95) L = 0 = ( N i= t ) /k i k (96) N g(k) = k + T 0 N T 2(k) T (k) g (k) = k 2 T 3(k) T (k) [T 2 (k)] 2 [T (k)] 2 (97) T 0 = ln t i T (k) = i= i= t i k T 2 (k) = [ln t i t k i ] T 3 (k) = [ln t i ln t i t k i ] (98) i= i= k n k n+ = k n g(k n) g (k n ) (99) k ( N i= = t ) /k i k (00) N 3
3. Jckknife (JckKnife ) 2 3 N x (i) ˆθ i N ˆθ (i) n ˆθ (i) N ˆθ ( ) ˆθ ( ) = N ˆθ (i) (0) i= 4 N x (i) ˆθ N ˆθ ( ) jckknife θ θ = N ˆθ (N ) ˆθ ( ) (02) 5 θ (SE) (SE) = N (ˆθ(i) N ˆθ ) 2 ( ) (03) i= 4. bootstrp 2 3 N x (i) ˆθ N N θ (i) θ (i) (bootstrp ) B bootstrp ˆ θ = B B θ (i) (04) i= 4 bootstrp bootstrp (percentile method) 4
5. 2 3 N x ɛ N F (x) ( )q u ɛ u ɛ Φ q = F (x ɛ ) = Φ(u ɛ ) (05) 4 N F (, M ) F M(= N ) F = ( ) M 2 u ɛ M + (06) F 2ɛ ( ) M FM (2ɛ) 2 = u ɛ M + (07) 5 β ɛ ɛ 0 = ( β 0 ) /N (β 5%) (08) 5 ɛ ɛ 0 ɛ ɛ 0 ( ) ɛ > ɛ 0 ( ) (09) 5