: ( ) - PDF 無料ダウンロード

: 7 3...........................................................................................3............................................. 3.4............................ 4.5............................. 7 8................................ 8. ARp p, Auto-regressive Model..................... 9.3 MAq q, Movig-average Model.....................4 ARMAp, q Autoregressive Movig-average Model......................5.................................... 3.6.................................... 4 3 7 3................................... 7 3............................................. 3.3............................................ 4 4....................................... 4.............................................. 6 4.3 3.............................. 9 5 3 5........................................ 3 5............................................... 38 5.3.............................................. 45

4 II,, 7 4,, 5, 966, 3, 986 E.L. Lehma, George Casella: Theory of Poit Estimatio, Secod Editio, Spriger, 998 S.I. Resick: Extreme Values, Regular Variatio ad Poit Processes, Spriger, 987, : ISM :,, 6 Breima, L.: Probability, Addiso-Wesley, 968. Classics i applied mathematics, 7, Society for Idustrial ad Applied Mathematics, 99. Reprit Durrett, R.: Probablity Theory ad Examples, 4th ed., Cambridge Uiversity Press,.

. x i, y i, i =,,, x = s x = s xy = x i, i= x i x = i= x i x = x x, i= x i xy i y = i= x i y i x y = xy x y, i= r xy = s xy s x s y, s x = s x r xy. r xy = a > a r ax+b,cy+d = r xy. x i α + βx i y i Q = i= α = α, β = β : {y i α + βx i } y y i y = α + βx = Q α = y i α + βx i = y α βx i= = Q β = x i y i α + βx i = xy αx βx i= α + βx i O x i x α + βx = y x y αx + βx = xy x x α β =. xy β x y + xy = = s xy s y x x s = r xy, α = y x s βx s y = y r xy x x s x y = α + βx x y x y α = y βx, β s y = r xy y y = s βx x x y y s y = r xy x x s x y = x. x, y. x y y = α + β x, y x x = α + β y β β > r xy, s y /s x β, β β β x, y α, α, β, β

ŷ i = α + βx i y i e i = y i ŷ i e i =, α = y βx β = s xy s x e i = i= x i e i = i= i= y i ŷ i = y α + βx =, i= i= x i e i = x i y i ŷ i = xy αx + βx = xy y βxx βx = s xy βs x =. : = = y i y = i= i= e i + ŷ i y = i= e i + e i α + βx i y + e i + i= i= i= i= e i + e i ŷ i y + ŷ i y i= ŷ i y +. i= i= ŷ i y R = = R R = r xy y i y = s y i= ŷ i y = i= i= { α + βx i α + βx } = β x i x = i= i= r xy s y s x s x.. x, y, s x, s y, s xy, β, α, R,, x i, y i = i, i i =,,..., : i 4 = + + 3 + 3? 3 i 3 4 5 6 x i 4 5 5 y i 5 4 8 6 3 i= i 3 4 5 6 7 8 x i 3 6 8 9 6 7 3 5 y i 4 7 8 9 4 5 5 6, 3. x i, x i, y i, i =,,, Q = {y i α + β x i + β x i } α, β, β Q α = Q = Q = β β i= x x α x x x x β = x x x x β y x y x y X T X x x X = x x x XX T = x x x x, x x x x x x x y y y X. = x y. x y y

y = α + β x + β x x, x, y e i = y i ŷ i = y i α + β x i + β x i e i =, i= x i e i =, i= x i e i =. R / k =. / = e i, = y i y, k i= i=.3 x, y, y,, x 5, y 5, y 5 x i = 3, x i =, yi = 5, x i x i = 4, x i y i =, x i y i = 8, x i =, x i =, yi = 6 y x, x x i, y i, i =,,, α y = α + βx { i = d i = i = x i, d i, y i y = α + β d + β x y = α + β + β x, y = α + β x β y = α + βx x i, d i x i, y i y = α + β x + β dx y = α + β + β x, y = α + β x i=.4. α y = α + βx. β y = α + βx 3. 3 4 4 d α y = α + βx.3 y = αx β log y = log α + β log x. y = log y, x = log x y = αe βx log y = log α + βx. y y = log y y = eα+βx dy β > = βy y < y < + eα+βx dx y y y = log y y y F y = F α+βx y = F y y = α+βx. F x = Φx N, ex F x = + ex 3

.5 x, y eα+βx y = β > α, β + eα+βx x..4.7 3.5 4.8 y % % 5% 8% 9%.4.4. x i, ε i Y i Y i = α + βx i + ε i, i =,,,.3 ε,, ε ε i N, σ..4 α, β σ : β = s xy s x = i= x i xy i Y i= x i x, α = Y βx, σ = i= Y i α + βx i..5..4 α, β, σ α α N, σ + x s x x s α x = N, σ β β x s x s β x α N α, σ + x, β Nβ, σ. s x s x x x x. σ χ. χ σ 3 α σ β σ : α, β ε,, ε c i = x i x i= x i x i= c i = c i x i = s x i x x i x i= i= i= = x x s x = β = i= x i xy i Y i= x i x = α = Y βx = c i α + βx i + ε i i= α + βx i + ε i i= β + c i Y = β + i= c i ε i x = α + i= c i ε i, i= ix c ε i..4 α, β E[ α] = α, E[ β] = β. c i = V β = c i V ε i = s σ, i= x V V α = c ix εi = c ix + c i x σ = + x i= i= i= i= s x s x s x = s x σ, 4

Cov α, β = ix c c i V ε i = x s x i= + x s = x x s x σ e i = Y i α + βx i i= e i =, i= c ie i = χ 3 : /,, /, x x,, x x A s x s x σ ε. = A ε ε,, ε ε i N, σ ε = ε i, ε = x i x ε i, s x α α = ε x ε, β β = ε, s x s x σ = } {ε { i α α β βx i = ε i ε x i x } ε = = s x i= i= i= i= ε. ε ε i = ε i i= i= i=3 ε i. α, β ε, ε σ ε 3,, ε 3.4. α, β t Z N, Y χ T = Z Y/ t α t α : T t P T t α = α. α : Z = α α σ N,, Y = V α σ χ Z Y T = Z Y/ = α α V α σ /σ = α α σ + x s x t. ε α α t ε/ σ + x s x α α + t ε/ σ + x s. x β : Z = β β σ N,, Y = σ V β χ Z Y T = Z Y/ = ε β σ β t ε/ β β V β σ /σ = s x β β σ s x t. β β σ + t ε/.. σ σ = r xy s y σ = i= e i = = = R 5 s x

: = r xy s y. ε H : β = β, H : β β H T = β β t. t t ε/ t σ s x t > t ε/ H t t ε/ H H : β = β, H : β > β t > t ε H t t ε H H : β = β, H : β < β t < t ε H t t ε H.6. σ α, β 95% 3.7. 3 H : β =, H : β > 5%.4.3 x + Y + Ŷ+ α, β Ŷ+ = α + βx + α, β ε,, ε Y + Ŷ+ = α α β βx + + ε +. E[Y + Ŷ+] = E[ α α] x + E[ β β] + E[ε + ] =, V Y + Ŷ+ = V ε + α α β βx + = V ε + α βx + = V ε + + V α βx +, ε + α, β = σ + V α + x + Cov α, β + x +V β = σ + σ + x xσ s + x + x s + x + x { = σ + + x + x } Y + Ŷ+ N, σ { + + x + x } s x s x σ σ Y + Z = Ŷ+ N,, W = V Y + Ŷ+ T = Z W/ = Y + Ŷ+ V Y + Ŷ+ σ /σ = σ σ σ s x χ Z W Y + σ Ŷ+ { + + x + x s x } t. ε Y + Ŷ + t ε σ { + + x + x } s x Y + Ŷ+ + t ε σ { + + x + x } s x Ŷ+ σ 6

.8. 3.7 x 8+ = 4 95% 3.5. χ : X,..., X i.i.d. N, X + + X χ χ. t : Y, Z Y χ, Z N, T = Z t t. Y/ F : X, Y X χ m, Y χ W = X/m Y/ F m m, F. I 3.7. X,..., X Nµ, σ X = X i N µ, σ i= U = X i X i= σ U = σ X i X χ. i= 3 X U σ. x ε µ σ σ x uε/ µ x + uε/, uα N, α. u ε σ u χ ε/ σ χ α χ α u ε/, σ. T = X µ U / t, x, u χ ε µ u u x t ε/ µ x + t ε/, t α t α I X,..., X m, Y,..., Y Nµ, σ, Nµ, σ X, Y, SX, S Y σ σ Z = X Y µ µ σ /m + σ / N, x, y µ µ ε x y uε/ σ m + σ µ µ x y + uε/ 7 σ m + σ,

σ, σ u, u p m P, p P P N p, p p m, P N p, p p P P N p p, p p m + p p. p, p p p ε p p p p uε/ m + p p p p p p + uε/ p p m + p p. 9 8 3 3 5 : 4 95% : p, p P, P P N p, p p, P 9 N p, p p 8 P = 5 9 P + 4 9 P P P P 5 N 9 p + 4 5 9 p p p 4 p p, +. 9 9 9 8 p = 3 9, p = 3 8 5 9 p + 4 5 9 p p p 4 { p p.3857 ± u.5 + =.3696 ±.8 = 9 9 9 8.34.34 p.386..9 6 48 6 8 56 6 6 95% 3 6 6 4 : 5 95% 3. Y t, t =, ±, ±,... t. Y t, t =, ±, ±,... statioary E[Y t ] = µ CovY t, Y t h = γ h, CovY t, Y t = V Y t = γ 8

. γ h = CovY t, Y t h h ρ h = γ h = CovY t, Y t h h γ V Y t : X t, t =, ±, ±,..., i.i.d. {a } = a < Y t = a i X t i E[X t ] = µ, V X t = σ i= E[Y t ] = a i E[X t i ] = µ a i, i= i= i= CovY t, Y t h = Cov a i X t i, a j X t h j = a i a j CovX t i, X t h j j= i= j= = σ a h+j a j, CovX t i, X t h j i = h + j i=.3 {y t } t= {Y t} t= y = i= y t γ h = t=h+ y t yy t h y h ρ h = γ h t=h+ = y t yy t h y γ t= y t y h. ARp p, Auto-regressive Model {Y t } p ARp Y t = ϕ + ϕ Y t + ϕ Y t + + ϕ p Y t p + ε t. ϕ, ϕ,..., ϕ p ε t Y t, Y t,..., ε t, ε t,... E[ε t ] =, V ε t = σ {Y t } ARp. ϕx = ϕ x + ϕ x +... + ϕ p x p = * ARp p =, Y t = ϕ + ϕ Y t + ε t ϕx = ϕ x /ϕ > ϕ <.. t t, t,... Y t = ϕ + ϕ Y t + ε t = ϕ + ϕ ϕ + ϕ Y t + ε t + ε t = ϕ + ϕ + ϕ Y t + ε t + ϕ ε t = ϕ + ϕ + ϕ ϕ + ϕ Y t 3 + ε t 3 + ε t + ϕ ε t = ϕ + ϕ + ϕ + ϕ 3 Y t 3 + ε t + ϕ ε t + ϕ ε t = = ϕ + ϕ + + ϕ + ϕ + Y t + ε t + ϕ ε t + + ϕ ε t. ϕ < Y t = ϕ ϕ + ϕ i ε t i. AR MA. i= {Y t } ARp. µ = E[Y t ] = ϕ + ϕ µ + ϕ µ + + ϕ p µ Y t µ = ϕ Y t µ + + ϕ p Y t p µ + ε t. * L: LY t = Y t. Y t = ϕ L + ϕ L + + ϕ p L p Y t + ϕ + ε t, ϕly t = ϕ + ε t 9

Y t µ, Y t µ, γ = ϕ γ + ϕ γ + + ϕ p γ p + σ γ = ϕ γ + ϕ γ + + ϕ p γ p γ = ϕ γ + ϕ γ + + ϕ p γ p. γ p = ϕ γ p + ϕ γ p + + ϕ p γ γ p+ = ϕ γ p + ϕ γ p + + ϕ p γ γ p+t = ϕ γ p+t + ϕ γ p+t + + ϕ p γ t+ + ϕ p γ t t γ µ, γ, γ,..., γ p ϕ = ϕ + + ϕ p µ ρ = ϕ + ϕ ρ + + ϕ p ρ p ρ ρ ρ p ϕ ρ = ϕ ρ + ϕ + + ϕ p ρ p ρ.. = ρ ρ p ϕ........3 ρ p = ϕ ρ p + ϕ ρ p + + ϕ p ρ p ρ p ρ p = ρ h = γ h /γ AR: Y t = ϕ + ϕ Y t + ε t µ = ϕ + ϕ µ µ = ϕ. ϕ γ = ϕ γ + σ, γ = ϕ γ, γ = ϕ γ, γ = σ ϕ, γ = ϕ σ ϕ. γ h = ϕh σ ϕ, ρ h = ϕ h. AR: Y t = ϕ + ϕ Y t + ϕ Y t + ε t ϕx = ϕ x ϕ x = i D = ϕ + 4ϕ, ϕ = + ϕ ϕ >, ϕ = ϕ ϕ >. ii D = ϕ + 4ϕ <, ϕ ± ϕ + 4ϕ i ϕ ϕ < ϕ > ϕ ϕ = + + 4ϕ = ϕ + { ϕ + 4ϕ } ϕ ϕ 4ϕ ϕ p = ϕ > AR ϕ = ϕ ϕ > ϕ, ϕ,,,,, ϕ = 4 ϕ ϕ ϕ - ϕ : µ = ϕ + ϕ µ + ϕ µ + µ = ϕ + ϕ. : Y t µ = ϕ Y t µ + + ϕ Y t µ + ε t γ = ϕ γ + ϕ γ + σ, γ = ϕ γ + ϕ γ, γ = ϕ γ ϕ ϕ γ = ϕ γ + ϕ γ, γ = + ϕ γ ϕ γ = ϕ γ + ϕ γ γ γ = ϕ ϕ γ + + ϕ γ + σ, ϕ ϕ γ = σ ϕ ϕ ϕ ϕ + ϕ = ϕ σ + ϕ ϕ ϕ >.

. AR Y t = 3.3Y t +.4Y t + ε t, E[ε t ] =, V ε t = 3 µ = E[Y t ], γ, γ, γ, γ 3, ρ, ρ, ρ 3. AR Y t = + 3 Y t + 6 Y t + ε t, µ = E[Y t ] 3 σ = γ, γ, γ, γ 3.3 AR Y t = + ay t +.Y t + ε t a AR Y t =.5Y t + by t + ε t b.3 MAq q, Movig-average Model {Y t } q MAq Y t = θ + ε t θ ε t θ ε t θ q ε t q.4 θ, θ,..., θ q ε t Y t, Y t,..., ε t, ε t,... E[ε t ] =, V ε t = σ {Y t } MAq MAq E[Y t ] = θ γ = V Y t = + θ + + θ qσ, γ = CovY t, Y t = Covε t θ ε t θ ε t θ q ε t q, ε t θ ε t θ ε t 3 θ q ε t q = θ + θ θ + θ θ 3 + + θ q θ q σ. γ = θ + θ θ 3 + + θ q θ q σ,, γ q = θ q σ, γ = q +. ρ h ρ =, ρ = θ + θ θ + θ θ 3 + + θ q θ q + θ + + θ q, ρ = θ + θ θ 3 + + θ q θ q + θ + +,. θ q ARp {Y t } Y t = ξ + ε t + i= ξ iε t i MA. ARp MA Y t = ξ + ε t + ξ ε t + ξ ε t + Y t = ϕ + ϕ Y t + ϕ p Y t p + ε t = ϕ + ϕ ξ + ε t + ξ ε t + ξ ε t 3 + + ϕ ξ + ε t + ξ ε t 3 + ξ ε t 4 + + ξ = + + ϕ p ξ + ε t p + ξ ε t p + ξ ε t p + + ε t = ϕ + ϕ + + ϕ p ξ + ε t + ϕ ε t + ϕ ξ + ϕ ε t + ϕ ξ + ϕ ξ + ϕ 3 ε t 3 +. ϕ ϕ + + ϕ p = µ, ξ = ϕ, ξ = ϕ ξ + ϕ = ϕ + ϕ, ξ 3 = ϕ ξ + ϕ ξ + ϕ 3 =. Y t ε t. θx = θ x + θ x +... + θ q x q = MAq AR.3 θx = MAq. Y t MA, Y t = θ + ε t θ ε t µ = E[Y t ], γ = V Y t, γ = CovY t, Y t θ, θ, σ

: δ, δ, ω γ, γ µ = θ = δ, γ = σ + θ = ω + δ, γ = θ σ = δ ω. θ + θ = δ + δ, θ + δ δ + θ = θ δ θ δ = θ δ = µ, γ, γ θ x = x =., θ θ θ.4 MA Y t = 3 + ε t aε t + aε t a a.5 MA Y t = + ε t + 5 6 ε t 6 ε t, E[ε t ] =, V ε t = µ = E[Y t ], γ, γ, γ, γ 3.6 MA Y t = + ε t 4 ε t 4 ε t, ε t N, 3 γ, γ, γ, γ 3 Y t Y t, Y t+.7 AR Y t = +.8Y t +.3Y t +ε t MA Y t = ξ +ε t +ξ ε t +ξ ε t + ξ, ξ, ξ, ξ 3.8 AR Y t = 3 + 5 6 Y t 6 Y t + ε t Y t MA Y t = µ + i= a i ε t i {a i } a i+ = 5 6 a i+ 6 a i, a =, a = 5 6 LY t = Y t ϕly t µ = ε t /ϕx x.4 ARMAp, q Autoregressive Movig-average Model {Y t } p, q ARMAq, p Y t = c + ϕ Y t + ϕ Y t + + ϕ p Y t p + ε t θ ε t θ ε t θ q ε t q.5 c, ϕ,..., ϕ p, θ,..., θ q ε t Y t, Y t,..., ε t, ε t,... E[ε t ] =, V ε t = σ {Y t } ARMAq.4 ϕx = ϕ x+ϕ x +...+ϕ p x p = ARMAp, q θx = θ x + θ x +... + θ q x q = ARMAp, q 3 θx = ϕx = ARMAp, q. ARMA, Y t = ϕ Y t + ε t θ ε t ϕ, θ σ = V ε t ρ h h =,,

: Y t ϕ Y t = ε t θ ε t V Y t ϕ Y t = V Y t ϕ CovY t, Y t + ϕ V Y t = γ + ϕ ϕ γ, V ε t θ ε t = σ + θ, γ + ϕ ϕ γ = σ + θ. γ = CovY t, Y t+ = CovY t, ϕ Y t + ε t+ θ ε t = ϕ CovY t, Y t + Covϕ Y t + ε t θ ε t, ε t+ θ ε t = ϕ γ θ σ. γ + ϕ ϕ ϕ γ θ σ = σ + θ γ = σ + θ ϕ θ ϕ, γ = σ ϕ + ϕ θ ϕ θ θ ϕ = σ θ ϕ ϕ θ ϕ. ρ = γ = θ ϕ ϕ θ γ + θ ϕ. θ h γ h = CovY t, Y t+h = CovY t, ϕ Y t+h + ε t+h θ ε t+h = ϕ γ h. γ h = ϕ h γ ρ h = ϕ h ρ = θ ϕ ϕ θ + θ ϕ θ h..9 ARMA, Y t = + 3 Y t ε t +ε t, E[ε t ] =, V ε t = 4 µ = E[Y t ], γ h = CovY t, Y t h.5 : ρ, ρ, ρ 3, Y t = ϕ + ϕ Y t + ϕ Y t + ε t ρ = ϕ ρ = ϕ + ϕ ρ ρ = ϕ 3 + ϕ 3 ρ + ϕ 33 ρ ρ = ϕ ρ + ϕ ρ = ϕ 3 ρ + ϕ 3 + ϕ 33 ρ ρ 3 = ϕ 3 ρ + ϕ 3 ρ + ϕ 33 ϕ hh Y t Y t h =.3 ϕ i ϕ hi. AR, MA ϕ, ϕ, ϕ 33 AR Y t = 3.3Y t +.4Y t + ε t, cf... AR Y t = + 3 Y t + 6 Y t + ε t, cf... 3 MA Y t = + ε t + 5 6 ε t 6 ε t, cf..5. {y t } h Y +h : Y X E[Y X] E[Y gx ] gx = E[Y X] E[fX X] = fx, X Y E[X Y ] = E[X].7 [ARp ] Y t = ϕ + ϕ Y t + ϕ Y t + + ϕ p Y t p + ε t Y + Ŷ+ Ŷ + = E[Y + Y, Y,...] = E[ϕ + ϕ Y + ϕ Y + + ϕ p Y p+ + ε + Y, Y,...] = ϕ + ϕ Y + ϕ Y + + ϕ p Y p+ 3

ε + Y, Y,... E[ε + ] = Y + = Ŷ+ + ε + Y + = ϕ + ϕ Y + + ϕ Y + + ϕ p Y p+ + ε + = ϕ + ϕ Ŷ+ + ε + + ϕ Y + + ϕ p Y p+ + ε +, Ŷ + = E[Y + Y, Y,...] = ϕ + ϕ Ŷ + + ϕ Y + + ϕ p Y p+ Ŷ+3 = ϕ + ϕ Ŷ + + ϕ Ŷ + + ϕ 3 Y + + ϕ p Y p+3 [MAq ] Y t = θ + ε t θ ε t θ ε t θ q ε t q Ŷ + = E[Y + ε, ε,...] = θ θ ε θ q ε + q θ q ε + q, Ŷ + = E[Y + ε, ε,...] = θ θ ε θ q ε + q,. Ŷ +q = E[Y + ε, ε,...] = θ θ q ε, Ŷ +h =, h = q +, q +,.... ε +,..., ε +h ε, ε,... e, e,... p.-. AR Y t = + 3 Y t + ε t, ε t N,, Y T =.7 5 ŶT + E[Y T + ŶT + Y T =.7] ŶT + E[Y T + ŶT + Y T =.7] Y T + Y T + 95%. MA Y t = + ε t 3 ε t 3 ε t, ε t N,, 4 ε T = 3, ε T = 3, ε T = 3 [ ŶT + E Y T + ŶT + εt = 3, ε T = 3, ε T = ] [ 3 ŶT + E Y T + ŶT + εt = 3, ε T = 3, ε T = ] 3 Y T + Y T + 95% 3.6.7 X,..., X i.i.d. X,..., X X, X,..., X X j j X fx, F x X = mi{x,..., X }, X = max{x,..., X } = m X m, = m {X m + X m+ } X,..., X X,..., X g gt, t,, t = {!ft ft ft t < t < < t 4

x < y ft ft ft k dt dt dt k = {F y F x}k x<t <t < <t k <y k! F x X X j f Xj x =! = t < <t j <x ft ft j dt dt j fx! j!! j! F xj fx F x j x<t j+ < <t i < j X i, X j x < y f Xi,X j x, y = ft j+ ft dt j+ dt! i!!j i!! j! F xi fxf y F x j i fy F y j, x y f Xi,X j x, y = U,..., U U, U k Betak, k + cf. pp. cf. pp.7 8, p.36.3 7% 6 5% Exλ, Γa, λ, χ, Betaa, b λ a X Γa, λ X f X x = Γa xa e λx x > x a Exλ Γ, λ χ Γ, b X, Y X Γa, λ, Y Γb, λ c > : X i cx Γa, λ/c, ii X + Y Γa + b, λ, iii Betaa, b, X + Y iv a = m/, b = / W = Y/ X/m F m X X + Y = + Y/X = + m W F Fm Beta m,.5 X,..., X i.i.d. Ex/λ λ X + + X χ. a b i, ii S = X + + X P χ ε λ S χ ε = ε λ S = X + + X s λ ε s χ ε λ s χ ε H : λ = λ, H : λ λ ε s = x + + x λ s < χ ε/ χ ε/ < λ s H.6 5

χ ε/ < λ s < χ ε/ H.7.6 X,..., X N i.i.d. Ex/λ j T j = X j, j =,..., < N, T, T,, T ft, t,, t N! [ ft,, t = N! λ exp { } ] t j + N t < t < t < < t λ j=.8 λ λ = { T j + N T }. 3 λ λ λ χ. : j= P T k t = e λ t lλ = log ft,, t λ 3 s < / E[e s λ/λ ] = s χ E[e s λ/λ ] = E[e s λ { N! = N! λ e s λ j= Tj+N T} ] t dt t e s λ t dt t e = = s. = 4 < N.8 s λ t dt e s λ N +t dt t S = T + + T + N T P χ ε λ S χ ε = ε. S = T + + T + N T λ ε s χ ε λ s χ ε.9 H : λ = λ, H : λ λ ε s = t + + t + N t.6,.7 X N X x,, x s = x + + x + N X λ ε.9 H : λ = λ, H : λ λ ε.6,.7.4, 95% 73, 838, 95,, 355, 4, 638 3 638 95% 3 3 7 4 5% 6

3 3. 3. {X t } t < t < < t < t x, x,, x, x P X t x X t = x, X t = x,, X t = x = P X t x X t = x {X t } : P X = x, X = x,, X = x = P X = x X = x,, X = x P X = x,, X = x = P X = x X = x P X = x,, X = x. = P X = x X = x P X 3 = x 3 X = x P X = x X = x. P X t+ = y X t = x = p xy, f Xt+s X t y x = f X t,x t+s x, y = ps, x, y f Xt x t X t S = {,,, m} S p p m p ij = P X t+ = j X t = i P =.. P p m p mm p p m p ij = P X t+ = j X t = i P =.. - + p ij = m k= P X t+ + = j X t = i = p m p mm p ik p kj m P X t+ = k X t = ip X t+ + = j X t+ = k k= P X = i = p i P X =,, P X = m = P X =,, P X = mp = p,, p m P. π = π,, π m π = πp π i, j p ij > π π m i, j p ij > lim P = π π m P X t+ x X = x, X = x,, X t = x t, X t = x t = P X t+ x X t = x t, X t = x t X t Y t = X t, X t+ Y t 7

3. X t {,, 3} P = / / / / P X = i X = j = p ji p, 3 p p : p = p p + p p + p 3 p 3 = + + =, p = + + = 4, p 3 = + + = 4 3p = p p + p p + p 3 p 3 = + 4 + 4 = 4. P, i/, i/. P U P = UDU * D, i/, i/ P = UD U D, i/, i/ a, b, c i p = a + b + c i p ±i = e ±iπ/ α, β, γ ± i = e ±iπ/ = cos π ± i si π { p = α + β cos π + γ si π P = p ij P = p ij }. 3. = p = α + β, = p = α + γ, = p = α 4 β α = /5, β = 4/5, γ = /5. P X = = p = { 4 5 + 5 cos π 5 si π }. 3. 3. X t p, p 3, 3 p 3 p 3 : p 3 3. 3. {,, 3, 4} b, c 3..4.6.7.3.7.3 a.7.3 b..5.3 c..5.3..4.5.3.6....4.3.5.5..8.4.6 : π = π, π, π 3, π 4 πp = π.4π +.π 3 +.5π 4 = π,.6π +.7π = π,.3π +.4π 3 = π 3,.5π 3 +.5π 4 = π 4, 3 π = π / = π 3 = π 4 π +π +π 3 +π 4 = π = /5, /5, /5, /5 3. 3. b, c * U 8

3. {X t } t E[X t+ X t, X t,, X, X ] = X t, a.s. 3. cf..7 3. X E[E[Y X]] = E[Y ], E[aY + bz X] = ae[y X] + be[z X], a.s. a, b 3 E[gXhY X] = gxe[hy X], a.s. E[gX X] = gx, a.s. 4 X, Y E[Y X] = E[Y ], a.s. 5 E[E[Z X, Y ] X] = E[Z X], a.s. {X t } t > s E[X t X s, X s,, X ] = X s E[X s+ X s, X s,, X ] = E[E[X s+ X s+, X s, X s,, X ] X s, X s,, X ] = E[X s+ X s, X s,, X ] = X s 3. 5 3. t = s + t = s 3. E[X t ] = E[X s ] : s t > s s s t a ξ, ξ,, i.i.d. P ξ i = = P ξ i = = t Zt a = a + ξ + ξ + + ξ t a a = Z t = Zt { y = x, x + Z t Z P Z t+ = y Z t = x = E[Z t+ Z t, Z t,, Z ] = E[Z t Z t, Z t,, Z ] + E[ξ t+ Z t, Z t,, Z ] = Z t + E[ξ t+ ] = Z t 3. 3, 4 3.3 Z t < s < t, a, b E[Z t Z s ], V Z t Z s, 3 P Z a+b = a b, 4 P Z 5 = 3, Z 8 = : 4 Z t 3.4 ξ, ξ,, i.i.d. P ξ i = = P ξ i = = Z t = ξ + ξ + + ξ t E[Zt+ t + ξ t, ξ t,, ξ ] = Zt t. Zt t ξ t, ξ t,, ξ p.36 3. 9, 3. 3.5 9

3. 3.3 N t, t λ N = N t P oλt t, P N t = k = λtk e λt, k =,,,.... k! < t < t < < t N t, N t N t,, N t N t 3 t > s > N t N s N t s. p.3-7 N t+h N t P N t+h N t = oh, lim = h h,, 3 p.33 3.3 N t λ < s < t, k, l =,,, P N s = k, N t = k + l, E[N s N t ], 3 CovN s, N t. : N s N t N s N s P oλs, N t N s N t s P oλt s P N s = k, N t = k+l = P N s = k, N t N s = l = P N s = kp N t N s = l = λsk λt s l e λt. k! l! N P oλ E[N] = V N = λ E[N t N s ] = E[N t N s N s ] + E[N s ] = E[N t N s ]E[N s ] + V N s + E[N s ] = λt sλs + λs + λs = λ st + λs. 3 CovN s, N t = E[N s N t ] E[N s ]E[N t ] = λ st + λs λsλt = λs. 3.4 N t λ N t λt : s < t N t N s N u, u s. E[N t λt N u, u s] = E[N t N s + N s λt N u, u s] = E[N t N s ] + N s λt = λt s + N s λt = N s λs 3.5 N t λ < s < t a E[N t N t + ], b E[N s N t ], c E[N s N t ], d E[N t N s = ], e E[N s N t = ] T i = i t t t coutig process T, T T, T 3 T, t T, T T, T 3 T, Exλ t k t k + T k+ > t T Γ, λ P t k = P T k+ > t = λ k+ t k! x k e λx dx = = k λt j e λt. 3.3 j! 3.6 T, T T, T 3 T Exλ P 3T < T, P 3T < T < T 3, P T < T 3 j=

Poisso T k+ Γk +, λ Y = λt k+ χ k+ X P oλ 3.3 t = P X k = P T k+ > = P Y > λ P X k = P X k = P Y > λ = P Y λ, Y χ k Poisso χ s ε λ χ s+ ε s x,..., x, s = x k k= H : λ = λ, H : λ λ ε λ < χ s ε/ λ > χ s+ ε/ H χ s ε/ < λ < χ s+ ε/ H 3.3 W t, t W = W t N, t t. < t < t < < t W t, W t W t,, W t W t 3 t > s > W t W s W t s. t W t 3.5 W t < s < t W t f Wt x, W s, W t f Ws,W t x, y, 3 E[W s W t ], 4 E[W t W u, u s] = W s, W t : W s W t W s W s N, s, W t W s W t s N, t s f Wt x = e x t. fws,w t W sx, z = e x s e z t s πt πs πt s f Ws,W tx, y = e x s πs e y x πt s t s. 3 E[W s W t ] = E[W s W t W s + W s ] = E[W s ]E[W t W s ] + E[W s ] = s. 4 W t W s W u, u s. E[W t W s + W s W u, u s] = E[W t W s W u, u s] + W s = E[W t W s ] + W s = W s 3.7 W t < s < t, α R a E[W 3 t ], b E[W 4 t ], c E[e αw t ], d E[W t e αw t ], e E[W s W t ], f E[W t W s ], g E[e αw t W s ], h E[W s e αw t ], i P W t > W s >

4 X fx E[gX] = gxfx dx [ ] X, X,... i.i.d. X i X gx, gx,... i.i.d. gx i gx lim gx i = E[gX] i= X x, x,... N U U, E[e U ] = gxfx dx N N gx i i= e x dx U,.63,.346,.465,.796,.44,.58,.6877,.9943,.77,.668 u,..., u N, N = E[e U ] = e x dx N N e u i.3.5859,.43676 Excel R.4665 i= gx i E[gX] N, V gx V gx gx i E[gX] = N, + o V gx i= cf. i= 4. 4.. U F x X X = F U, F u = if{ x ; F x u } X x F x F u = F u X = F U F x P X x = P F U x = P U F x = F x. X P X k = x k = p k, k =,,, N X U p X = x p < U p + p X = x p + p < U p + p + p 3 X = x 3.. p + p + + p N < U p + p + + p N = X = x N 4.

4. X Ex U U, E[X] E[+X ] X F x = e x x > F u = log u X = log U U.84.3.76.35.6 U X.836.3566.47.437.743 U X + X + X.353.737.4.699.856 U =.84 V X = log.84 = log.6 =.36.47 =.836, + X = +.836 =.3533 U X, + X E[X].836 +.3566 +.47 +.437 +.743 =.8446 5 E[ + X ].353 +.737 +.4 +.699 +.856 =.654 5 4. U X U X 3 4 5 E[X] X. X. U.5.3.54.35.8 X 3 4 5 P.3.7..8. log U 4. cgy cgy c 4. 4. X f. : y fy cgy P X x = P Y x U fy P Y x, U fy cgy = = cgy P U fy cgy x fy cgy fy cgy gy dudy gy dudy 3

= x fy gy dy cgy = fy gy dy cgy c c x fy dy = fy dy x fy dy. 4. 4. X = Y P U fy = cgy c T 3 P T = k = k, k =,,..., c c E[T ] = k= k k c c = c { = c c } p.4-9 4. Z N, X = Z Y Ex X fx = π e x / x > Y gx = e x x > fx gx = π ex x / = π e [x ]/ f e g = π. c = e/π fx cgx = exp x x [ ] U, U U, [ ] Y = exp { x }. F Y y = e y F Y u = log u Y = log U [ 3] U exp { Y / } X = Y U, U U Y X F Y y = e y F Y u = log u. Y = log U Y = e Y / U =.84 U.84.3.76.35.6 U.8.77.69.57.77 Y.836.3566.47.437.743 Y.746.85.939.85.77 Y = log.84 = log.6 =.36.47 =.836, Y = e.836 / = e.347 = e.35 = e. 3 e.5 =.948 3.95 =.746 V VI *3 U Y, Y Y U E[X].836 +.3566 +.47 +.437 =.75 4 4.3 X Γ 3, Y Exλ λ λ 4. U, U Y Y X X *3 e., e. Y % 4

4.4 fx = 3x x 4, < x < Y U, c Y.5.3.54.35.8 X Y U U, U.8.77.69.57.77 X 4..3 F,, F F F x = h i F i x, i= h i =, h i > Y = 3 U, U U, i= U h Y = F U h < U h + h Y = F U h + h < U h + h + h 3 = Y = F3 U Y F i 4. : h < k P h U k, F i U x = P h U kp U F i x = k hf i x 4.3 f X x = P Y x = P U h, F U x + P h < U h + h, F U x +P h + h < U, F 3 U x = h F x + h F x + h 3 F 3 x. { 3x 3 + x < x < U, U X X U.3.94.78.55. U.94.56.6.9.9 : F x = x 4, F x = x F X x = 3 4 F x + 4 F x < x < U 3/4 X = U /4, 3/4 < U X = U / U =.3 U /4 X = F U =.94 /4 =.9846, U =.94 /4 g x = g x [, 5] f X x = 3 g x + 3 g x X 4.3 U, U { X {.5e.5x x > e x 3 g x =, g x < x = f X x = 3 g x + 3 g x 5

X U, U =.3,.75,.8,.35 X 4. X, X,... i.i.d. gx i θ, σ θ = gx i E[ θ ] = E[gX ] = θ i= V θ = i= V gx i = σ, E[ θ θ ] = σ. θ θ c/ c/ c pp.4-3 4.. 4.3 f, g CovfX, gx. : x y fx fygx gy, x y fx fygx gy fx fy gx gy X, Y E[fX fy gx gy ] = E[fXgX] E[fXgY ] E[fY gx] + E[fY gy ] = E[fXgX] E[fX]E[gY ] E[fY ]E[gX] + E[fY gy ] X, Y = E[fXgX] E[fX]E[gX] E[fX]E[gX] + E[fXgX] X, Y = E[fXgX] E[fX]E[gX] = CovfX, gx. U U, hx h x CovhU, h U = CovhU, h U X F F U i, i =,,..., U, i.i.d. U i U, X i = F U i, Y i = F U i F X i, Y i θ M = M M i= {gx i + gy i } = M M {gf U i + gf U i } E[ θ M ] = θ θ gx gx gf u CovgF U i, gf U i V θ M = M = M i= M V [gf U i + gf U i ] i= M { } V [gf U i ] + CovgF U i, gf U i + V [gf U i ] i= 6

M M i= { V [gf U i ] + V [gf U i ] } = M V gx = V θ M. 3 F U i, F U i X θ M θ θ M M M 4.4 si π x dx a M U, U,, U M V θ M b M U, U,, U M, U, U,, U M V θ M c 4. U θ M θ M, M = 5, : U i U gx = si π x θ M = M gu i M i= θ M = M E[gU i ] = V θ M = M si π x dx = π, E[gU i ] = M i= M {gu i + g U i } i= V gu i = M V gu = M si π x dx = 4 π M.947565. E[{gU i + g U i } ] = E[gU i ] + E[g U i ] + E[gU i g U i ] = + + si π x si π x dx = + si π x cos π x dx = + π V θ M = M M V gu i + g U i = V gu + g U 4M i= = [E[{gU + g U} ] E[gU + g U] ] 4M = + 4M π 6 π M.7744754. θ 5 =.84 =.47, 5 θ.84 + 4.36 5 = =.645. U.5.3.54.35.8 si π U.78.454.75.5.79.84 si π U.997.89.66.853.96 4.36 4.6 x dx a M U, U,, U M V θ M b M U, U,, U M, U, U,, U M V θ M c 4. U θ M θ M, M = 5, 4.. θ 3 = {gx i + chx θ h } E[hX i ] = θ h h i= V θ 3 = V [gx + chx θ h] = { } V gx + c CovgX, hx + c V hx 7

= [ { V hx c + CovgX, hx V hx } CovgX, hx ] + V gx c = c CovgX, hx = V hx V θ 3 = {V gx V hx CovgX, } hx V hx V gx = V θ 4.3 3 V θ CovgX, hx = = CorrgX, hx, V θ V hxv gx gx hx CorrgX, hx Ĉov[gX, hx] = gx i ĝxhx i ĥx, i= Var[hX] = hx i ĥx, ĝx = c c = θ 3 4.5 i= Ĉov[gX, hx] Var[hX] = ĝx gx i, i= Ĉov[gX, hx] Var[hX] ĥx θ h ĥx = hx i. e x dx a U, U,, U V θ b hu = U V θ 3 3 c 4. U θ θ, = 5, : gu = e U θ = gu i θ 3 i= i= E[gU i ] = e u du = e, E[gU i ] = e u du = e V θ = V gu = e e = e 3 e.43567. = CovgU, hu {gu i + chu θ h } 4.3 c = V hu V hu = E[U ] E[U] =, CovgU, hu = xe x dx e x dx = e V θ 3 = CovgU, ] hu [V gu = [ e e e ] V hu = 3 e7e 9 =.3949. VarU =.3393, Ĉov[eU, U] =.463 ĉ =.3567, θ h =.5 θ 5 = 6.733 =.3466, 5 θ 3 5 = θ U.5.3.54.35.8 5 + ĉ ĥu θ h e U.5.35.76.49.97 6.733 =.3466.3567.84.5 =.6396. 4.7 x dx a U, U,, U V θ i= 8

b hu = U V θ 3 c hu = U V θ 3 3 d 4. U θ 5 b θ 5 4.3 3 4.3. d A,..., A d A j p j H :p,, p d = p,, p d, H :p,, p d p,, p d. A j j d j p j T = j= p j = 4.4 H T d χ II α 4.4 t t < χ d α H t χ d α H p j < 5 p j 5 4.6 5% A : B : O : AB = 38. :.8 : 3.8 : 9.4 : A, B, O, AB p, p, p 3, p 4 H : p, p, p 3, p 4 =.38,.8,.38,.94, H H. 5% 4 T χ 3.5 = 7.847 A B O AB 36 6 33 94 A, B, O, AB 38, 8, 38, 94 t = 36 38 38 + 6 8 8 + 33 38 38 + 94 94 94 =.644. H 4.8 5% A : B : O : AB = 38. :.8 : 3.8 : 9.4 A B O AB 8 3 4 7 4.9 4 B4, / 3 4 5% 5 3 33 3 9 9

4.3. N A, B A.A,, A r B, B,, B s r s A i B j f ij H : A, B α f ij 5 f i f j /N 5 r s f ij f i f j χ = N > χ f i f j ϕα H i= j= N χ < χ ϕ α H ϕ = r s A B B B B s A f f f s f A f f f s f........ A r f r f r f rs f r f f f s N 4. H A i B j f i f j /N χ χ = i j 4.4 ϕ = r s : χ χ = r i= j= s f ij f i f j N f i f j N = f f f f N f f f f. 4.5 3 Yates : χ f ij 5 N χ f f f f N =, f f f f = 4 Fisher : 3 f ij χ H : A, B f i, f j r s r i= A i B j f ij f i! s j= f j! N! r s i= j= f ij! p 4.7 A 5 A 5% 75 6 35 75 4 5 5 5 : H :, H H. 5% = T χ.5 = 3.845 3

χ = 75 4 6 75 5 35 5 5 =.45. H 4. 3 5% 3 5% 3 8 3 4 38 4 6 4 58 6 7 7 9 9 3

5 5. [LC] E.L. Lehma, G. Casella: Theory of Poit Estimatio, Secod Editio, Spriger, 998 II 5. δ > lim P X X < δ = X X X X i prob. Y F Y y = P Y y X X F X c lim F X c = F X c X X i law 5. X, X,... i.i.d. S = X i i= lim x xp X > x = a S a i prob. E[ X ] < S E[X ] i prob. E[X ] < µ = E[X ], σ = V X S µ N, σ i law θ Θ X, X,..., X i.i.d. fx θ px θ = P θ X = x P θ, E θ A θ θ fx θ fx θ. A A = { x fx θ > } θ A3 Θ θ L θ x = i= fx i θ, l θ x = i= log fx i θ x = x, x,... θ = θ x,..., x : L θ x = max θ Θ L θ x l θ x = max θ Θ l θ x θ = θ X,..., X 5. A A3 θ θ X = X, X,.... : L θ X > L θ X P θ L θ X > L θ X, as, i= i= log fx i θ fx i θ < log fx [ i θ fx i θ E θ log fx θ ] fx θ Jese log x < [ E θ log fx θ ] [ fx θ ] < log E θ = log fx θ fx θ 3 i prob. fx θ fx θ fx θ dx =

5.3 Jese fx E[fX] fe[x]. f x > X : µ = E[X] fx c R fx cx µ + fµ E[fX] E[cX µ + fµ] = ce[x] µ + fµ = fµ. f x > fx > f µx µ + µ, x µ, P X = µ = 5.4 A A3 x Θ θ fx θ θ f x θ l θ x = i= f x i θ fx i θ = 5. θ = θ x,..., x θ X,..., X θ i prob. 5. θ θ θ θ : a > [θ a, θ + a] Θ S = { x ; l θ x > l θ a x ad l θ x > l θ + a x } 5. P θ X S. x S θ θ a, θ + a l θ x l θ x = P θ θ θ < a P θ X S. θ a l θ x θ θ θ θ x θ Lehma-Casella [LC] p.54 5.5 X a > Z Z Z /X Z/a : c F Z z c/a F Z/a z/a F Z z = F Z/a z/a F Z c lim P Z /X c/a = P Z c ε > δ > x c < δ = F Z x F Z c < ε 4 F Z c, c F Z c δ < c < c < c < c + δ lim F Z c i = F Z c i i =, X a i prob. N N N = F Z c i F Z c i < ε 4 i =,, P A c < ε 4 33

{ A = X a < a } c + δ, δ = mi{c c, c c, } A c < c δ < c a X < c + δ < c X > P P Z c { Z P c } { A + P A c = P Z c } X a X a a X A + P A c Z c P X a P Z c + δ + P A c P Z c + ε 4 P Z c + ε, P Z c ε. N P Z /X c/a F Z c < ε 5.6 X i prob. {Y } tight, ε > M > if P Y M ε X Y i prob. : ε > M > P Y > M < ε δ > N N N P X δ < ε N M P X Y δ = P X Y δ, Y M + P Y > M P M X δ, Y M + P Y > M P X δ + P Y > M < ε. M A A3 X X i A4 fx θ θ C 3. [ ] ] [{ A5 E θ θ log fx θ =, Iθ := E θ [ } ] θ log fx θ = E θ θ log fx θ,. A6 c > Mx x A cf. A θ θ < c 3 θ 3 log fx θ Mx E θ [MX] <. fx θ dx θ A5 fx θ dx = θ θ fx θ dx = fx θ dx =, θ f log fx θ = fx θ θ x θ, log fx θ = θ fx θ [ ] [ E θ θ log fx θ = E θ E θ [ θ log fx θ fx θ dx =. 5. θ f θ x θ fx θ f θ x θ f ] fx θ θ X θ = θ fx θ dx =, 5.3 ] [ ] [{ = E θ fx θ θ fx θ f } ] E θ fx θ θ X θ 5.4 [{ = θ fx θ f } ] dx E θ fx θ θ X θ [{ } ] = E θ θ log fx θ = Iθ Iθ Fisher Cramér-Rao 34

5.7 X, X,... i.i.d. A A6 5. θ = θ x,..., x θ X θ N, /Iθ i law 5.5 5. θ θ 5.5 θ X = θ X,, X : x = x,, x l θ x θ Taylor l θ x = l θ x + θ θ l θ x + θ θ l θ x θ = θ + h θ θ, < h <, = θ X θ = A5 l θ X l θ X θ X θ l θ X. l θ X = i= θ log fx i θ N, Iθ i law. A5 l θ X = [ ] θ log fx i θ E θ θ log fx θ = Iθ i prob. i= 5.4 θ θ i prob. ε > N N P θ θ θ c < ε/ θ θ θ θ A6 l θ X { θ θ <c} 3 θ 3 log fx i θ { θ θ <c} i= MX i E θ [MX] i= i prob. M = E θ [MX] P θ l θ X M + P θ θ θ c + P θ l θ X { θ θ <c} M + < ε 5.5, 5.6 Z N, Iθ θ X θ Iθ Z Iθ Z N, Iθ i law θ X θ, Iθ 5. X, X,... i.i.d. X i Exλ, λ >, λ λ : fx λ = λe λx l λ x = λ log fx i λ = λ x + + x = i= 35

λ = X, X = X + + X. Fisher [ ] Iλ = E λ log fx λ = λ. λ N λ, λ 5. Gep, < p <,, 5 p 95% * 4 : fx p = p x p, x =,,,..., p = l p x = p i= log fx i p = x + + x p + p = + X, X = X + + X. Fisher [ ] [ Ip = E p log fx p X = E p + ] p p = + X N pp p, p = / + 5 =.9538 = p p + p p = pp. pp p p p ±.96 p p =.9538 ±.96.476.9538 =.9538 ±.47 = {.95645.9483 95%.9483 p.9565 5. a > θ > X, X,... i.i.d. X i fx θ = aθ a x a e θxa, x >, θ θ k θ = θ,..., θ k Fisher Iθ Iθ = I θ I k θ.., I k θ I kk θ I ij θ = E θ [{ θ i log fx θ 5.4 }{ }] [ log fx θ = E θ θ j θ i θ j log fx θ ], i, j =,..., k. A A6 5.8 A5 Iθ, Iθ 5.8 p. X, X,... i.i.d. θ j l θ x = i= θ j log fx i θ =, j =,..., k 5.6 *4 [IK] 4,, 5 36

θ = θ x,..., x θ X θ N, Iθ i law 5.7 5.6 θ θ 5.7 θ X θ, Iθ k β 5.3 α, β > X, X,... i.i.d. X i F x α, β = x + β x α, β α, β : fx α, β = αβ α. x + β α+ α log fx α, β = + log β logx + β, α α log fx α, β = α, β log fx α, β = α β α + x + β, α β log fx α, β = β x + β, β log fx α, β = α β + α + x + β. E[X + β k αβ α ] = x + β α++k dx = α α + kβ k, k > α, Fisher Iα, β = α α + β α. Iα, β α, β α + β α + β α N, α α + αα + α + β β α + α + β αα + α + β α 5. X, X,... i.i.d. µ, σ fx µ, σ { = πσ x exp log x } µ σ, x >, µ, σ µ, σ α 5.9 p. 4 k X = X,, X k θ = θ,, θ k, Σ θ Σ g k G = gx,, X k G gθ, g Σ g g = g θ θ g θ k θ 5.8 θ g θ gθ, g Iθ g µ σ σ : k = a, b X, Y N, µ σ σ ax + by aµ + bµ : V ax + by = a V X + ab CovX, Y + b V Y = a b σ σ a. 5.8 σ σ b gx, x G gθ, θ = g θ X, X X θ + g θ X, X X θ 5.9 37

X, X = θ, θ + hx θ, X θ, < h <, X θ i prob. X θ i prob. g X θ, X g θ, θ i prob. g X θ θ, X g θ, θ i prob. θ X θ, X θ,, Σ Y, Y 5.9 G gθ, θ 5.8 g θ, θ Y + g θ, θ Y θ θ i law 5.4 5. µ = p 95% p p : 5. p = + X p p N, pp p gp = p g p g p gp, p g p pp g p = pp p = p p 95% p x + x.5 +.5 g p ± u.5 p = x ± u.5 =.5 ±.96 =.5 ±.449 95%.455 µ.545 5.3 5. µ, σ σ = σ : e µ+ σ 5. [R] S.I. Resick: Extreme Values, Regular Variatio ad Poit Processes, Spriger, 987 [TS], : ISM :,, 6 5. p. 55 {X } a >, α >, b, β R a.s. X, Y X b a X i law, X β α Y i law, 5. a lim = a >, α b β lim = b 5. α a, b Y ax + b 38

X a.s. P X = c = c a > b R ax + b Y X Y x b F Y x = P ax + b x = F X a cf. [R] pp.7 8 5. Y X β α = a X b + b β a X + b β α a α α α X, Y a.s. 5. 5. Fisher-Tippett p. 5 X, X,... i.i.d. M = max{x,, X } a.s. Y c >, d R M d c Y i law 5. Y { exp x α, x >, Fréchet Φ α x =, x. { exp x α, x, Weibull Ψ α x =, x > Gumbel Λx = exp e x. α > Φ α x, Ψ α x, Λx α = 5. : X i, Y F x = P X i x, Hx = P Y x M F x 5. Hx x R 5.3 [t]...4.6.8 4 4 M d P x = F c x + d Hx,. 5.3 c M[t] d [t] P x = F c [t] x + d [t] [t] Hx,. 5.4 c [t] [a] a M [t] F x [t] M[t] d P x = F c x + d { [t] F = c x + d } [t]/ Hx t c 5.5 39

5.4, 5.5 {Hx} t a.s. Y t M [t] d [t] c [t] Y i law, M [t] d c Y t i law, lim c c [t] = γt >, d d [t] lim = δt 5.6 c [t] γt, δt t > γty t + δt Y : { Hx } t = P Yt x = P Y γtx + δt = Hγtx + δt. 5.7 t st { Hx } st = Hγstx + δst. { Hx } st = { Hx s } t = { Hγsx + δs } t = H γt{γsx + δs} + δt. x 5. γst = γsγt, δst = γtδs + δt, s, t > 5.8 γe = e θ, θ R, 5.8 s, t > N γs = γs t = s γt / = γt / r Q γe r = γe r = e θ r = e r θ 5.6 γt *5 γt = t θ, t >. 5.9 a θ = : γt 5.8 δst = δs + δt. δe = c r Q δe r = rδe = rc δt δt = δe log t = c log t, t >, 5.7 { } t Hx = Hx + c log t, t >. 5. c = Hx =, Y a.s. c Hx { Hx } t t c < x R Hx = 5. Hx + c log t =, t >. x R Hx = Hx x R Hx <. x R Hx = x R Hx = x R Hx >. 5. x = {H} t = Hc log t. c = /c >, d R exp{ e d } = H, x = c log t t > < x < Hx = { H } e x/c = exp{ e d e c x } = Λc x + d. *5 [BGT] pp.4 5 [BGT] N.H. Bigham, C.M. Goldie ad J.L. Teugels: Regular Variatio, Cambridge Uiversity Press, 987. 4

b θ > : 5.8 t, s γtδs + δt = γsδt + δs δs γs = δt γt, δs c t γs δt = δs γt γs = c tθ {Hx} t = Ht θ x + c t θ = Ht θ x c + c 5. x = c Hc t = Hc Hc =. Hc = x > c Hx > {Hx} t = Ht θ x c + c Hc = as t +. a = Hx > a t as t + Hx Hx Hc = < Hc < Hc = 5. H t θ + c = Ht θ c c + c = {Hc } t =, t >, H H Hc = H p > Hc = exp{ p α } α = /θ x < c x c = t θ t = c x α 5. Hx = Hx c + c = H t θ + c = {Hc } t = exp{ pc x α } = Ψ α px c. c θ < : b c 5. Hc =, Hc = x < c Hx < Hc = lim t Ht θ x c + c = lim t {Hx} t = Y = c a.s. Hc =. b < H + c H + c = exp{ p α } α = /θ x > c x c = t θ t = x c α 5. Hx = Hx c + c = Ht θ + c = {H + c} t = exp{ px c α } = Φ α px c. 5. X a.s. F x a >, c >, b, d R x R F ax + b = F cx + d a = c, b = d : a c Gx = F cx + d F ax + b = F c a c x + b d c + d = G a c x + b d c α = a c, β = b d Gx = Gαx + β c Gx = Gαx + β = Gααx + β + β = Gα x + α + β 4

α > Gx = G = Gα{α x + α + β} + β = = Gα x + α + + β 5. α x + α α β = G α x + β α β α F x a.s. Gx , α α, x + β α <, α =. β 5. {, β >, p = Gx = Gx + β, β <, β = 5.5 X, X,... i.i.d. c >, d R 3 5. Y Φ α, Ψ α, Λ { βx β, x, Parate : f X x = c = /β, d =. β >, x <, { b x b, < x <, Beta, b : f X x = c = /b, d =.,, 3 Exλ c = λ, d = log. λ : Φ α x, Ψ α x, Λx < x < 5.3 lim {F c x + d } { x β, x, X i F x =, x <. x F c + d =. x > /β x > {F c x + d } = { /β x β } = x β e x β = Φ β x. Φ β x, 3 5.4 5.5, 3 5.6 9 X, X,..., X M M 8 8 cf. 5.5 3 : x F Xi x = e x/9 P M 8 8 = P M 8 < 8 = e 8/9 8 =.9947. M 9 log 5.5 3 P x e e x,, 9 M8 9 log 8 8 9 log 8 P M 8 8 = P exp{ e 9+log 8 } 9 9 = exp{ 8e 9 } =.994639. 4

5.5 X, X,..., X i.i.d. 5.5 β = 9 7 Parate M M 8 8 : 8 = 7. cf. 5.5 e =.36788 5. σ >, µ R { H ξ,µ,σ x = exp + ξ x µ σ /ξ }, + ξ x µ >, 5.3 σ GEV geeralized extreme value distributio ξ = } H,µ,σ x = lim H ξ,µ,σ x = exp { e x µ σ, < x <, 5.4 ξ ξ, µ, σ µ =, σ = H ξ x GEV Φ α x, Ψ α x, Λx x Fréchet Φ α x = H /α, ξ = /α > /α Gumbel Λx = H x, ξ = x + Weibull Ψ α x = H /α, ξ = /α < /α 5.3 α >, Z Ex X = log Z Gumbel Λ M X t = E[e tx ] = Γ t. E[X] = γ, V X = π /6. γ = lim + + + log =.5775664 Euler X = Z /α Fréchet Φ α E[X ] = Γ, α α >. 3 X = Z /α Weibull Ψ α E[X ] = Γ +, + α α >. : X P X x = P Z e x = M X t = E[e tx ] = E[Z t ] = e x e z dz = e e x = Λx, z t e z dz = Γ t X ψ X t = log M X t ψ X = E[X], ψ X = V X Weierstrass Γx = xeγx p= + x e x/p p ψ X t = log Γ t = log t γ t ψ Xt = t + γ ψ Xt = p= { /p + t/p + } p t + p + t. p= { log p= + t p = t + γ p= t }, p { p p + t }, 43

E[X] = ψ X = + γ V X = ψ X = + 5.6 5.3, 3 p= { p } = + γ = γ, p + p= p + = k = π 6. k= 5.3 5. X i F 5. Y H c >, d R ξ Hx = H ξ x F H ξ maximum domai of attractio F MDAH ξ H ξ Φ α, Λ, Ψ α Poisso [R], [TS] [TS], pp.35 36 F x = F x 5.6 Fréchet Fréchet Φ α x = exp x α, x >, α > F MDAΦ α ω F =, F x = x α lx, lx c = F /, d = M /c Φ α i law : fat-tailed Cauchy fx = /[π + x ], x R, c = /π, α = Parate F x Kx α, K, α >, c = K /α fx = αβ Γβ log xβ x α, x >, α, β >, c = [log β /Γβ] /α Weibull Weibull Ψ α x = exp x α, x <, α > F MDAΨ α ω F <, F ω F x = x α lx, lx c = ω F F /, d = ω F M ω F /c Ψ α i law : fx =, < x <, c = /, d =, α = ω F F x = Kω F x α, ω F K /α x ω F, K, α >, c = K /α, d = ω F fx = Ba,b xa x b, < x <, a, b >, Betaa, b c = [ ΓaΓb+ ] /b, Γa+b d =, α = b 44

Gumbel Gumbel F MDAΛ Λx = exp e x, x R ω F, F x = cx exp x gt α F at dt, α F < x < ω F, x ω F cx c >, gx, a x. d = F /, c = ad M d /c Λ i law : fx = Γλ xλ e x, x >, λ > Γλ, c =, d = log + λ log log log Γλ fx = π exp x, x R N, c = log /, d = log [log log + log4π]/ log fx = log x µ πσx exp σ, x >, µ R, σ > LNµ, σ c = σ log / d, d = exp { µ + σ [ log fx = [ λ πx exp λx µ 3 µ x ], x >, µ, λ >, c = µ /λ, d = µ { log 3 log log + log ω F < F x = K exp a ω F x, x < ωf, a, K >, c = a/[logk], d = ω F a/ logk log log +log4π log } λ 4πµ + λ µ /λ 7 [MFE] A.J. McNeil, R. Frey, P. Embrechts: - -,, 8. ]} 5.3 M cf. 5. S = X + + X σ := V X < S E[X ] N, σ i law X k Cauchy fx = ϕ Xk t = e t Y = S ϕ Y t = E[e ity ] = E[e i t X k ] = k= ϕ Xk t/ = e t/ = e t k= π + x Y Cauchy Y Cauchy i law S 5.4 X k N X X,..., X k a k >, b k X + + X k a k X + b k 45

5.4 Z a >, b S b Z i law Z a 5.5 Z Z α, β, m, m ϕ Z t ϕ Z t = exp {iβt + m e itx itx dx + x x +α + m e itx α < α < c R, d > θ 9 exp { ict d t α + iθ t t ta πα }, α, ϕ Z t = exp { ict d t + iθ t }, t α log t α =, } itx dx + x x +α.5.5 [B] L. Breima, Probability, Addiso-Wesley, 968. Classics i applied mathematics, 7, Society for Idustrial ad Applied Mathematics, 99. Reprit 3 7 [D] Durrett, R.: Probablity Theory ad Examples, 4th ed., Cambridge Uiversity Press,. 5.6 X, X,... i.i.d. P X > x i lim = θ [, ], x P X > x ii < α < Lx P X > x = x α Lx. S = X + + X a = if{x; P X > x /}, b = E[X { X a }] a.s. Z S b Z i law Z 5.3 5.5 m = αθ, m = α θ, β 5.33 Ltx Lx t > lim x Lx = log x x α α a 5.7 X 5.6 ii X = X δ > C δ > x δ > y x δ P X > y C δ y α+δ. E[X ] =. C > x > x x x : x δ > x x δ P X > x P X > x yp X > y dy Cx P X > x. = α Lx Lx α+δ P X > x δ P X > x δ α+δ y x δ y/x δ y P X > x δ P X > x δ α+δ C δ y α+δ 46

C δ = x δ / α+δ P X > x δ δ > α + δ < Ix = E[X ] = x yp X > y dy [ ] yc δ y α+δ C δ dy = x δ α + δ y α+δ =. x δ yp X > y dy t > s > Itx Isx = tx ε > T > u T sx t yp X > y dy = x up X > ux du. 5.6 s P X > ux P X > u εx α t Itx Isx εx α up X > u du = εx α {It Is}, t > s > T. lim t It = lim if t s yp X > y dy = E[ X ]/ = It t Itx It Itx εx α, ε > lim if t It x α. 5.7 5.6 s = Ix Itx Ix = + x P X > x Ix t x 5.7 t > t α lim if x Itx Ix = + t lim if x x > x x x P X > x Ix P X > ux u P X > x du + x P X > x Ix t. x P X > x x P X > x, lim if t α Ix x Ix t. t α t 5.6 : lim P X > a = P X > a ε > ii a P X + ε α > +εa +ε P X > a = lim P X > a +ε lim if / ii x > i x > P X > xa = P X > xa P X > a P X > a x α, 5.8 P X > xa = P X > xa P X > xa P X > xa θx α, 5.9 x I ε = {k N; k, X k > εa } µ ε = E[X {εa < X a }], µ ε = E[X { X εa }], S ε = k I ε X k, T ε = S b S ε µ ε = {X k { Xk εa } µ ε}, k= 47

T ε X ε k = X k { Xk εa } E[T ε/a ] = V T ε/a = V X ε /a E[X ε /a ] = ε xp X > xa dx = a εa yp X > y dy εa 5.7 5.8 xp X ε /a > x dx E[T ε/a ] a Cεa P X > εa Cε α. 5.3 S ε #I ε = k S ε/a F ε x = P X /a x X /a > ε k #I ε B, p ε, p ε = P X > εa, F ε ψε t [ ] E[e its ε/a ] = E E[e its ε/a #I] ε = ψt ε k p ε k k p ε k = { + ψt ε p ε }. k= P X > xa Fx ε = P X > εa P X xa P X > εa ψ ε t ψ ε t := ε α p ε ε α ε ε e itx θα dx + εα xα+ θx/ε α, x > ε, θ x /ε α, x < ε, ε αx α dx = ε α E[e its ε/a ] e ε α ψ ε t = exp e itx θα dx x α+ + 5.9 µ ε/a = E[X /a {εa< X a }] E[exp{itS ε µ ε/a }] ε ε e itx θα ε xθα dx x α+ + dx x α+ e itx θα dx x α+ ε x θα dx x α+ { exp e itx θα dx x α+ + + e itx θα dx x α+ + ε e itx itxθα dx ε x α+ e itx itx θα dx } x α+ e itx itx t x / x < α < 5.3 e itx itx dx x α+, ε 5.3 { exp iβt + e itx { β = θ α e itx itx itx + x θα dx x +α + e itx x dx + x x +α + 5.6 x dx } + x x x +α dx x α+ } itx + x θα dx x +α, 5.3 5.33 48

5.8 ε > h ε hε, g ε gε hε h, gε g ε {ε } h ε h g ε g : N < N <... N m h /m h/m /m g /m g/m /m < N ε =, N m < N m+ ε = /m h ε h h /m h/m + h/m h /m + h/m h m h ε h g ε g 5.6 : 5.8 h ε = E[T ε/a ], g ε = E[exp{itS ε µ ε/a }] {ε } 5.3 h ε, g ε 5.3 T ε /a i L, T ε /a i prob. 5.3 t = II 6.3 5.3 Y S ε µ ε /a Y i law S b a = T ε a + S ε µ ε a Y i law X i prob. Y Y i law X + Y Y i law 5.5 X k F a >, b S b a α, Z F x X k α Z 5.6 X i i, ii α Lévy X t+s X t X s Brow Poisso Poisso [S] :,, 999. 49