, ARMAP, Q) CARMAP, Q) DARMAp, q) CARMAP, Q) CARMAP, Q) DARMAp, q) DARMAp, q), CARMAP, Q) CARMAP, Q) Chan and Tong 1987), He and Wang 1989), Br
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1 41,, ARMA Can We Have Correspondence between Discrete-Time ARM A Process and Continuous-Time ARM A Process? Mituaki Huzii ARMA Huzii 006) Huzii 006) ARMA3, ) We assume that a time series is a continuous-time weakly stationary process. We observe the process at equal sampling intervals and construct a discrete-time weakly stationary process. Does the discrete-time weakly stationary process satisfy a DARMAdiscrete-time autoregressive moving average) model, if the continuous-time weakly stationary process satisfies a CARMAcontinuous-time autoregressive moving average) model? And, conversely, we assume we have a discrete-time weakly stationary process which satisfies a DARMA model. Then can we have a continuous-time weakly stationary process, which satisfies a CARMA model and whose sampled process satisfies the DARMA model? In this paper, we review the papers, which have been published by now and treat of these two problems mainly the latter problem), and show more rigorous formulations of the problems and more general answers to these two problems, by using the resuls shown in Huzii 006) and adding new results. : DARMA, CARMA, 1. ARMAp, q) DARMAp, q) huzii@indsys.chuou.ac.jp).
2 , ARMAP, Q) CARMAP, Q) DARMAp, q) CARMAP, Q) CARMAP, Q) DARMAp, q) DARMAp, q), CARMAP, Q) CARMAP, Q) Chan and Tong 1987), He and Wang 1989), Brockwell 1995), Brockwell and Brockwell 1999) n {X n } E X n <, EX n = m m = 0, EX n+j Xn = γ j j X n = 1/ expπnµi)dzµ), γ j = 1/ 1/ 1/ expπjµi)f d µ)dµ 1.1) Brockwell and Davis 1991)) EdZµ)d Zµ ) = 0µ µ ), E dzµ) = f d µ)dµ f d µ) f d µ) f d µ) {X n } µ DARMAp, q) p φ k X n k = k=0 q θ l ɛ n l 1.) l=0 {X n } {ɛ n } 0 1 n {φ k } {θ l } t φ p 0, θ q 0 1.1) 1/ 1/ expπnµi) p φ k exp πkµi) ) dzµ) = k=0 1/ expπnµi) q θ l exp πlµi) ) dz ɛ µ) 1/ l=0 Z ɛ µ) {ɛ n } 1.1) Zµ) E dz ɛ µ) = 1dµ = fd ɛµ)dµ {X n}
3 ARMA 43 f d µ) q f d µ) = l=0 θ l exp πlµi) p k=0 φ k exp πkµi) 1.3) Φz) = p k=0 φ kz k = 0 Θz) = q l=0 θ lz l = 0 DAR DMA Φz) = 0 {1/β τ ; 1 τ p} z > 1 X n ɛ n {ɛ n j ; j 0} {X n j ; j 0} {X n } γ j {β τ } p j γ j = g τ β τ τ=1 1.4) {g τ } t {X t } E X t <, EX t = m m = 0, EX t+h Xt = R h h X t = expπtλi)dzλ), R h = expπhλi)f c λ)dλ 1.5) Brockwell and Davis 1991)) EdZλ)d Zλ ) = 0λ λ ), E dzλ) = f c λ)dλ {X t } CARMAP, Q) k=0 a k d k X t dt k = Q l=0 d l ) dbt b l dt l dt 1.6) Brockwell 1995)) a 0 = 1 {a k }, {b l } {B t } EB t B t1 )B t4 B t3 ) = 0t 1 < t t 3 < t 4 ), EB t B u ) = t u a P 0 b Q 0 1.6) Brockwell 1995) t 0 X t = b U t, du t = AU t dt + edb t.
4 b = b 0, b 1,, b P 1 ), b Q+ = b Q+3 = = b P 1 = 0) e = 1 0, 0,, 0, ), P ) a P ) U = U t, U 1) P 1) t,, U t, A =.... P P ) a P a 1 a P a a P a P 1 a P {U t } {B t } U 1) t dt = du t, U ) t dt = ddu t ), Aω) = a k ω k, Bω) = k=0 Q b l ω l l=0 Aω) = 0 Bω) = 0 CAR CMA X t B t {B t u ; u 0} {X t u ; u 0} R h f c λ) R h = τ=1 ) G τ exp α τ h. 1.7) {α τ } CAR {G τ } {X t } α j j α j = ᾱ j, G j = Ḡj Q f c λ) = l=0 b lπλi) l P k=0 a kπλi) k 1.8) Priestley 1981, Vol. 1), Yaglom 1987) DARMAp, q) CARMAP, Q) CARMAP, Q) p, q
5 ARMA 45 DARMAp, q) Priestley 1981, Vol. 1) DARMAp, q) P, Q CARMAP, Q) DARMAp, q) CARMAP, Q) Huzii 006) Chan and Tong 1987) DARMA1, 0) 1.) p = 1, q = 0 f d µ) = f c µ + k) 1.9) k= f c λ) 0 < φ 1 < 1 Priestley 1981, Vol. 1) CARMA1, 0) DARMA1, 0) 1 < φ 1 < 0 1.9) f c λ) CARMA1, 0) CARMA1, 0) CARMA, 1) DARMAp, q) p > 1 He and Wang 1989) DARMAp, q) CARMAP, Q), P = p + p p p k=0 φ kz k = 0 Brockwell 1995) Brockwell and Brockwell 1999) He and Wang 1989) Brockwell and Brockwell 1999) f d µ) q l=0 θ l exp πlµi) µ 0 CARMAP, Q) Brockwell 1995) DARMA, 0) DAR DARMA, 0) CARMA, 1) CARMA4, ) He and Wang 1989) DARMA, 0) CARMA, 1) CARMA4, ) DARMA CARMA
6 Phillips 1959) Robinson 1978) CARMAP, Q) DARMAp, q) Jones 1981) CARMAP, Q). 1 n n = 0, ±1, ±, ) = 1 n 1 {X n } X n ) γ j DARMA, CARMA,.1 {X n } f d µ) {φ k ; 0 k p} {θ l ; 0 l q} φ p 0, θ q 0 1.3) {X n } DARMAp, q) p > q f d µ) 1/ 1/ log f d µ)dµ > X n {X j ; j n 1} Φz) = 0 ξ Θz) = 0 ξ ξ > 1, ξ > 1 Priestley 1981, Vol.
7 ARMA 47. {X t } f c λ) {a k ; 0 k P } {b l ; 0 l Q} a P 0, b Q 0 1.8) {X t } CARMAP, Q) P > Q f c λ) log f c λ) 1 + λ dλ > ) X t {X u ; u < t} Aω) = 0, Bω) = 0 {α k }, {α k } α k, α k Priestley 1981, Vol. CARMAP, Q) 1.6 X t P B t DARMAp, q) CARMAP, Q) CARMAP, Q) DARMAp, q) {Y n } {X t }.3 K K n 1, n,, n K Y n1, Y n,, Y nk ) X n1, X n,, X nk ) {Y n } {X t } embeddable) 1 {X t } P QQ < P ) CARMAP, Q) {X n } p qq < p) DARMAp, q) {Y n } p qq < p) DARMAp, q) P QQ < P ) CARMAP, Q) {X t }
8 GDARMAp, q) GCARMAP, Q) 3.1 Priestley 1981, Vol. 1 CARMAP, Q) DARMAp, q) 3.1 CARMAP, Q) {X t } {X n } DARMAP, P 1) 3.1 Aω) = 0 {α k } 1.7) {X n } f d µ) γ j j j 0) 1 z expα 1 ))1 z expα )) 1 z expα P )) = φ k z k {φ 0 = 1), φ k ; 1 k P } γ j = P l=1 G l expα l j) φ k γ j k = 0 j P ) k=0 j P 1/ 1/ k=0 P expπjµi) φ k exp πkµi)) fd µ)dµ = 0 k=0 j P 1/ 1/ expπjµi) φ k exp πkµi) fd µ)dµ = 0 {θ l ; 0 l P 1} k=0 φ k exp πkµi) P 1 fd µ) = θ k exp πkµi), a.e., k=0 {X n } DARMAP, P 1) k=0
9 ARMA {X t } {X n } K K {t 1, t,, t K } X t1, X t,, X tk ) K CARMAP, Q) DARMAp, q) GCARMAP, Q) GDARMAp, q) {Y n } GDARMAp, q) P Q {Y n } GCARMAP, Q) {X t } 0 h R h λ f c λ) 3. {Y n } GDARMAp, q) {X t } GCARMAP, Q) {Y n } {X t } a) b) a) γ j = R j j b) µ, 1/ < µ 1/, f d µ) = f c µ + k). 3.1) k= Priestley 1981, Vol. 1 {Y n } GDARMAp, q) GCARMAP, Q) Huzii 006) f c λ) f c λ) f c λ) = = Bπλi) Aπλi) 1 C l πλi α l. 3.) l=1 {α l } CAR {C l } {α l ; 1 l P } f c λ) {C l } {α l }
10 GDARMAp, q) GCARMAP, Q) GDARMAp, q) {φ k } Φ1/z) = p k=0 φ k1/z) k = 0 {β k }, β k = 1/ξ k, GCARMAP, Q) {a k } Aω) = P k=0 a kω k = 0 {α k } [β l, β l ] = expαl ), expα l ) ) = β l βl ) m 1 =, 1 β l βl m=0 0 1 expα l u + ᾱ l u)du = α l + ᾱ l [β l, β l ] expα l ), expα l )) {X t } 0 f d µ) 3.1) f c λ) f d µ) {β l ; 1 l p} f d µ) = p Ψ l 1 β l exp πµi) l=1 = p exp πjµi) Ψ l β j l j=0 + 1 j= l=1 exp πjµi) p Ψ l [β l, β l ] l =1 p l =1 Ψ l β j l p Ψ l [β l, β l ]. 3.3) 3.) f c λ) 3.1) l=1 f d µ) = = = = + k= k= f c µ + k) 1 C l πµ + k)i α l l=1 C l exp ) ) πµ + k)i α l u du k= j=0 l=1 0 exp πjµi) C l expα l j) 1 j= l=1 l =1 l =1 C l expαl ), expα l ) ) exp πjµi) C l expᾱ l j ) C l expαl ), expα l ) ) 3.4) l=1
![1) f c λ) f d µ) {β l ; 1 l p} f d µ) = p Ψ l 1 β l exp πµi) l=1 = p exp πjµi) Ψ l β j l j=0 + 1 j= l=1 exp πjµi) p Ψ l [β l, β l ] l =1 p l =1 Ψ l β j l p Ψ l [β l, β l ]. 3.3) 3.](/docs-images/46/20539429/images/page_10.jpg ") f c λ) 3.")
11 ARMA 431 Huzii 006) 3..1 Ψ l 0 1 l p) 3.. {β l } β l Φz) = 0 Θz) = 0 3.3) 3.4) 3.3 Huzii 006)) 3.4) a), b), c) a) P p b) P = p expα l ) = β l 1 l p) c) P > p, 1 l p β l = expα l ) p + 1 l P l 1 l p l I l α l = α l + πi l i 3.4 Huzii 006)) {Y n } {X t } GDARMAp, q) GCARMA P, Q) {C l, 1 l P } {Y n } {X t } a) P = p 1 l p l p Ψ Ψ p l l =1 l [β l, β l ] = C C l l =1 l expαl ), expα l ) ) b) P > p 1 l p l p Ψ Ψ l l =1 l [β l, β l ] = j Ll) C P C j l =1 l expαj ), expα l ) ) Ll) = {j : expα j ) = β l } DARMAp, q) β l1 l 1 β l1 = β l1, Ψ l1 = Ψ l1 CARMAP, Q) α l l α l = ᾱ l, C l = C l 3.4 a), b) [β l, β l ] expα l ), expα l ) ) GDARMAp, q) GCARMAP, Q) p = 1 p = p 3
12 p = GDARMA1, 0) GCARMAP, Q) GDARMA1, 0) f d µ) = θ φ 1 exp πiµ) = θ φ 1 + φ 1 cosπµ), {Y n } Y n + φ 1 Y n 1 = θ 0 ɛ n {Y n } Φz) = 1 + φ 1 z Φz) = 0 1/β 1 β 1 = φ 1 Ψ 1 = θ 0 θ 0 > 0 GCARMA1, 0) β 1 > 0 φ 1 < b) 3.4 a) α 1 = logβ 1 ), Ψ 1 [β 1, β 1 ] = C 1 expα 1 ), expα 1 )) [β 1, β 1 ] C 1 = ± expα 1 ), expα 1 )) Ψ α 1 ) 1 = ± 1 β1 )θ 0 GCARMA1, 0) β 1 < 0 β 1 = expα 1 ) α c) α 1 = log β 1 + πi, α = log β 1 πi 3.4 b) C 1 = C = r 1 expπνi) r 1 r 1 = Ψ 1 [β 1, β 1 ] expα1 ), expα 1 ))G, 1 1 α1 = log β 1 ) ) G 1 1 = 1 + cosπθ 1 1) cos πθ ν) expπθ 1 1i) = α 1 + πi α 1 ) + π) ν ν r 1 GCARMA, 1) Chan and Tong 1987) β 1 < 0 GCARMA, 1) 3.3. GDARMA, 1) GCARMAP, Q) GCARMA, 1) P
![4 a) α 1 = logβ 1 ), Ψ 1 [β 1, β 1 ] = C 1 expα 1 ), expα 1 )) [β 1, β 1 ] C 1 = ± expα 1 ), expα 1 )) Ψ α 1 ) 1 = ± 1 β1 )θ 0 GCARMA1, 0) β 1 < 0 β 1 = expα 1 ) α 1 3.](/docs-images/46/20539429/images/page_12.jpg "3 c) α 1 = log β 1 + πi, α = log β 1 πi 3.")
13 1) β 1, β 0 ARMA a) β = β 1, Ψ = Ψ 1, α = ᾱ 1, β 1 = expα 1 ), C = C 1 l = 1, l = GDARMA, 1) a) F 1, F Ψ 1 = Ψ 1 expπη 1 i), α 1 = α1 + α1 i, C 1 = r 1 expπν 1 i), expπδ 1 1i) = 1 expα 1) cosα1 ) + expα1) sinα1 )i 1 expα 1 ) cosα1 ) 1 ) expπθ 1 1i) = α1 + α1 i α 1 ) + α1 ) F 1 = Ψ 1 [β 1, β 1 ]F1 + F1 i), F = F 1 α 1, α 1, F 1, F 1 r 1 0, 1/ < η 1, ν 1 1/ 3.4 a) α1, α1, η 1 U num U den cotπθ 1 1), 3.5) ) ) U num = 1 expα1) sin πη 1 + δ 1 1), U den = 1 expα1 ) cosα 1 ) + exp4α 1 ) ) ) + 1 expα1) cos πη 1 + δ 1 1), 3.4 a) C 1, C C 1 = r 1 expπν 1 i) C = C 1 r 1 ν 1 Ψ 1 [β 1, β 1 ]F1 = r1 expα 1 ), expα1) ) 1 + cos πν 1 + θ ) ) 1 1) cosπθ 1 1), Ψ 1 [β 1, β 1 ]F1 = r1 expα 1 ), expα1) ) sin πν 1 + θ ) ) 1 1) cosπθ 1 1) Ψ 1 = expπ 0.15i) Ψ 1 = 1, η 1 = 0.15), β 1 = exp + 0.5i), α 1 =, α 1 = 0.5) 3.5) ) GDARMA, 1) Y n 0.4Y n Y n = 1.41ɛ n 0.6ɛ n 1 f c λ) f c λ) = r1 cosπν 1 ) ) πλi) + 4 cosπν 1 ) sinπν 1 ) πλi) + 4 πλi)
![1 expα 1 ) cosα1 ) 1 ) expπθ 1 1i) = α1 + α1 i α 1 ) + α1 ) F 1 = Ψ 1 [β 1, β 1 ]F1 + F1 i), F = F 1 α 1, α 1, F 1, F 1 r 1 0, 1/ < η 1, ν 1 1/ 3.4 a) α1, α1, η 1 U num U den cotπθ 1 1), 3.](/docs-images/46/20539429/images/page_13.jpg "5) ) ) U num = 1 expα1) sin πη 1 + δ 1 1), U den = 1 expα1 ) cosα 1 ) + exp4α 1 ) ) ) + 1 expα1) cos πη 1 + δ 1 1), 3.")
14 r 1, ν 1 ) Sol 1 =.04, 0.11) Sol = 11.68, 0.3). f c λ) Sol 1 f c λ) = 3.1 πλi) πλi) + 4 πλi) CMA Sol f c λ) = 3.1 πλi) 5.16 πλi) + 4 πλi) CMA Brockwell 1995) He and Wang 1989) {Y n } GDARMA, 0) β 1, β 0 3.5) Brockwell 1995) 3.6) ) β 1, β β 1, β 0 β 1 = β β 1, β β 1, β 3.3 b) α 1 = log β 1, α = log β 3.4 a) GDARMA, 1) 3.4 a) F 1 = Ψ 1 F = Ψ Ψ l [β 1, β l ], 3.6) l =1 Ψ l [β, β l ] 3.7) l =1 C = expα 1), expα 1 )) expα 1 ), expα )) C 1 + C 1 = expα ), expα )) expα ), expα 1 )) C + F 1 1, expα 1 ), expα )) C 1 3.8) F 1 expα ), expα 1 )) C 3.9) C 1, C F 1 + F 0 F 1 < 0, F < 0 F 1 > 0, F > 0 C 1, C ) C 1, C 0 ± C 1, C )
![3 b) α 1 = log β 1, α = log β 3.4 a) GDARMA, 1) 3.4 a) F 1 = Ψ 1 F = Ψ Ψ l [β 1, β l ], 3.6) l =1 Ψ l [β, β l ] 3.](/docs-images/46/20539429/images/page_14.jpg "7) l =1 C = expα 1), expα 1 )) expα 1 ), expα )) C 1 + C 1 = expα ), expα )) expα ), expα 1 )) C + F 1 1, expα 1 ), expα )) C 1 3.8) F 1 expα ), expα 1 )) C 3.")
15 ARMA a) C 1, C GCARMA, 1) F 1, F F 1, F, β 1, β C 1, C ) GCARMA, 1) Huzii 006)) 3.4 b) GCARMAP, Q) P GCARMA3, ) Ψ 1 =, Ψ = 1, β 1 = exp ) = α 1 = ), β = exp.1) = α =.1) F 1 =.04096, F = C 1, C ) 4 GCARMA, 1) GCARMA, 1) Ψ 1, Ψ, β 1 β β = exp 5) = C 1, C ) GCARMA, 1) GCARMA3, ) GCARMA4, 3) Huzii 006)) 3) β 1, β Huzii 006) GDARMA3, ) GCARMAP, Q) GDARMAp, q) p 3 GCARMAP, Q) GCARMAP, Q) β 1, β, β 3 β 1 > β > β 3 C 1, C, C a) GCARMA3, ) GDARMA3, ) 3.4 a) 3 F l = Ψ l Ψ l [β l, β l ], 1 l 3 l =1 expα k ), expα l ) ) = α k,l, 1 k, l 3, α k, 1 k 3, α k,l = α l,k
16 a) C 3 = C 3 = C 3 = 1 F 1 α 1,1 C 1 α 1, C, 3.10) α 1,3 C 1 α 1,3 α 1,3 1 F α,1 C 1 α, C, 3.11) α,3 C α,3 α,3 1 C 3,1 ± C 3, ), 3.1) α 3,3 C 3,1 = ) α 3,1 C 1 + α 3, C C 3, = α 3,1 C 1 + α 3, C ) + 4α3,3 F 3 C 1, C, C ) 3.11) C = 1 B 11 C 1 + B ) 1 + B 11 C 1 + B 1 ) A 11 C 1 C + D ) C = 1 B 11 C 1 + B ) 1 B 11 C 1 + B 1 ) A 11 C 1 C + D ) α, A 11 = α ) 1,, B 11 = α,1 α 1,1, α,3 α 1,3 α,3 α 1,3 B 1 = F 1 α 1,3, D 11 = F 1 α 1,3 B 11 + F α,3 A 11, 3.10) 3.1) C = 1 B 1 C 1 + B ) + D 1 C 1 + D ) A 1 C 1 C + D ) C = 1 B 1 C 1 + B ) D 1 C 1 + D ) A 1 C 1 C + D )
17 ARMA 437 A 1 = α 1, α1,3 α 1,α,3, α 1,3 α 3,3 B 1 = α 1,1α 1, α 1,3 B = F 1 α 1,3 α 1,1α,3 α 1,, α 1,3 α 3,3 α 3,3 α,3 α ) 1,, α 3,3 α 1,3 D 1 = α 1,1α,3 α 1,, D = F 1α,3, α 1,3 α 3,3 α 3,3 α 1,3 α 3,3 D 3 = F 1 α 1, α,3 α 1,3 4α3,3 α ) 1,1α,3 4α 1,3 α3,3 ) + F 3 α 1, α 1,α,3. α 3,3 α 1,3 α 3,3 α 1,3 3.13) 3.14) 3.15) 3.16) C 1, C ) C 1, C 0 C 1, C ± 3.1 A 11 < 0, B 11 < 0, A 1 > 0, B 1 > 0, D 1 > 0 B 1 B D > 0, < 0, > 0, F 1 F 1 F 1 B 1 + D 1 > 0, B 1 D 1 > 0 B + D F 1 < 0, B D F 1 < C 1 C ) C < 3.15) C, 3.13) C < 3.16) C, 3.14) C > 3.15) C, 3.14) C > 3.16) C β 1 = 0.8, β = 0.6, β 3 = 0.4, Ψ 1 = Ψ = Ψ 3 = 1.0 f d µ) = exp πµi) exp πµi) exp πµi)
18 β 1 = 0.8, β = 0.6, β 3 = 0.4, Φ 1 = Φ = Φ 3 = 1.0 Y n 1.8Y n Y n 0.19Y n 3 = 3ɛ n 3.6ɛ n ɛ n {Y n } 39.5 C ) 3.15) ) 5.0 C 1.5 C 1, C ) C 1, C, C 3 ) 1.1, 1.6, 1.48) 10.35, 34.86, 8.38) 1.1, 1.6, 1.48) GCARMA3, ) f c λ) = = 1.1 πλi log πλi log πλi log πλi) 4.1 πλi) 0.95 πλi) πλi) πλi) CMA GCARMA3, ) CMA 10.35, 34.86, 8.38) f c λ) = πλi log πλi log πλi log 0.4 = 3.87 πλi) πλi) 0.95 πλi) πλi) πλi) CMA )
19 ARMA ) C 1 C 1 C C 3.14) 3.16) C 1, C ) C 1, C ) β 1 = exp.0), β = exp 5.0), β 3 = exp 10.0), Ψ 1 =.0, Ψ = 1.0, Ψ 3 = ), 3.14), 3.15), 3.16) 8.0 < C < C ) 3.15) 3.16) 3.13) < C C 1 C 1 4 GDARMA3, ) GCARMA3, ) GCARMA3, ) 3.4 b) GCARMAP, Q), P CMA ζ 1, ζ,, ζ Q Bω) = b Q ω ζ 1 )ω ζ ) ω ζ Q ) ζ 1 ζ 1 = ζ 1 + ζ 1 i ζ 1, ζ 1 ζ 1 > 0 Bω) ζ 1 ζ 1 B rev ω) = b Q ω + ζ 1 )ω ζ ) ω ζ Q ) πλi + ζ 1 = ζ 1 ) + πλ ζ 1 ) = πλi ζ 1. f c λ) = Bπλi) Aπλi) = B rev πλi) Aπλi) 1 f rev λ) = C rev.l πλi α l l=1 = f rev λ) ). C rev.l 4. GDARMAp, q) GCARMAP, Q)
20 β 1 = exp.0), β = exp 5.0), β 3 = exp 10.0), Φ 1 =.0, Φ = 1.0, Φ 3 = 0.01, C 1 < 0. 3 β 1 = exp.0), β = exp 5.0), β 3 = exp 10.0), Φ 1 =.0, Φ = 1.0, Φ 3 = 0.01, C 1 > 0. GDARMAp, q) GCARMAP, Q) DAR CAR GDARMAp, q) 1 p 3 p = p = 3 p =
21 ARMA 441 Huzii 006) p = 3 GDARMAp, q) GCARMAP, Q) P, Q GCARMAP, Q) Brockwell and Brockwell 1999)) GCARMA P, Q) GDARMAp, q) Peter J. Brockwell A. A.1 1.6) CARMAP, Q) DARMAp, q) CARMAP, Q) DARMAp, q) R h h h α h R h = Oexpα h )) P = 1, Q = 0 1.6) a 1 dx t dt + a 0 X t = b 0 db t dt A.1) {B t } A.1) db t /dt A.1) δ > 0 A.1) [t, t + δ] t+δ a 1 X t+δ X t ) + a 0 X u du = b 0 B t+δ B t ) t A.)
22 {X t } 1.5) {X t } t+δ X u du t h > 0 A.) X t h a 0, a 1 X t h {B t h τ B t h τ δ ; τ 0} B t+δ B t t+δ a 1 R h+δ R h ) + a 0 R h+u t du = 0 t A.3) A.3) δ δ 0 R h h > 0 1 h = 0 a 1 dr h dh + a 0R h = 0, h > 0) A.4) h > 0 R h = G 11 expα 1 h) A.5) α 1 = a 0 /a 1 a 0 /a 1 > 0 G 11 R 0 h < 0 R h = R h R h h f c λ) = exp πhλi)r h dh = G 11 α 1 ) 1 πλi α 1 ) λ O 1/λ P =, Q = 1 a d X t dt + a 1 dx t dt + a 0 X t = b 1 d dt ) ) dbt dbt + b 0 dt dt = G 11α 1 πλ) + α 1 A.6) A.7) CAR α 1, α 0 α = ᾱ 1 B t A.7) [t, t + δ 1 ] [t, t + δ ] δ 1 > 0, δ > 0) t+δ t t+δ u +δ 1 X u+δ1 du, X u1 du 1 du, u =t u 1 =u t+δ u=t B u+δ1 B u )du P = 1, Q = 0 X t h, h > 0 δ δ 0
23 ARMA 443 δ 1 δ 1 0 R h h h = 0 a d R h dh + a 1 dr h dh + a 0R h = 0, h > 0) R h = G 1 expα 1 h) + Ḡ1 expᾱ 1 h) A.8) A.9) G 1 R 0 R h R 1 h < 0 R h = R h f c λ) f c λ) = = exp πhλi)r h dh G 1 πλi α 1 + f c λ) = C 1 C 1 + πλi α 1 πλi ᾱ 1 Ḡ 1 πλi ᾱ 1 + G 1 πλi α 1 + Ḡ 1 πλi ᾱ 1 = b 1 πλi) + b 0 πλi α 1 )πλi ᾱ 1 ) C 1, b 0, b 1 A.10) 1.6) {X t } {α k ; 1 k P } 1.7) 1.8) Brockwell 1995) Yaglom 1987) R h h P = 1 P = f c λ) λ P = 1 P = O 1/λ P Q)) 1.6) R h 1.5) P λ P f c λ)dλ < A.11) 1953)) P Q P Q {C l ; 1 l P } C l = 0, l=1 l=1 C l ) l=1 C l l 1 =1,l 1 l α l1 = 0,, α l1 α l α lp 1 [P/] = 0 A.1)
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* abstract This paper attempts to show a method to estimate joint distribution for income and age with copula function. Further, we estimate the joint distribution from National Survey of Family Income
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A Feasibility Study of Direct-Mapping-Type Parallel Processing Method to Solve Linear Equations in Load Flow Calculations Hiroaki Inayoshi, Non-member
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