Consistent estimator in multivariate errors-in-variables model in the case of unknown error covariance structure
We consider a linear multivariate errors-in-variables model AX ? B, where the matrices A and B are observed with errors and the matrix parameter X is to be estimated. In the case of lack of information about the error covariance structure, we propose an estimator that converges in probability to X a...
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| Дата: | 2007 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2007
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3366 |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509443743023104 |
|---|---|
| author | Kukush, A. G. Polekha, M. Ya. Кукуш, О. Г. Полеха, М. Я. |
| author_facet | Kukush, A. G. Polekha, M. Ya. Кукуш, О. Г. Полеха, М. Я. |
| author_sort | Kukush, A. G. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:52:34Z |
| description | We consider a linear multivariate errors-in-variables model AX ? B, where the matrices A and B are observed with errors and the matrix parameter X is to be estimated. In the case of lack of information about the error covariance structure, we propose an estimator that converges in probability to X as the number of rows in A tends to infinity. Sufficient conditions for this convergence and for the asymptotic normality of the estimator are found. |
| first_indexed | 2026-03-24T02:41:11Z |
| format | Article |
| fulltext |
UDK 519.21
O. H. Kukuß, M. Q. Polexa (Ky]v. nac. un-t im. T. Íevçenka)
KONZYSTENTNA OCINKA U VEKTORNIJ MODELI
Z POXYBKAMY U ZMINNYX
PRY NEVIDOMIJ KOVARIACIJNIJ STRUKTURI POXYBOK
A linear multivariate errors-in-variables model AX B≈ is considered, where the data matrices A and
B are observed with errors and a matrix parameter X is to be estimated. In the situation of lack of
information about error covariance structure, an estimator is proposed that converges in probability to X
as the number of rows in A tends to infinity. Sufficient conditions for such convergence and for the
asymptotic normality of the estimator are found.
Rassmatryvaetsq vektornaq model\ s pohreßnostqmy v peremenn¥x AX B≈ , hde matryc¥ A, B
nablgdagtsq s pohreßnostqmy y neobxodymo ocenyt\ matryçn¥j parametr X. Pry uslovyqx,
kohda net dostatoçnoj ynformacyy o kovaryacyonnoj strukture pohreßnostej, predloΩena
ocenka, sxodqwaqsq po veroqtnosty k X, kohda kolyçestvo strok matryc¥ A stremytsq k bes-
koneçnosty. Ustanovlen¥ dostatoçn¥e uslovyq takoj sxodymosty, a takΩe dostatoçn¥e uslo-
vyq asymptotyçeskoj normal\nosty ocenky.
1. Vstup. Ostannim çasom intensyvno rozvyva[t\sq teoriq pereoznaçenyx sys-
tem linijnyx rivnqn\ vyhlqdu AX = B, de matryci A i B sposterihagt\sq z
poxybkamy, X — matryçnyj parametr, qkyj potribno ocinyty. Podibni zadaçi
vynykagt\ pry obrobci rezul\tativ ximiçnyx doslidiv, syhnaliv, identyfikaci]
dynamiçnyx system towo.
U vypadku, koly sukupna kovariacijna struktura ßumu vidoma z toçnistg do
staloho mnoΩnyka, v [1, 2] vstanovleno konzystentnist\ ocinky povnyx naj-
menßyx kvadrativ. U robotax [3, 4] rozhlqnuto vypadok, koly kovariacijna
struktura poxybok matryci A vidoma z toçnistg do odnoho mnoΩnyka, a podibna
struktura dlq matryci B — z toçnistg do inßoho mnoΩnyka. Pry c\omu ocinku
pobudovano za empiryçnymy momentamy druhoho porqdku na osnovi ide] klastery-
zaci]. Vperße cg ideg bulo zastosovano u [5] dlq linijno] skalqrno] modeli z
poxybkamy u zminnyx, pryçomu ocinka ©runtuvalas\ na momentax perßoho po-
rqdku. My takoΩ budemo vykorystovuvaty ideg klasterizaci] ta empiryçni mo-
menty perßoho porqdku.
Vvedemo nastupni poznaçennq: A — norma Frobeniusa matryci A , Ip —
odynyçna matrycq rozmiru p, E — symvol matematyçnoho spodivannq, tr — slid
matryci, λmin( )V ta λmax( )V — najmenße ta najbil\ße vlasni znaçennq mat-
ryci V, Op( )1 — poslidovnist\ stoxastyçno obmeΩenyx vypadkovyx velyçyn.
Opyßemo korotko budovu statti. U p.:2 rozhlqnuto model\ spostereΩen\ i
pobudovano ocinku. U p.:3 dovedeno ]] konzystentnist\, koly kil\kist\ rqdkiv
matryci A prqmu[ do neskinçennosti. Strohu konzystentnist\ ta asymptotyçnu
normal\nist\ ocinky vstanovleno u pp.:4, 5, a p.:6 mistyt\ vysnovky.
2. Model\ spostereΩen\ i pobudova ocinky. Rozhlqnemo model\
A X ≈ B , (1)
de matryci A m n∈ ×
R ta B m p∈ ×
R sposterihagt\sq, a X n p∈ ×
R — nevidoma
matrycq parametriv. Symvoliçnyj zapys (1) oznaça[, wo
A = A A+ ˜ , B = B B+ ˜ , A X = B . (2)
Tut A , B — nevypadkovi matryci, Ã ta B̃ — matryci, skladeni z poxybok
spostereΩen\. Naßa meta polqha[ v pobudovi konzystentno] ocinky matryçnoho
parametra X pry m → ∞ , qkwo nema[ dostatn\o] informaci] pro kovariacijnu
strukturu poxybok.
© O. H. KUKUÍ, M. Q. POLEXA, 2007
1026 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
KONZYSTENTNA OCINKA U VEKTORNIJ MODELI Z POXYBKAMY … 1027
Prypustymo, wo zadano t, t ≥ n, nezaleΩnyx kopij modeli (1):
A ( k ) X ≈ B ( k ) , k = 1, t , (3)
de A k m k n( ) ( )∈ ×
R , X n p∈ ×
R , B k m k p( ) ( )∈ ×
R , m ( k ) — deqki natural\ni çysla.
Model\ (3) oznaça[ nastupne: sposterihagt\sq matryci
A k A k A k( ) ( ) ˜ ( )= + , B k B k B k( ) ( ) ˜( )= + , 1 ≤ k ≤ t ,
pryçomu dlq nevidomyx nevypadkovyx matryc\ A k( ) , B k( ) vykonu[t\sq
A k X B k( ) ( )= , 1 ≤ k ≤ t .
Nexaj A ( k ) = [ ]( ), , ( )( )a k a km k
T
1 … , B ( k ) = [ ]( ), , ( )( )b k b km k
T
1 … i tak samo
poznaçatymemo rqdky matryc\ A k( ) , B k( ) , ˜ ( )A k , ˜ ( )B k . Ocinku matryci X
budu[mo, vykorystovugçy empiryçni momenty spostereΩen\ perßoho porqdku.
Nexaj
a kc( ) = 1
1m k
a ki
i
m k
( )
( )
( )
=
∑ ,
b kc( ) = 1
1m k
b ki
i
m k
( )
( )
( )
=
∑ ,
Ac = [ ]( ), , ( )a a tc c
T1 … , Bc = [ ]( ), , ( )bc c
Tb t1 … .
Dlq spostereΩen\ a kc( ), b kc( ) ma[mo
a k Xc
T ( ) ≈ b kc
T ( ) , 1 ≤ k ≤ t . (4)
Dlq oseredneno] modeli (4) ocinku X̂ budu[mo zvyçajnym metodom najmenßyx
kvadrativ, nextugçy naqvnistg poxybok u spostereΩennqx a kc( ) :
X̂ : = ( )A A A Bc
T
c c
T
c
† . (5)
Tut W
†
— obernena matrycq Mura – Penrouza do matryci W [6, c. 79].
3. Konzystentnist\ ocinky. Dali kil\kist\ rqdkiv m ( k ) matryci A ( k ) bu-
de neobmeΩeno zrostaty, tomu nastupni umovy nakladagt\sq na rqdky ãi , b̃i z
dovil\nym i ∈ N :
i) E ˜ ( )a ki = 0, E ˜ ( )b ki = 0 dlq bud\-qkyx i ≥ 1, k = 1, t ;
ii) isnu[ taka stala c, wo dlq dovil\nyx i ≥ 1 ta k = 1, t
E ˜ ( )a ki
2 ≤ c, E ˜ ( )b ki
2
≤ c;
iii) vypadkovi vektory [ ]˜ ( ), ˜ ( ) , ,a k b k i k ti
T
i
T ≥ ≤ ≤{ }1 1 nezaleΩni.
Poznaçymo Ac = [ ]( ), , ( )a a tc c
T1 … , mmin = min ( ), , ( )( )m m t1 … .
Nastupna umova zabezpeçu[ asymptotyçnu identyfikovanist\ oseredneno] mo-
deli spostereΩen\ (4):
iv) λmin min( )A A mc
T
c → ∞ pry mmin → ∞ .
Teorema551. Nexaj vykonugt\sq umovy i) – iv). Todi ocinka (5) [ slabko
konzystentnog, tobto X̂ X− p → 0 pry mmin → ∞ .
Dovedennq. 1°. NevyrodΩenist\ matryci A Ac
T
c . Analohiçno do matryci
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
1028 O. H. KUKUÍ, M. Q. POLEXA
Ac budemo vykorystovuvaty oseredneni matryci Ãc , B̃c . Poznaçymo V =
= A Ac
T
c . Ma[mo
A Ac
T
c = V A A A A A Ac
T
c c
T
c c
T
c+ + +˜ ˜ ˜ ˜ .
Z umovy iv) vyplyva[, wo A Ac
T
c > 0 pry mmin ≥ m0 . Tut i dali budemo vyko-
rystovuvaty nerivnosti dlq symetryçnyx matryc\ v sensi L\ovnera [7, c. 467]:
T > W oznaça[, wo T – W [ dodatno vyznaçenog, a T ≥ W — wo T – W [ ne-
vid’[mno vyznaçenog.
Pry mmin ≥ m0 ma[mo
A Ac
T
c ≥ V I V A A A A V Vn c
T
c c
T
c
1 2 1 2 1 2 1 2/ / / /( )˜ ˜+ +( )− − ; (6)
z umovy ii) otrymu[mo
E Ãc
2
=
O
m
( )
min
1
⇒ Ãc =
O
m
p( )
min
1
.
Dali,
V Ac
T−1 2/ = tr ( )/ /V A A Vc
T
c
− −1 2 1 2 = n ,
V A A Vc
T
c
− −1 2 1 2/ /˜ ≤ V A A Vc
T
c
− −1 2 1 2/ /˜ =
=
O
m
Vp( )
min
/1 1 2− =
O
V m
p( )
( )min min
1
λ
,
i za umovog iv) ce prqmu[ do 0 za jmovirnistg pry mmin → ∞ . Todi
V A A Vc
T
c
− −1 2 1 2/ /˜ = V A A Vc
T
c
− −1 2 1 2/ /˜ p → 0 pry mmin → ∞ .
OtΩe, z (6) vyplyva[, wo matrycq A Ac
T
c [ nevyrodΩenog z imovirnistg, qka
prqmu[ do::1 pry mmin → ∞ . Oskil\ky nas cikavyt\ asymptotyçna povedinka
ocinky (5), moΩemo vvaΩaty, wo A Ac
T
c [ nevyrodΩenog, wo pryvodyt\ do spro-
wenoho vyrazu dlq ocinky:
X̂ : = ( )A A A Bc
T
c c
T
c
−1 . (7)
2°. Peretvorennq ocinky. Z (7) pislq elementarnyx peretvoren\ ma[mo
X̂ X− = V I V R R V V R R R X R Xn
− − − − −+ +( ) + − ′′ −( )1 2 1 2
1 2
1 2 1 1 2
3 4 1 2
/ / / /( ) ,
(8)
de R1 = : ˜ ˜A A A Ac
T
c c
T
c+ = : ′ + ′′R R1 1 , R2 = : ˜ ˜A Ac
T
c , R3 = : A Bc
T
c
˜ , R4 = : ˜ ˜A Bc
T
c.
3°. ZbiΩnist\ zalyßkiv. Qk my baçyly v p.:1°,
V R V− −1 2
1
1 2/ / p → 0 pry mmin → ∞ .
Dali, R2 ≤ Ãc
2
=
O
m
p( )
min
1
, tomu
V R V− −1 2
2
1 2/ / =
O
V m
p( )
( )min min
1
λ
p → 0 pry mmin → ∞ .
OtΩe, pry mmin → ∞
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
KONZYSTENTNA OCINKA U VEKTORNIJ MODELI Z POXYBKAMY … 1029
I V R R Vn + +( )− − −1 2
1 2
1 2 1/ /( ) p → In .
Analohiçno
V V R− −1 2 1 2
3
/ / p → 0, V V R− −1 2 1 2
4
/ / p → 0.
Íukana zbiΩnist\ X̂ X− p → 0 pry mmin → ∞ vyplyva[ teper iz roz-
kladu (8) ta vstanovlenyx zbiΩnostej zalyßkiv.
ZauvaΩennq551. Osnovnog umovog [ umova iv), ale vona [ dosyt\ m’qkog.
Spravdi, rozhlqnemo odnovymirnyj skalqrnyj vypadok n = p = 1: bi = xai ,
bi = b bi i+ ˜ , ai = a ai i+ ˜ , i = 1, m . Ocinka matyme vyhlqd
x̂ =
m b
m a
ii
m
ii
m
−
=
−
=
∑
∑
1
1
1
1
= x
V xU
U
m m
m
+ −
+1
,
de
Vm : = 1 1
1 1
1
m
b
m
ai
i
m
i
i
m
˜
= =
−
∑ ∑
, Um : = 1 1
1 1
1
m
a
m
ai
i
m
i
i
m
˜
= =
−
∑ ∑
.
Pry odnakovyx dyspersiqx ãi ta odnakovyx dyspersiqx b̃i Vm p → 0 ta
Um p → 0 todi i lyße todi, koly m
m
ai
i
m
1
1
2
=
∑
→ ∞ . Ale ce qkraz i [ umova
iv) u c\omu skalqrnomu vypadku pry t = 1.
ZauvaΩennq552. Umovu ii) moΩna zaminyty slabßog:
ii) ′ 1 2 2
1
1
m
a k b ki i
i m
k t
E E˜ ( ) ˜ ( )+( )
≤ ≤
≤ ≤
∑ ≤ const.
Pry c\omu teorema zalyßatymet\sq spravedlyvog.
ZauvaΩennq553. Umovu iii) moΩna lehko poslabyty takym çynom:
iii) ′ dlq koΩnoho k = 1, t poslidovnist\ vypadkovyx vektoriv {[ ]˜ ( ), ˜ ( ) ,a k b ki
T
i
T
i ≥ 1} [ finitno zaleΩnog.
Nahada[mo, wo finitna zaleΩnist\ poslidovnosti vypadkovyx vektoriv { }zi
oznaça[ isnuvannq takoho nomera s, wo pry koΩnomu j ≥ 1 systemy vektoriv
{ }, ,z i ji = 1 ta { },z i j si ≥ + [ nezaleΩnymy. Umovu iii) ′ moΩna vykorysto-
vuvaty v modeli zi strukturnymy zv’qzkamy [2] pry nevidomij kovariacijnij mat-
ryci strukturnyx parametriv.
ZauvaΩennq554. Ocinku (5) moΩna zastosovuvaty na praktyci do modeli (2)
takym çynom. Rozbyttq matryci A na bloky
A = [ ]( ), ( ), , ( )A A A tT T T T1 2 … , t = n, A k m k n( ) ( )∈ ×
R , k = 1, n ,
slid ßukaty tak, wob (z ohlqdu na umovu iv) ) Φ( )( ), , ( )m m n1 … : = λmin( )A Ac
T
c
bulo maksymal\nym. Pry c\omu mmin = min ( ), , ( )( )m m n1 … ne povynno buty
malym; moΩna vymahaty, napryklad, wob mmin ≥ m n/ 2 . Zvyçajno, kil\kist\
blokiv t moΩna zadavaty rivnog n + 1, n + 2 towo. Zaznaçymo, wo pry t = n
ta nevyrodΩenij matryci Ac ocinka (5) sprowu[t\sq: X̂ = A Bc c
−1
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
1030 O. H. KUKUÍ, M. Q. POLEXA
4. Stroha konzystentnist\. Posylymo umovy ii) ta iv). VvaΩatymemo, wo
çysla m ( k ) zminggt\sq uzhodΩeno:
m ( k ) = fk ( m ) , 1 ≤ k ≤ t, de fk ( m ) → ∞ pry m → ∞ , k = 1, t . (9)
Todi budemo vymahaty vykonannq umov:
v) dlq fiksovanoho dijsnoho r > 1 isnu[ stala c taka, wo dlq vsix i ≥ 1
ta k = 1, t
E ˜ ( )a ki
r2 ≤ c, E ˜ ( )b ki
r2
≤ c;
vi) dlq fiksovanoho m0 ≥ 1 ta r z umovy v) pry m ( k ) = fk ( m ) , k = 1, t ,
vykonu[t\sq
m m
r
c
T
c
r
c
T
c
r
A A
A A m=
∞
∑
0
2
λ
λ
max
min min
( )
( )
< ∞ .
Teorema552. Nexaj çysla m ( k ) zminggt\sq zhidno z (9) ta vykonano umovy
i), iii), v) ta vi). Todi ocinka (5) [ stroho konzystentnog, tobto X̂ X− →
→ 0 pry m → ∞ majΩe napevno (m.n.).
Dovedennq. Budemo rozhlqdaty m ≥ m 0 , dlq qkyx V — nevyrodΩena
matrycq. Poçnemo z matryci
A Ac
T
c = V I V A A A A A An c
T
c c
T
c c
T
c+ + +( )−1( )˜ ˜ ˜ ˜ .
Z momentno] nerivnosti Rozentalq [8] ta umovy v) ma[mo
E Ãc
r2
≤ const ⋅ −m r
min ,
E V A Ac
T
c
r−1 2˜ ≤ V A m
r
c
r r− −⋅ ⋅1 2 2
const min ≤ const ⋅ λ
λ
max
min min
( )
( )
r
r r
V
V m2 .
Tomu za umovog vi)
m m
c
T
c
r
V A A
=
∞
−∑
0
1 2
E ˜ < ∞ ,
i za lemog Borelq – Kantelli ta nerivnistg Çebyßova V A Ac
T
c
−1 ˜ → 0 pry
m → ∞ m.n. Tak samo V A Ac
T
c
−1 ˜ → 0 pry m → ∞ m.n. TakoΩ ma[mo
E V A Ac
T
c
r−1 ˜ ˜ ≤ V A
r
c
r−1 2
E ˜ ≤ const ⋅ 1
λmin min( )r rV m
,
i z umovy vi) otrymu[mo
m m
c
T
c
r
V A A
=
∞
−∑
0
1E ˜ ˜ < ∞ .
Tomu za lemog Borelq – Kantelli V A Ac
T
c
−1 ˜ ˜ → 0 pry m → ∞ m.n.
Iz vstanovlenyx zbiΩnostej vyplyva[, wo m.n. pry m → ∞
A Ac
T
c = V I on( ( ))+ 1 ,
tomu z imovirnistg::1 matrycq A Ac
T
c [ nevyrodΩenog, poçynagçy z deqkoho
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
KONZYSTENTNA OCINKA U VEKTORNIJ MODELI Z POXYBKAMY … 1031
vypadkovoho nomera m m1 1= ( )ω , zvidky
ˆ ( )X A A A Bc
T
c c
T
c= −1
.
Poklademo
ˆ ˆ∆ = −X X . Pry m m≥ 1( )ω ma[mo
V A Ac
T
c
−1 ∆̂ = V A B V A A A A Xc
T
c c
T
c c
T
c
− −+ − −1 1˜ ˜ ˜ ˜( ) . (10)
U livij çastyni otrymu[mo V A A Ic
T
c n
− →1 , m → ∞ m.n., a prava çastyna prq-
mu[ do nulq pry m → ∞ m.n. Ce vyplyva[ iz vstanovlenyx vywe zbiΩnostej, a
takoΩ iz zbiΩnostej V A Bc
T
c
− →1 0˜ , V A Bc
T
c
− →1 0˜ ˜ , m → ∞ , m.n., qki do-
vodqt\sq analohiçno. Tomu z (10) bezposeredn\o otrymu[mo ßukane: ∆̂ → 0 ,
m → ∞ , m.n.
5. Asymptotyçna normal\nist\. U danomu punkti takoΩ vvaΩatymemo, wo
m ( k ) zminggt\sq zhidno z (9). Bil\ß toho, nexaj ci nomery zrostagt\
rehulqrno u nastupnomu sensi:
vii) pry koΩnomu k = 1, t isnu[ dodatna i skinçenna hranycq
λk : = lim
( )m k
m
f m→∞
.
TakoΩ vymahatymemo stabilizaci] u seredn\omu rqdkiv matryci Ac :
viii) a k a kc( ) ( )→ ∞ pry m → ∞ , pryçomu hranyçni vektory a∞( )1 , … , a t∞( )
[ linijno nezaleΩnymy.
Nexaj A∞ = a a t T
∞ ∞…[ ]( ), , ( )1 . ZauvaΩymo, wo za umovy viii) matrycq V =
= A Ac
T
c prqmuvatyme do dodatno vyznaçeno] matryci V∞ = A AT
∞ ∞ . Wodo po-
xybok vymahatymemo nezaleΩnosti à , B̃ ta stabilizaci] (u seredn\omu) kovari-
acijno] struktury:
ix) vypadkovi vektory { }˜ , ˜ , , ,a b i k ti i ≥ =1 1 [ nezaleΩnymy, pryçomu isnu-
gt\ hranyci
S ka( ) : = lim
( )
˜ ( ) ˜ ( )
( )
( )
m k
i i
T
i
m k
m k
a k a k
→∞ =
∑1
1
E ,
S kb( ) : = lim
( )
˜ ( ) ˜ ( )
( )
( )
m k
i i
T
i
m k
m k
b k b k
→∞ =
∑1
1
E ,
de S ka( ) , S kb( ) , k = 1, t , — dodatno vyznaçeni matryci.
Teorema53. Nexaj çysla m ( k ) zminggt\sq zhidno z (9) ta vykonano umovy
i), v) – ix). Todi pry m → ∞
m X X( )ˆ − d → V a k S k X S kk k
T
a k
T
b
k
t
∞
−
∞
=
+∑1 1 2 1 2
1
( ) ( ) ( )( )/ /λ γ ε pry m → ∞ ,
d e { }, , ,γ εk k k t= 1 — nezaleΩni vypadkovi vektory, γ k nN I∼ ( , )0 , εk ∼
∼ N I p( , )0 , k = 1, t .
Dovedennq. Z imovirnistg, wo prqmu[ do::1 pry m → ∞ , vykonu[t\sq
A A Xc
T
c
ˆ = A Bc
T
c, tomu
( ( )) ˆI o mn p+ 1 ∆ = V m R X R X R R− − ′′ − + +1
1 2 3 4( ) , (11)
de çleny Ri , ′′R1 taki sami, qk i v (8). Matrycq V V−
∞
−→1 1, m → ∞ ;
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
1032 O. H. KUKUÍ, M. Q. POLEXA
m R2 = m A Ac
T
c
˜ ˜ =
O
m
p( )1
p → 0, m → ∞ ;
analohiçno m R4 p → 0 , m → ∞ . Zalyßylos\ doslidyty zbiΩnist\ çleniv
′′R1 ta R3. Ma[mo
m R′′
1 = a k m
m k m k
a kc
k
t
i
T
i
m k
( )
( ) ( )
˜ ( )
( )
1
1 1= =
∑ ∑ .
Tut
m
m k k( )
→ λ , m → ∞ , i za central\nog hranyçnog teoremog (CHT) u for-
mi Lqpunova pry k = 1, t
1
1m k
a ki
i
m k
( )
˜ ( )
( )
=
∑ d → N S ka( , ( ))0 pry m → ∞ .
Tomu
m R′′
1 d → a k S kk k
T
a
k
t
∞
=
∑ ( ) ( )/λ γ 1 2
1
,
de γ k — vypadkovi vektory z teoremy::3. Nareßti,
m R3 = a k m
m k m k
b kc
k
t
i
T
i
m k
( )
( ) ( )
˜ ( )
( )
1
1 1= =
∑ ∑ ,
i znovu za CHT u formi Lqpunova pry k = 1, t
1
1m k
b ki
i
m k
( )
˜ ( )
( )
=
∑ d → N S kb( , ( ))0 pry m → ∞ .
Tomu
m R3 d → a k S kk k
T
b
k
t
∞
=
∑ ( ) ( )/λ ε 1 2
1
,
de εk — vypadkovi vektory z teoremy::3. Oskil\ky za umovog ix) ′′R1 ta R3
nezaleΩni miΩ sobog, to z (11) ta lemy Sluc\koho otrymu[mo ßukanu zbiΩnist\
poslidovnosti m ∆̂ .
Teoremu dovedeno.
Z ohlqdu na teoremu::3 na bazi ocinky X̂ moΩna pobuduvaty asymptotyçni
dovirçi elipso]dy dlq vektornoho parametra vec ( X ) . Dlq c\oho potribno vmity
konzystentno ocingvaty parametry hranyçnoho rozpodilu. Dlq λk nablyΩen-
nqm [ vidnoßennq
m
m k( )
(u teoremi::3 moΩna v qkosti m uzqty mmin ); zaly-
ßylosq pobuduvaty nablyΩennq dlq velyçyn a k∞( ), S ka( ) ta S kb( ) (nably-
Ωennqm dlq X [ konzystentna ocinka X̂ ). Otrymu[mo
a k∞( ) ≈ 1
1m k
a ki
i
m k
( )
( )
( )
=
∑ ≈ a kc( ).
Tut i dali nablyΩeni rivnosti oznaçagt\, wo riznycq miΩ pravog i livog çasty-
namy [ op( )1 pry m → ∞ . Za metodom momentiv ma[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
KONZYSTENTNA OCINKA U VEKTORNIJ MODELI Z POXYBKAMY … 1033
S kb( ) ≈ 1
1m k
b k b k a k Xi i
T
i
T
i
m k
( )
( ) ( ) ( ) ˆ
( )
−( )
=
∑ ,
S ka( ) ≈
1 1
1 1m k
a k a k X X X
m k
b k a ki i
T
i
m k
T
i i
T
i
m k
( )
( ) ( ) ˆ ˆ ˆ
( )
( ) ( )
( ) ( )
( )
= =
∑ ∑− † .
Ostann[ spivvidnoßennq otrymano pry umovi, wo X — matrycq ranhu n (zokre-
ma, neobxidno, wob vykonuvalas\ nerivnist\ n ≤ p ) .
6. Vysnovky. Rozhlqnuto vektornu model\ z poxybkamy u zminnyx pry vid-
sutnosti informaci] wodo kovariacijno] struktury poxybok. Osnovnym prypu-
wennqm bulo te, wo sposterihagt\sq nezaleΩni kopi] modeli. Na praktyci ce
oznaça[, wo dani spostereΩen\ moΩna rozdilyty na vidokremleni hrupy — klas-
tery.
Prote v praktyçnyx zadaçax naqvni pevni vidomosti pro strukturu poxybok.
Podal\ßi doslidΩennq budut\ sprqmovani na te, wob zmenßyty kil\kist\ klas-
teriv, vykorystovugçy dodatkovu apriornu informacig. Inßym naprqmom dos-
lidΩen\ bude pobudova kryterig zhody dlq modeli (2), vin bude uzahal\nennqm
vidpovidnoho kryterig dlq polinomial\no] modeli z poxybkamy u zminnyx [9].
1. Kukush A., Van Huffel S. Consistency of element-wise weighted total least squares estimator in
multivariate errors-in-variables model AX B= // Metrika. – 2004. – 59, # 1. – P. 75 – 97.
2. Kukush A., Markovsky I., Van Huffel S. Consistency of the structured total least squares estimator
in a multivariate errors-in-variables model // J. Statist. Planning and Inference. – 2005. – 133, # 2. –
P. 315 – 358.
3. Kukush A., Markovsky I., Van Huffel S. Estimation in a linear multivariate measurement error mo-
del with clustering in the regressor // Int. Rept 05-170. ESAT-SISTA (Leuven, Belgium, 2005).
4. Markovsky I., Kukush A., Van Huffel S. On errors-in-variables estimation with unknown noise
variance ratio // 14th IFAC Symp. System Identification (Newcastle, Australia, 2006). –
P. 317 – 323.
5. Wald A. The fitting of straight lines if both variables are subject to error // Ann. Math. Statist. –
1940. – # 11. – P. 284 – 300.
6. Seber DΩ. Lynejn¥j rehressyonn¥j analyz. – M.: Myr, 1980. – 456 s.
7. Marßal A., Olkyn Y. Neravenstva: teoryq y ee pryloΩenyq. – M.: Myr, 1983. – 572 s.
8. Härdle W., Kerkyacharian G., Picard D., Tsybakov A. Wavelets, approximation, and statistical
applications. – New York: Springer, 1998. – 244 p.
9. Cheng C.-L., Kukush A. A goodness-of-fit test in a polinomial errors-in-variables model // Ukr.
mat. Ωurn. – 2004. – 56, # 4. – S.:527 – 543.
OderΩano 16.03.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
|
| id | umjimathkievua-article-3366 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:41:11Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/48/56389c9aa1ea38c45d8c78c41d9d3348.pdf |
| spelling | umjimathkievua-article-33662020-03-18T19:52:34Z Consistent estimator in multivariate errors-in-variables model in the case of unknown error covariance structure Конзистентна оцінка у векторній моделі з похибками у змінних при невідомій коваріаційній структурі похибок Kukush, A. G. Polekha, M. Ya. Кукуш, О. Г. Полеха, М. Я. We consider a linear multivariate errors-in-variables model AX ? B, where the matrices A and B are observed with errors and the matrix parameter X is to be estimated. In the case of lack of information about the error covariance structure, we propose an estimator that converges in probability to X as the number of rows in A tends to infinity. Sufficient conditions for this convergence and for the asymptotic normality of the estimator are found. Рассматривается векторная модель с погрешностями в переменных $AX \approx B$, где матрицы $A$, $B$ наблюдаются с погрешностями и необходимо оценить матричный параметр $X$. При условиях, когда нет достаточной информации о ковариационной структуре погрешностей, предложена оценка, сходящаяся по вероятности к $X$, когда количество строк матрицы $A$ стремится к бесконечности. Установлены достаточные условия такой сходимости, а также достаточные условия асимптотической нормальности оценки. Institute of Mathematics, NAS of Ukraine 2007-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3366 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 8 (2007); 1026–1033 Український математичний журнал; Том 59 № 8 (2007); 1026–1033 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3366/3475 https://umj.imath.kiev.ua/index.php/umj/article/view/3366/3476 Copyright (c) 2007 Kukush A. G.; Polekha M. Ya. |
| spellingShingle | Kukush, A. G. Polekha, M. Ya. Кукуш, О. Г. Полеха, М. Я. Consistent estimator in multivariate errors-in-variables model in the case of unknown error covariance structure |
| title | Consistent estimator in multivariate errors-in-variables model in the case of unknown error covariance structure |
| title_alt | Конзистентна оцінка у векторній моделі з похибками у змінних при невідомій коваріаційній структурі похибок |
| title_full | Consistent estimator in multivariate errors-in-variables model in the case of unknown error covariance structure |
| title_fullStr | Consistent estimator in multivariate errors-in-variables model in the case of unknown error covariance structure |
| title_full_unstemmed | Consistent estimator in multivariate errors-in-variables model in the case of unknown error covariance structure |
| title_short | Consistent estimator in multivariate errors-in-variables model in the case of unknown error covariance structure |
| title_sort | consistent estimator in multivariate errors-in-variables model in the case of unknown error covariance structure |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3366 |
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