The 8 reference contexts in paper E. Goryainova R., V. Goryainov B., В. Горяинов Б., Е. Горяинова Р. (2016) “Устойчивые оценки параметра авторегрессионного уравнения со случайным коэффициентом // Stable Parameter Estimation for Autoregressive Equations with Random Coefficients” / spz:neicon:technomag:y:2014:i:2:p:407-415

  1. Start
    1171
    Prefix
    Î ̨ ̃‚ ̊ÂÒÎÓ‚‡:ÓˆÂÌ͇̇ËÏÂÌ ̧ ̄ËıÍ‚‡‰‡ÚÓ‚;‡ÒËÏÔÚÓÚË ̃ÂÒ͇ˇÓÚÌÓÒËÚÂÎ ̧̇ˇ ̋ÙÙÂÍÚË‚ÌÓÒÚ ̧;‡‚ÚÓ„ÂÒÒËÓÌÌÓÂÛ‡‚ÌÂÌËÂ;ÒÎÛ ̃‡ÈÌ ̊ÂÍÓ ̋ÙÙˈËÂÌÚ ̊;Ó·‡ÒÚ̇ˇÓˆÂÌ͇ ¬‚‰ÂÌË ¬Ó ÏÌÓ„Ëıӷ·ÒÚˇı̇ÛÍËË ÚÂıÌËÍË(ÒÏ.,̇ÔËÏÂ,
    Exact
    [1, 2, 3]
    Suffix
    ) ̇·Î ̨‰ÂÌˡÓÔËÒ ̊‚‡ ̨ÚÒˇ Û‡‚ÌÂÌËÂχ‚ÚÓ„ÂÒÒËË Xt= ΦtXt−1+εt,t= 0,±1,±2, . . . ,(1) ‚ ÍÓÚÓÓÏÍÓ ̋ÙÙˈËÂÌÚΦtˇ‚ΡÂÚÒˇÒÎÛ ̃‡ÈÌ ̊Ï.¬ ̇˷ÓÎÂÂ‡ÒÔÓÒÚ‡ÌÂÌÌÓÏÒÎÛ ̃‡Â Φt=a+ηt,t= 0,±1,±2, . . . ,(2) „ ‰ Âηt,t= 0,±1,±2, .
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  2. Start
    1672
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    ., | ÔÓÒΉӂ‡ÚÂÎ ̧ÌÓÒÚ ̧ÌÂÁ‡‚ËÒËÏ ̊ıÓ‰Ë̇ÍÓ‚Ó‡ÒÔ‰ÂÎÂÌÌ ̊ı ÒÎÛ ̃‡ÈÌ ̊ı‚ÂÎË ̃ËÌÒ ÌÛ΂ ̊ÏËχÚÂχÚË ̃ÂÒÍËÏËÓÊˉ‡ÌˡÏËEηt= 0. ŒÒÌÓ‚ÌÓÈÁ‡‰‡ ̃ÂÈ ‡Ì‡ÎËÁ‡Û‡‚ÌÂÌˡ(1) ˇ‚ΡÂÚÒˇÓˆÂÌË‚‡Ìˇ‚ÚÓ„ÂÒÒËÓÌÌÓ„ÓÔ‡‡ÏÂÚ‡a. “‡‰ËˆËÓÌÌ ̊ÏÏÂÚÓ‰ÓÏÓˆÂÌË‚‡Ìˡˇ‚ΡÂÚÒˇÏÂÚӉ̇ËÏÂÌ ̧ ̄ËıÍ‚‡‰‡ÚÓ‚
    Exact
    [4]
    Suffix
    , ÍÓÚÓ ̊ÈÔË ‡ÁÛÏÌ ̊ı Ó„‡ÌË ̃ÂÌˡı̇ ‚ÂÓˇÚÌÓÒÚÌ ̊ÂÒ‚ÓÈÒÚ‚‡ÔÓˆÂÒÒÓ‚ηt,εtÔÓÁ‚ÓΡÂÚÔÓÎÛ ̃ËÚ ̧ÒÓÒÚÓˇÚÂÎ ̧Ì ̊Â Ë ‡ÒËÏÔÚÓÚË ̃ÂÒÍËÌÓχΠ̧Ì ̊ÂÓˆÂÌÍË. ¬ ̃‡ÒÚÌÓÏÒÎÛ ̃‡Â,Í Ó „ ‰ ‡ÍÓ ̋ÙÙˈËÂÌÚΦtÌÂÒÎÛ ̃‡ÂÌ(ηt= 0), ÒÛ ̆ÂÒÚ‚Û ̨ÚÓˆÂÌÍË,ËÏ ̨ ̆ËÂ‚Ó ÏÌÓ„ËıÒÎÛ ̃‡ˇı·-ÓÎ ̧ ̄Û ̨ ̋ÙÙÂÍÚË‚ÌÓÒÚ ̧, ̃ÂÏ ÓˆÂÌ͇̇ËÏÂÌ ̧ ̄ËıÍ‚‡‰‡ÚÓ‚.
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  3. Start
    2134
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    ̊ı Ó„‡ÌË ̃ÂÌˡı̇ ‚ÂÓˇÚÌÓÒÚÌ ̊ÂÒ‚ÓÈÒÚ‚‡ÔÓˆÂÒÒÓ‚ηt,εtÔÓÁ‚ÓΡÂÚÔÓÎÛ ̃ËÚ ̧ÒÓÒÚÓˇÚÂÎ ̧Ì ̊Â Ë ‡ÒËÏÔÚÓÚË ̃ÂÒÍËÌÓχΠ̧Ì ̊ÂÓˆÂÌÍË. ¬ ̃‡ÒÚÌÓÏÒÎÛ ̃‡Â,Í Ó „ ‰ ‡ÍÓ ̋ÙÙˈËÂÌÚΦtÌÂÒÎÛ ̃‡ÂÌ(ηt= 0), ÒÛ ̆ÂÒÚ‚Û ̨ÚÓˆÂÌÍË,ËÏ ̨ ̆ËÂ‚Ó ÏÌÓ„ËıÒÎÛ ̃‡ˇı·-ÓÎ ̧ ̄Û ̨ ̋ÙÙÂÍÚË‚ÌÓÒÚ ̧, ̃ÂÏ ÓˆÂÌ͇̇ËÏÂÌ ̧ ̄ËıÍ‚‡‰‡ÚÓ‚.Õ‡ÔËÏÂ,Ã-ÓˆÂÌÍˡ‚Ρ ̨ÚÒˇÔ‰ÔÓ ̃ÚËÚÂÎ ̧ÌÂÂÓˆÂÌÍË̇ËÏÂÌ ̧ ̄ËıÍ‚‡‰‡ÚÓ‚,ÂÒÎËεtËÏÂÂÚ ‡ÒÔ‰ÂÎÂÌË“ ̧ ̨ÍË
    Exact
    [5]
    Suffix
    . ¬ ‰‡ÌÌÓÈ‡·ÓÚÂÔÓÒÚÓÂÌ ̊Ã-ÓˆÂÌÍËÔ‡‡ÏÂÚ‡a=EΦt, ‰Ó͇Á‡Ì‡Ëı ÒÓÒÚÓˇÚÂÎ ̧ÌÓÒÚ ̧Ë ‡ÒËÏÔÚÓÚË ̃ÂÒ͇ˇÌÓχΠ̧ÌÓÒÚ ̧, ̃ÚÓ ÔÓÁ‚ÓÎËÎÓ‚ ̊ ̃ËÒÎËÚ ̧‡ÒËÏÔÚÓÚË ̃ÂÒÍÛ ̨ÓÚÌÓÒËÚÂÎ ̧ÌÛ ̨ ̋ÙÙÂÍÚË‚ÌÓÒÚ ̧ ̋ÚËıÓˆÂÌÓÍÔÓ ÓÚÌÓ ̄ÂÌË ̨Í ÓˆÂÌÍÂ̇ËÏÂÌ ̧ ̄ËıÍ‚‡‰‡ÚÓ‚. 1. œÓÒÚ‡Ìӂ͇Á‡‰‡ ̃Ë ƒ‡ÎÂÂ‚Ò ̨‰ÛÔ‰ÔÓ·„‡ÂÚÒˇ, ̃ÚÓEεt= 0‰Îˇ Î ̨·Ó„Ót= 0,±1,±2, .
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  4. Start
    3014
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    Œ·ÓÁ̇ ̃ËÏ ̃ÂÂÁFkσ-‡Î„·ÛÒÓ· ̊ÚËÈ,ÔÓÓʉÂÌÌÛ ̨ÏÌÓÊÂÒÚ‚ÓÏÒÎÛ ̃‡ÈÌ ̊ı‚ÂÎË ̃ËÌ ηt,εt,t≤k. ŒˆÂÌ͇̇ËÏÂÌ ̧ ̄ËıÍ‚‡‰‡ÚÓ‚a∗nÔ‡‡ÏÂÚ‡aÔÓ Ì‡·Î ̨‰ÂÌˡÏX0,X1, . . . ,Xn ÒÎÛ ̃‡ÈÌÓ„ÓÔÓˆÂÒÒ‡Xt, ÓÔËÒ ̊‚‡ÂÏÓ„ÓÛ‡‚ÌÂÌËÂÏ(1), ÓÔ‰ÂΡÂÚÒˇÍ ‡ Í ÚÓ ̃ ͇ÏËÌËÏÛχ ÙÛÌ͈ËË
    Exact
    [6]
    Suffix
    LLS(a) = ∑n t=1 (Xt−E(Xt|Ft−1))2, „ ‰ ÂE(Xt|Ft−1)| ÛÒÎÓ‚ÌÓÂχÚÂχÚË ̃ÂÒÍÓÂÓÊˉ‡ÌËÂEXtÓÚÌÓÒËÚÂÎ ̧ÌÓσ-‡Î„· ̊Ft−1. œÓÒÍÓÎ ̧ÍÛE(Xt|Ft−1) =aXt−1, ÚÓ LLS(a) = ∑n t=1 (Xt−aXt−1)2. ¬ [4] ÔÓ͇Á‡ÌÓ, ̃ÚÓ ÔË‚ ̊ÔÓÎÌÂÌËË(3){(4)ÓˆÂÌ͇̇ËÏÂÌ ̧ ̄ËıÍ‚‡‰‡ÚÓ‚ˇ‚ΡÂÚÒˇ ÒÓÒÚÓˇÚÂÎ ̧ÌÓÈË ‡ÒËÏÔÚÓÚË ̃ÂÒÍËÌÓχΠ̧ÌÓÈ.
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  5. Start
    3184
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    ,Xn ÒÎÛ ̃‡ÈÌÓ„ÓÔÓˆÂÒÒ‡Xt, ÓÔËÒ ̊‚‡ÂÏÓ„ÓÛ‡‚ÌÂÌËÂÏ(1), ÓÔ‰ÂΡÂÚÒˇÍ ‡ Í ÚÓ ̃ ͇ÏËÌËÏÛχ ÙÛÌ͈ËË[6] LLS(a) = ∑n t=1 (Xt−E(Xt|Ft−1))2, „ ‰ ÂE(Xt|Ft−1)| ÛÒÎÓ‚ÌÓÂχÚÂχÚË ̃ÂÒÍÓÂÓÊˉ‡ÌËÂEXtÓÚÌÓÒËÚÂÎ ̧ÌÓσ-‡Î„· ̊Ft−1. œÓÒÍÓÎ ̧ÍÛE(Xt|Ft−1) =aXt−1, ÚÓ LLS(a) = ∑n t=1 (Xt−aXt−1)2. ¬
    Exact
    [4]
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    ÔÓ͇Á‡ÌÓ, ̃ÚÓ ÔË‚ ̊ÔÓÎÌÂÌËË(3){(4)ÓˆÂÌ͇̇ËÏÂÌ ̧ ̄ËıÍ‚‡‰‡ÚÓ‚ˇ‚ΡÂÚÒˇ ÒÓÒÚÓˇÚÂÎ ̧ÌÓÈË ‡ÒËÏÔÚÓÚË ̃ÂÒÍËÌÓχΠ̧ÌÓÈ.—ÓÒÚÓˇÚÂÎ ̧ÌÓÒÚ ̧ÓÁ̇ ̃‡ÂÚ, ̃ÚÓ ÔÓÒΉӂ‡ÚÂÎ ̧ÌÓÒÚ ̧a∗nÔËn→ ∞ÒıÓ‰ËÚÒˇÔÓ ‚ÂÓˇÚÌÓÒÚËÍ ËÒÚËÌÌÓÏÛÁ̇ ̃ÂÌË ̨a0Ô‡‡ÏÂÚ‡a, Ú.
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  6. Start
    4260
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    1),(6) „ ‰ Âρ| ÌÂÍÓÚÓ‡ˇÙÛÌÍˆËˇ.›Ú‡ ÓˆÂÌ͇ˇ‚ΡÂÚÒˇ‡Ì‡ÎÓ„ÓÏÃ-ÓˆÂÌÓÍ‚ ‡‚ÚÓ„ÂÒÒËÓÌÌ ̊ı ÏÓ‰ÂΡıÒ ÌÂÒÎÛ ̃‡ÈÌ ̊ÏËÍÓ ̋ÙÙˈËÂÌÚ‡ÏË.¬ ̃‡ÒÚÌÓÏÒÎÛ ̃‡Â,ÍÓ„‰‡ρ(x) =x2, ÔÓÎÛ ̃‡ÂÚÒˇ ÓˆÂÌ͇̇ËÏÂÌ ̧ ̄ËıÍ‚‡‰‡ÚÓ‚. ƒÎˇÔÓÒÚÓÂÌˡÓ·‡ÒÚÌ ̊ıÓˆÂÌÓÍ‚ ͇ ̃ÂÒÚ‚ÂρÓ· ̊ ̃ÌÓ‚ ̊·Ë‡ÂÚÒˇ ̃ÂÚ̇ˇÙÛÌÍˆËˇ, ÍÓÚÓ‡ˇÎË·ÓÓ„‡ÌË ̃Â̇,ÎË·Ó‡ÒÚÂÚ̇ ·ÂÒÍÓÌ ̃ÌÓÒÚËωÎÂÌÌÂÂ, ̃ÂÏx2. Շ˷ÓÎÂÂ‡ÒÔÓÒÚ‡ÌÂÌÌ ̊Ïˡ‚Ρ ̨ÚÒˇρ-ÙÛÌÍˆËˇ’ ̧ ̨·Â‡
    Exact
    [7]
    Suffix
    ρH(x) =    x2,|x|≤k; 2k|x|−k2,|x|> k, Ë ·Ë‚ÂÒ“ ̧ ̨ÍË ρT(x) =    1− ( 1− ( x k )2)3 ,|x|≤k; 1,|x|> k. ›ÚËÙÛÌ͈ËËÁ‡‚ËÒˇÚÓÚ Ô‡‡ÏÂÚ‡k >0, ËÁÏÂÌÂÌËÂÍÓÚÓÓ„ÓÔÓÁ‚ÓΡÂÚ„ÛÎËÓ‚‡Ú ̧ ÒÚÂÔÂÌ ̧Ó·‡ÒÚÌÓÒÚËÓˆÂÌÓÍ. 2.
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  7. Start
    6351
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    ÛÂÚ, ̃ÚÓ |αn(b)|≤ 1 2 E [∣∣∣ ∣∣∣ρ′′ ( η1X0+ε1− τbX0 √ n ) −ρ′′(η1X0+ε1) ∣∣∣ ∣∣∣X2 0b 2 ] . œÓ ̋ÚÓÏÛ,‡ Ú‡ÍÊ‚ ÒËÎÛÌÂÔÂ ̊‚ÌÓÒÚËË Ó„‡ÌË ̃ÂÌÌÓÒÚËρ′′ÔÓ ÚÂÓÂÏÂÀ·„‡Ó χÊÓËÛÂÏÓÈÒıÓ‰ËÏÓÒÚËE|αn(b)|→0ÔËn→∞, ÓÚÍÛ‰‡‚ ̊ÚÂ͇ÂÚ, ̃ÚÓαn(b) =op(1)ÔËn→∞ (Á‰ÂÒ ̧Ë ‚ ‰‡Î ̧ÌÂÈ ̄ÂÏop(1)ÓÁ̇ ̃‡ÂÚÔÓÒΉӂ‡ÚÂÎ ̧ÌÓÒÚ ̧ÒÎÛ ̃‡ÈÌ ̊ı‚ÂÎË ̃ËÌ,ÒıÓ‰ˇ ̆ËıÒˇ Í ÌÛÎ ̨ÔÓ ‚ÂÓˇÚÌÓÒÚË).»Á ÒÚ‡ˆËÓ̇ÌÓÒÚËË ̋„Ó‰Ë ̃ÌÓÒÚËζ1tÒΉÛÂÚ
    Exact
    [8, p. 181]
    Suffix
    , ̃ÚÓ ÒÛ ̆ÂÒÚ‚ÛÂÚÔ‰ÂÎÔÓ ‚ÂÓˇÚÌÓÒÚË B= lim n→∞ Bn=E [ ρ′′(η1X0+ε1)|X20 ] . œÓ ̋ÚÓÏÛ L(b) =−Anb+ 1 2 Bb2+γn(b), „ ‰ Âγn(b) =op(1)ÔËn→∞. Œ·ÓÁ̇ ̃ËÏ ̃ÂÂÁ ̃bn= An B ÚÓ ̃ ÍÛÏËÌËÏÛχÙÛÌ͈ËË−Anb+ 1 2 Bb2. œÓ͇ÊÂÏ, ̃ÚÓ ÔÓÒΉӂ‡ÚÂÎ ̧ÌÓÒÚ ̧ ̃bn‡ÒËÏÔÚÓÚË ̃ÂÒÍËÌÓχΠ̧̇ˈbn− ̃bn=op(1), ÓÚÍÛ‰‡·Û‰ÂÚÒΉӂ‡Ú ̧ ‡ÒËÏÔÚÓÚË ̃ÂÒ͇ˇÌÓχΠ̧ÌÓÒÚ ̧ÔÓÒΉӂ‡ÚÂÎ ̧ÌÓÒÚˈbn.
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  8. Start
    7327
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    —Ήӂ‡ÚÂÎ ̧ÌÓ,ÔÓÒΉӂ‡ÚÂÎ ̧ÌÓÒÚ ̧ ̃bnˇ‚ΡÂÚÒˇ‡ÒËÏÔÚÓÚË ̃ÂÒÍËÌÓχΠ̧ÌÓÈÒ ÌÛ΂ ̊ÏχÚÂχÚË ̃ÂÒÍËÏÓÊˉ‡ÌËÂÏË ‰ËÒÔÂÒËÂÈ(7). “ÂÔÂ ̧‰Ó͇ÊÂÏ, ̃ÚÓˆbn− ̃bn=op(1)ÔËn→∞. “ ‡ Í Í ‡ ÍL(b)‚ ̊ÔÛÍ·,ÚÓ ‚ ̊ÔÛÍÎÓÈ·Û‰ÂÚ Ë ÔÓÒΉӂ‡ÚÂÎ ̧ÌÓÒÚ ̧ÙÛÌ͈ËÈL(b) +Anb, ÍÓÚÓ‡ˇÔËn→ ∞ÒıÓ‰ËÚÒˇÔÓ ‚ÂÓˇÚÌÓÒÚËÍ ‚ ̊ÔÛÍÎÓÈÙÛÌ͈ËËBb2. œÓ ̋ÚÓÏÛ
    Exact
    [10]
    Suffix
    sup b∈K |γn(b)|=op(1)‰Îˇ Î ̨·Ó„ÓÍÓÏÔ‡ÍÚ‡K⊂RÔË n→∞. œÓ͇ÊÂÏ, ̃ÚÓ ‰Îˇ Î ̨·Ó„Óδ >0 lim n→∞ P { |ˆbn− ̃bn|> δ } = 0,(8) ̃ÚÓ ‡‚ÌÓÒËÎ ̧ÌÓˆbn− ̃bn=op(1). Œ·ÓÁ̇ ̃ËÏ ̃ÂÂÁUnÓÚÂÁÓÍ[ ̃bn, ̃bn+δ]. “ ‡ Í Í ‡ Í ÔÓÒΉӂ‡ÚÂÎ ̧ÌÓÒÚ ̧ ̃bnÒı‰Ó‰ËÚÒˇÔÓ ‡ÒÔ‰ÂÎÂÌË ̨,ÚÓ ÒÛ ̆ÂÒÚ‚ÛÂÚÚ‡ÍÓÈÍÓÏÔ‡ÍÚK⊂R, ̃ÚÓ P{Un⊂K}ÒÍÓÎ ̧Û„Ó‰ÌÓ·ÎËÁÍ‡Í Â‰ËÌˈÂ.œÓ ̋ÚÓÏÛÔËn→∞ ∆n= sup b∈Un |γn(b)|=op(1).
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