Влияние аномальных наблюдений на оценку наименьших квадратов параметра авторегрессионного уравнения со случайным коэффициентом.

DOI: 10.18698/1812-3368-2016-2-16-24
??? 519.234.3
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?.?. ????????1 ,
?.?. ?????????2
1
???? ??. ?.?. ???????, ??????, ?????????? ?????????
e-mail: vb-goryainov@bmstu.ru
???????????? ????????????????? ??????????? ??????? ????? ??????????,
??????, ?????????? ?????????
e-mail: el-goryainova@mail.ru
2
??????? ????????? ???????? ?????? ?????????? ????????? ????????? ?????????????????? ????????? ?? ????????? ????????????? ??? ??????? ?????????? ??? ?????????? ???????? ? ???????????. ???????? ?????????????
????????? ??????????? ??????????? ??????? ?????? ?? ??????????????????
?????????, ????????? ???????????? ?????????????, ????????? ????????????
???????? ? ?????????? ?????? ??????????. ???????? ??????????? ???????????????? ?????? ? ??????? ???????????, ???????? ??????? ??? ??????????.
????????, ??? ?????? ????? ?????? ????????? ?? ??????????? ????????????
?????? ???????? ?????????.
???????? ?????: ????????????????? ?????? ?? ????????? ?????????????,
?????????? ???????, ??????????? ???????????????? ? ??????? ???????????,
?????????? ???????, ?????????? ???????.
THE INFLUENCE OF ANOMALOUS OBSERVATIONS ON THE LEAST
SQUARES ESTIMATE OF THE PARAMETER OF THE AUTOREGRESSIVE
EQUATION WITH RANDOM COEFFICIENT
V.B. Goryainov1 ,
E.R. Goryainova2
1
Bauman Moscow State Technical University, Moscow, Russian Federation
e-mail: vb-goryainov@bmstu.ru
2
National Research University Higher School of Economics,
Moscow, Russian Federation
e-mail: el-goryainova@mail.ru
The study tested robustness properties of the least squares estimate of the parameter
of the autoregressive equations with random coefficients in the presence of additive
or replacement outliers in the observations. We investigated the folowing parmeters:
the relation of the functional of the least squares estimate with the autoregression
parameter; the variance of the autoregressive coefficient; the variance of the
innovation process and parameters of the observations process. Moreover, we
calculated the gross-error sensitivity of the least squares estimate and investigated the
conditions for its boundedness. The findings of the research illustrate that the estimate
is always biased except in the degenerate case of zero autoregressive parameter.
Keywords: random coefficient autoregressive model, influence functional, gross-error
sensitivity, additive outliers, replacement outliers.
????????. ? ????????? ???????? ???? ?? ????? ???????? ??????? ?????????? ????? ?????? ?????????? ??????, ????? ?????????
16
ISSN 1812-3368. ??????? ???? ??. ?.?. ???????. ???. ????????????? ??????. 2016. ? 2
??????????? ????????? ???????? ?????? [1]. ???? ?? ????? ??????? ? ?????? ????????????? ?? ?????????? ?????????????? [2].
??? ?????????? ?????????? ???? ?????? ?????? ???????????? ????? ?????????? ?????????, ?????? ??? ????????? ?????????????? ? ????????????? ????????????? ??????????? ???? ?????????????
? ?????????????? ?????????? ?????? [2]. ?????? ?? ???????? ?????? ??????????? ? ????????????. ???????? ? ??????????? ??????
??????????? ????????? ??????? ????????, ?????????? ?????????.
???????? ?????????? ?????, ??? ????? ??????????? ???????? ?????????? ??????????. ? ??????????????? ????? ?????? ??? ????? ????????
? ?????? ??????????????? ??????, ????? ? ??????????? ?????? ?????????? ??????????? ???? ?????????? ???????? ?????? ?? ????????? ?
??????????? ??????????. ?????????????? ????? ?????? ??????????? ?????? ?????????? ??????? ??????, ???????????? ??? ??????
??????????? ?????????? ? ?????? [3] ? ??????????? ? ??????? ???
????? ? ?????? [4].
? ????????? ?????? ??????? ??????????????? ????????? ??????
?????????? ????????? ??? ????????? ??????? ???????? ? ??????????? ??????????? ????, ???????????? ????????????????? ??????????
?? ?????????? ??????????????. ??? ????? ???????? ?? ??????????
??????? ? ??????????? ??? ????????? ? ??????????? ?? ??????????
?????????????????? ????????? ? ?????????? ?????? ??????????? ??????????.
??????? ?????????????. ?????????? ?????????? ??? Xt , ??????????????? ????????? ?????????????
Xt = (?0 + ?t )Xt?1 + ?t .
(1)
? ????????? (1) ????????????????? ??????????? ?0 + ?t ???? ?????
???????????? ????????? ?0 ? ?????????? ???????? ?t . ???? ?t = 0, ??
????????? (1) ?????????? ??????? ????????????????? ??????????.
????? ???????????, ??? ??? ?????? t = 0, � � . . . ? ?????????
??????? ?t ? ?t ???? ??????? ?????????????? ????????
E?t = 0,
E?t = 0
(2)
Dt = ? 2 < ?,
(3)
? ???????? ?????????
D?t = ? 2 < ?,
??????????????? ???????
? 2 + ? 2 < 1.
(4)
????? ???????????, ??? ????????? ???????? {?t , ?t , t=0, � � . . .}
??????????. ??? ?????????? ???? ??????? ?????????? ????????????
ISSN 1812-3368. ??????? ???? ??. ?.?. ???????. ???. ????????????? ??????. 2016. ? 2
17
? ???????????? ??????? ????????? (1), ???????????? ? ???? ??????
X
?????? ? ???????????? ??????? ???? Xt =
?i ?t?i , ??? ?0 = 1 ?
i=0
?i =
i?1
Y
(a + ?i?j ), i = 1, 2, . . . [2, 5]. ????? ??? Xt ????? ????????
j=0
?????? ???????????? ??????? ????????? (1).
?????? ??????????? ??????????. ? ?????? ?????????? ?????
???????? ?????????????? ??? ?????? ???????????? ??????????:
1) ??????????; 2) ??????????; 3) ????????????? [6].
? ?????????? ?????? ?????? ???????? Xt ??????????? ??????? Yt
????
Yt = Xt + ?t ?t ,
(5)
??? ?t ? ????????? ??????? ? ???????????? ??????????,
P{?t = 1} = ?,
P{?t = 0} = 1 ? ?,
0 < ? < 1.
(6)
??????? ???????, ?? ?????????? Xt ????????? ??????? ? ???????????? ? ????????????? ?????? ?t , ??????? ????? ????????????????
??? ????????? ???? ????????? ????? ?????????????? ??????????. ???????????, ??? ?t ? ??????? ? ???????????? ??????????, ????? ???
???? ?t ???????? ????????????? F? ? ???????? ?????????? ??2 = D?t .
? ?????????? ?????? ?????????? Yt ????? ???
Yt = (1 ? ?t )Xt + ?t ?t ,
(7)
??? ???????? ?t ? ? ??????????? (6), ?.?. ? ???????????? ? ?????? ???????? Xt ??????????? ??????? ?t . ????? ???????, ?????????? ??????
????????? ?????? ????? ????????????? ?????????? ? ???????????? ?.
?????? ? ??????? (5)?(7) ??????? ?t ???????? ????????? ? ??????????, ??????????? ???????, ??? ????????? ???????????? ???????????
???? Xt . ???????????, ??? ????????? ???????? Xt , ?t ? ?t ?? ???????
???? ?? ????? ? ???????? ????????????? ? ??????? ??????. ???????,
??? ? ??????? (5)?(7) ?????? ?t ? ????????????? ?????? ??????? t
?????? ?????? ?? ??????????? ??????? ? ???? ?? ?????? ??????? ?
?? ?????? ?? ??? ??????????? ??????????.
????????????? ?????? ???????? ? ????????????? ??????, ???????? ?????? ????????? ?????????????????? ????: ???????? ?????????????, ???????? ????????????? ? ??????????? ???????? ? ????????
????????????? ? ??????????????????? ??????????? ????????. ? ?????? (1) ????????????? ?????? ? ?????? ??? ???????????? (??????????????) ???????? ?t , ????????????? ? ???, ??? ??????? ?t ?????
?? ?????? ??????????, ? ???????????? ?????????? ????????????? (4),
18
ISSN 1812-3368. ??????? ???? ??. ?.?. ???????. ???. ????????????? ??????. 2016. ? 2
?????????? ????? ?????????????? ????? ? ??????????
x2
x2
1
1
e? 2? 2 + ? ?
e? 2?2 , 0 ? ? ? 1, ? > ?. (8)
2??
2??
?????????????????? ????????? ???????, ??????? ?????????????
?????, ????????? ???????? ?? ???????? ??????????? ?????????????????? ?????????????? ?????????? ??????? ? ?????????? ? 2 ????????? (0,01?0,15) ????? ? ?????????????? ?????????? ??????? ?
?????????? ? 2 > ? 2 . ????? ????? ??????????? ????????????? ?????? ??? ??????? ?? ????? ???????????? ??????? (1), ? ??????? Xt ?
??? ??????? ??????? ?? ??? ??????????? (???????). ???????, ???
????????????? ?????? ???????????? ?? ?????? ?? ??????? ??????????, ?? ? ?? ???????????. ????? ???????, ? ????????????? ??????
Yt = Xt , ??? Xt ????????????? (1), ? ??????? ????????? ?????????????
??????????? f (x) ????????? ???????? ?t ????? ??? (8).
?????? ?????????? ?????????. ???? ?? ???????? ????? ???
???????????? ????????? (1) ? ?????????? ??? ????????? ?0 ?? ??????????? Y0 , Y1 , . . . , Yn . ???????? ???????????????? ??????? ?????????? ????????? ?0 ???????? ????? ?????????? ?????????. ??????
?????????? ????????? ??n ????????? ?0 ???????????? ??? ????? ???????? ???????
n
X
(Yt ? ?Yt?1 )2 ,
(9)
Ln (?) =
f (x) = (1 ? ?) ?
t=1
???, ??? ?? ?? ?????, ??? ??????? ?????????
(10)
Sn (?) = 0,
???
Sn (?) =
?L0n (?)
=
????? ????????? (10), ????????
??n =
n
X
t=1
n
P
(Yt ? ?Yt?1 )Yt?1 .
(11)
Yt Yt?1
t=1
n
P
t=1
.
2
Yt?1
???? ? (6) ????????? ? = 0 , ?.?. Yt = Xt ??? ???? t, ?? ??? ?????????? ??????? (2)?(4) ?????? ??n ????????????, ?.?. ? ??????????? n
????????? ?? ??????????? ? ????????? ???????? ????????? ?0 [2].
??????????? ??????????? ??????. ???? ? 6= 0 ? (6), ?? ?????? ??n
?? ??????? ???? ?????????????. ???????????, ??? ? ???? ?????? ?????????? ?????? lim ??n = ?(?). ????????, ??? ?????? ??n ??? ?????,
n??
??? ?????? ???????? ?(?) ? ?0 .
ISSN 1812-3368. ??????? ???? ??. ?.?. ???????. ???. ????????????? ??????. 2016. ? 2
19
????????? ????? IF (?(?), F? ) ??????????? ?(?) ?? ? ? ????:
IF (?(?), F? ) = ?0 (0). ??????????? IF (?(?), F? ) ?????????? ???????????? ??????? ?????? ??n . ?????????? ??????? IF (?(?), F? ) ???????
?? ??????????? ???????? ?(?), ??????? ????????????? ??????????? F?
??????? ?t ? ???????? ??????????? ???????? ???????? ?????? ?????????? ???????????????? ???????? ?(?) ? ?0 ??????????? ????????
?(?) ?????? ??n : ?(?) ? ?0 = IF (?(?), F? )? + o(?), ? ? 0.
????????? ????? K ????????? ????????? ??????? ?????????????
??????????? F? ????????? ???????? ?t . ?????? ?????????? ?????????,
???? ??????????? ???????????????? GE(?(?), K) ? ??????? ???????????, ???????????? ??? GE(?(?), K) = sup IF (?(?), F? ) ????? ??F? ?K
??????.
?????????? ??????????? ??????? ?????? ?????????? ?????????. ??????? ???????? ?????????? ??????? ??? ?????????? ?????? ???????????? ??????????.
??????? 1. ????? ????????? ??????? (2)?(4) ? ??????????
Y0 , Y1 , . . . , Yn ?????????????????? ????????? (1) ??????????? ??????? (5), (6). ????? ?????????? ??????? ?????? ?????????? ?????????
??n ????????? ?0 ????? ???
IF (?(?), F? ) = ?
?0 E?02
.
EX02
(12)
J ????????? ????????? ?????????????????? Xt , ?t ? ?t ????????
????????????? ? ?????????????, (??. ?????? [8]) ????????????? ?
????????????? ????? ????? ??????????????????
n
n
1X
1X 2
?1n =
Yt Yt?1 , ?2n =
Y , n = 1, 2, . . .
(13)
n t=1
n t=1 t?1
???????? ?????? ??????? ?????, ??? ???????????? ??????????????????? [8] ?????????? ??????? lim ?1n = E(Y1 Y0 ), lim ?2n = EY02 .
n??
n??
???????
E(Y1 Y0 )
?1n
=
.
(14)
?(?) = lim ??n = lim
n??
n?? ?2n
EY02
?????????? ? (14) ????????? ??? Yt ?? (5), ???????? (6) ? ????????????? ??????? ? Xt , ?t ? ?t , ???????? E(Y1 Y0 ) = E(X1 + ?1 ?1 )(X0 +
+ ?0 ?0 ) = E(X1 X0 ) + ? 2 (E?0 )2 , EY02 = E(X0 + ?0 ?0 )2 = EX02 + ?E?02 .
?????????????,
E(X1 X0 ) + ? 2 (E?0 )2
?(?) =
.
EX02 + ?E?02
???????, ???
E(X1 X0 ) = E((?0 + ?1 )X0 + ?1 )X0 = ?0 EX02 .
20
(15)
ISSN 1812-3368. ??????? ???? ??. ?.?. ???????. ???. ????????????? ??????. 2016. ? 2
???????
?(?) =
?
?0 EX02 + ? 2 (E?0 )2
EX02 + ?E?02
?0 E?02
d
=?
. I
IF (?(?), F? ) =
?(?)
EX02
d?
?=0
(16)
(17)
?????? ?????? ?????????? ??????? ??? ?????????? ????????.
??????? 2. ????? ????????? ??????? (2)?(4) ? ??????????
Y0 , Y1 , . . . , Yn ?????????????????? ????????? (1) ??????????? ??????? (6), (7). ????? ?????????? ??????? ?????? ?????????? ?????????
??n ????????? ?0 ????? ???
IF (?(?), F? ) = ?
?0 (EX02 + E?02 )
.
EX02
(18)
J ??? ??, ??? ? ??? ?????????????? ??????? 1, ???????, ??? ?(?)
????? ??? (14). ????????? ? (14) ????????? ???????? Y1 ? Y0 ???????????? ?? ??????? (7), ? ?????? (6) ? ????????????? ??????? Xt , ?t
? ?t , ??????????
E(Y1 Y0 ) = E((1 ? ?1 )X1 + ?1 ?1 )(((1 ? ?0 )X0 + ?0 ?0 ) =
= (1 ? ?)2 E(X1 X0 ) + ? 2 (E?0 )2 = (1 ? ?)2 ?0 EX02 + ? 2 (E?0 )2 ,
???????
EY02 = E((1 ? ?0 )X0 + ?0 ?0 )2 = (1 ? ?)EX02 + ?E?02 .
?(?) =
(1 ? ?)2 ?0 EX02 + ? 2 (E?0 )2
,
(1 ? ?)EX02 + ?E?02
(19)
?????? ???????? ??????????? ??????? 2.
???? ??????? ??????????? ????????????? ???????, ?? ??????
?????????? ????????? ???????? ?????????????. ?????????????, ? ????
E(X1 X0 )
?????? Yt = Xt , ??????? ?(?) =
. ?????????????, ?????? ?(?)
EX02
??????????? ????????? (16) ? (19), ? ??????? ? = 0, E?0 = 0, E?02 = 0,
?.?. lim ??n = ?0 .
n??
?????? ??????????? ???????. ?? ?????? (12) ? (19) ???????,
??? ??? ??????? ?????????? ??? ?????????? ???????? ?t ?????? ?????????? ????????? ????? ????????? ?????? ?? ???????????
?????? ?0 = 0, ? ???????? ????? ?????? ?????????????. ??? ???????? ????? ??? ??????, ??? ?????? ????????? D?t ???????, ?????????
D?t = E?t2 + (E?t )2 . ????? ????, ? ??????????? ????????? D?t ?????????? IF (?(?), F? ) ????????????? ??????????, ??? ??? ??? ???? ???????
???????? GE(?(?), K) = ? ?? ????????? ???? ????????? ???????? ?
ISSN 1812-3368. ??????? ???? ??. ?.?. ???????. ???. ????????????? ??????. 2016. ? 2
21
???????? ??????????, ? ?????? ?????????? ????????? ?? ???? ????????? ?? ????? ?????????.
?????? ??????????? IF (?(?), F? ) ?? ?????????? ?0 , ?, ?. ? ???????????? ? (1) ? ????????????? ???????? X0 ?? ??????? ?1 ? ?1
????? EX12 = E((?0 + ?1 )X0 + ?1 )2 = E(?0 + ?1 )2 EX02 + ? 2 = (?20 +
+? 2 )EX02 +? 2 . ??????? Xt ????????????, ??????? EX12 = EX02 , ?????
EX02 = (?20 + ? 2 )EX02 + ? 2 , ??????
EX02
?2
=
.
1 ? ?20 ? ? 2
?????????? ??? ????????? ? (16) ? (19), ??? ?????????? ?????? ???????? ????????
?0 ? 2 + ? 2 (1 ? ?20 ? ? 2 )(E?0 )2
;
? 2 + ?(1 ? ?20 ? ? 2 )E?02
?0
IF (?(?), F? ) = ? 2 (1 ? ?20 ? ? 2 )E?02 ,
(20)
?
? ??? ?????????? ?????? ?
(1 ? ?)2 ?0 ? 2 + ? 2 (1 ? ?20 ? ? 2 )(E?0 )2
?(?) =
;
(1 ? ?)? 2 + ?(1 ? ?20 ? ? 2 )E?02
?0
IF (?(?), F? ) = ? 2 [? 2 + (1 ? ?20 ? ? 2 )E?02 ].
(21)
?
???????? ??????? (20), ?????????? ???????? ??????????? ???????
IF (?(?), F? ) ??????????? ?? ???? ? ??????????? ???????? ? 2 ? ????????????? ? ???????? ? 2 ?? ??????????? ?????????? ???????? 1 ? ?20 ,
??? ??????? ??????????? ???????? ?????????????? ???????? Xt . ????
?? ?????? ?????? ?????????????? ???? ??????????? ???, ??? ??? ??????? ????????? ? 2 , ? 2 ? ????????????? ?????????????? ????????
E?02 ?????? ??????? ???????? ? 2 ? ? 2 ???????? ??????? ????????
????????? ???????? ?????? ??n , ????? ? ??? ????????? ???????? E?02
???????????? ???????.
? ???????????? ??????????
???????? ????????? ?0 ?? ??????????
?? ?????????? ???????? 1 ? ? 2 ?????????? ???????? IF (?(?), F? )
??????? ?????????????
r ?? ???? ?? ????????????? ????????, ????????1 ? ?2
, ? ????? ??????????? ?? ????.
???? ? ????? ?0 =
3
?????????????, ? ?????????? ?????? ???????????? ??????????
?????? ?????????? ????????? ????????? ????????????? ???????????? ???? ??? ??????? ????????? ? 2 , ?0 ? 0 ? ? 2 ? 1 ? ?20 .
? ?????????? ?????? ???????????? ?????????? (??. (21)) ?????? ?????????? ????????? ????????? ????????????? ????????????
?(?) =
22
ISSN 1812-3368. ??????? ???? ??. ?.?. ???????. ???. ????????????? ??????. 2016. ? 2
???? ??? ?0 ? 0, ????????? ???????? |IF (?(?), F? )| ????????? ?????????? ? ??????????? |?0 |. ???????? |IF (?(?), F? )| ? ????????????
? 2 ? ? 2 ??? ? ? ?????????? ?????? ????? ???????????, ?? ??? ?? ??
????.
??????. ?????? ?????????? ????????? ????????? ?????????????????? ????????? ?? ????????? ????????????? ???????? ?????????
???? ? ?????????? ??????????? ???????. ? ?????????? ?????? ???
?????????? ???? ??? ?0 ? 0 ? ???????? ????????? ????????, ? ? ?????????? ?????? ?????? ? ???? ???????: ??? ? 2 ? ?, ??? ? 2 ? 1 ? ?20
? ??? ?0 ? 0. ?? ??????????? ????????? ??????? ?????? ??????????
????????? ????????? ?? ????????.
??????????
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York: Springer-Verlag, 2003. 551 p.
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New York: Springer, 1982. 154 p.
3. Hampel F.R. The influence curve and its role in robust estimation // J. Amer. Statist.
Assoc. 1974. Vol. 69. No. 346. P. 383?393.
4. Martin R.D., Yohai V.J. Influence functionals for time series. With discussion // Ann.
Statist. 1986. Vol. 14. No. 3. P. 781?855.
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Chichester: Wiley, 2006. 403 p.
7. Wilcox R.R. Introduction to Robust Estimation and Hypothesis Testing. Amsterdam:
Elsevier, 2012. 690 p.
8. White H. Asymptotic theory for econometricians. London: AP, 2001. 273 p.
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[1] Fan J., Yao Q. Nonlinear Time Series: Nonparametric and Parametric Methods. N.Y.,
Springer-Verlag, 2003. 551 p.
[2] Nicholls D.F., Quinn B.G. Random coefficient autoregressive models: an
introduction. N.Y., Springer, 1982. 154 p.
[3] Hampel F.R. The influence curve and its role in robust estimation. J. Amer. Statist.
Assoc., 1974, vol. 69, no. 346, pp. 383?393.
[4] Martin R.D., Yohai V.J. Influence functionals for time series. With discussion. Ann.
Statist., 1986, vol. 14, no. 3, pp. 781?855.
[5] Aue A., Horva?th L., Steinebach J. Estimation in random coefficient autoregressive
models. J. Time Ser. Anal., 2006, vol. 27, no. 1, pp. 61?76.
[6] Maronna R.A., Martin D., Yohai V. Robust Statistics: Theory and Methods.
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[8] White H. Asymptotic theory for econometricians. London: AP, 2001. 273 p.
?????? ????????? ? ???????? 21.09.2015
ISSN 1812-3368. ??????? ???? ??. ?.?. ???????. ???. ????????????? ??????. 2016. ? 2
23
???????? ???????? ????????? ? ?-? ???.-???. ????, ????????? ??????? ??????????????? ?????????????? ???? ??. ?.?. ??????? (?????????? ?????????, 105005,
??????, 2-? ?????????? ??., ?. 5).
Goryainov V.B. ? Dr. Sci. (Phys.-Math.), Professor of Mathematical Modelling
Department, Bauman Moscow State Technical University (2-ya Baumanskaya ul. 5,
Moscow, 105005 Russian Federation).
????????? ????? ??????????? ? ????. ???.-???. ????, ?????? ???????????? ?????????? ?? ?????????? ????????????? ???? ????????????? ?????????????????? ???????????? ??????? ????? ?????????? (??? ???, ?????????? ?????????, 101000,
??????, ??. ?????????, ?. 20).
Goryainova E.R. ? Cand. Sci. (Phys.-Math.), Assoc. Professor of Faculty of Economic
Sciences, Department of Mathematics, National Research University Higher School of
Economics (Myasnitskaya ul. 20, Moscow, 101000 Russian Federation).
??????? ????????? ?? ??? ?????? ????????? ???????:
???????? ?.?., ????????? ?.?. ??????? ?????????? ?????????? ?? ?????? ?????????? ????????? ????????? ?????????????????? ????????? ?? ????????? ????????????? // ??????? ???? ??. ?.?. ???????. ???. ???????????? ?????. 2016. ? 2.
C. 16?24. DOI: 10.18698/1812-3368-2016-2-16-24
Please cite this article in English as:
Goryainov V.B., Goryainova E.R. The influence of anomalous observations on the least
squares estimate of the parameter of the autoregressive equation with random coefficient.
Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman
Moscow State Tech. Univ., Nat. Sci.], 2016, no. 2, pp. 16?24.
DOI: 10.18698/1812-3368-2016-2-16-24
24
ISSN 1812-3368. ??????? ???? ??. ?.?. ???????. ???. ????????????? ??????. 2016. ? 2