P
mavzusidagi Respublika ilmiy-amaliy anjuman materiallari. Namangan 2025-yil.
378
MODELLARDA NOSTANDART BAHOLASH USULLARI
Mirzayev Toxirjon Saloxetdinovich
Namangan davlat pedagogika instituti
toxirjonmirzayev4@gmail.com
Namangan davlat pedagogika institute
Annotatsiya. Ushbu maqolada avtoregressiya modellari uchun eng kichik
kvadratlar baholagichlaridan farq qiluvchi alternativ parametr baholovchilari taklif
etilad
aylana ustida joylashgan hollarda, eng kichik kvadratlar baholagichlari odatda
-qarshi ravishda, taklif
etilgan nostandart baholagichlar aksariyat kritik holatlarda oddiyroq asimptotik
kichik kvadratlar usuli, nostandart baholash usuli, normal taqsimot qonuni.
NONSTANDARD ESTIMATION METHODS IN ONE-DIMENSIONAL
AND SPATIAL FIRST-ORDER AUTOREGRESSION MODELS
Mirzayev Toxirjon Saloxetdinovich
Namangan state pedagogical institute
toxirjonmirzayev4@gmail.com
Namangan state pedagogical institute
Abstract. The article proposes alternative parameter estimators for autoregression
that differ from the least squares estimates. In unstable (critical) cases, where the
characteristic equation's roots lie on the unit circle, least squares estimators generally
exhibit a complex asymptotic distribution. In contrast, the proposed nonstandard
estimators tend to have a simpler asymptotic distribution in most critical cases.
Keywords: One-dimensional autoregression model, limit theorem, Wiener process,
least squares method, nonstandard estimation method, normal distribution law.
toxirjonmirzayev4@gmail.com
P
mavzusidagi Respublika ilmiy-amaliy anjuman materiallari. Namangan 2025-yil.
379
INTRODUCTION
The report presents nonstandard approaches to constructing estimators in various
autoregressive models. The estimation equations are formulated using the recursive
relationships that define the original process, with each case considered separately. A
similar approach can also be applied to derive classical least squares estimators, without
relying on the conventional method of minimizing the corresponding sum of squares with
respect to the original values. The necessity of developing such estimators arises from the
fact that least squares estimators in critical (unstable) cases exhibit complex asymptotic
distributions, which are generally expressed in terms of functionals of the standard Wiener
process. From an applied perspective, critical cases are of particular interest. For example,
in the case of a first-order autoregressive process, i.e., when
,
1
Y
Y
k
k
k
significant
attention has been devoted to tabulating the complex asymptotic distribution when
1
.
In this article, we consider such distributions and study their asymptotic estimators.
PROBLEM FORMULATION
We consider the following autoregressive models.
Definition [10].
An autoregressive scheme
p
of the order
(
( ))
AR p
is a relationship of the
form
0
1
,
p
k
j
k j
k
j
X
X
(1)
where are
,
0,1,...,
j
j
p
constants, and
k
random variables are called autoregressive
noise or simply noise.
In the case of
0
0
0
X
a first-order autoregressive model, it will take the following
form
1
,
1, 2,...
k
k
k
X
X
k
,
(2)
P
mavzusidagi Respublika ilmiy-amaliy anjuman materiallari. Namangan 2025-yil.
380
where
k
are independent identically distributed random variables (i.i.d.r.v.) [19] with
2
1
1
0
0,
1,
E
E
X
an initial state.
The known estimate of the parameter
,
obtained from
n
observations using the least
squares method has the form
2
1
1
1
1
n
n
n
k
k
k
k
k
X
X
X
)
. (3)
In the work [11] this estimate is given as the serial correlation coefficient.
If
i
normally distributed, then estimate (3) coincides with the maximum likelihood
estimate.
The complex structure of the estimate
n
)
makes it difficult to find the limit
distribution even in the case of normally distributed noise.
.
i
It is known [12[, [15] that
n
n
)
has a normal limit distribution when
1, 1 ,
and for
1
the limit distribution
is complex and, moreover, when
1
it becomes dependent on the noise distribution.
.
i
For example, when
1
the statement [14-15] holds. at
n
1
2
2
0
1
1
(1) 1
( )
,
2
n
n
w
w x dx
)
( 4)
where is
( )
w x
a standard Wiener process [20], and the symbol
denotes weak
convergence of the corresponding distributions.
In [16], [25] another, simpler in structure, estimate of the parameter was proposed.
.
Simple summation of (2) over
t
from
k
to
n
leads to the following estimate [25]
,
...
.
...
k
n
n k
k
n
X
X
(5)
It has been shown ([25]) that when
n
2
,
1
1
( )
1
(
)
1
,
2
n k
c
c
P n
c
x
arctg
x
Where
1 2
1/ 2
1 3(1
)
lim
1,
( )
(1
)(1 2 ) 3
,
.
2
1 2
c
n
k
c
c
c
c
c
n
c
If
1
c
, so then [24]
1/ 2
,
2
1
1
(
)
1
.
1
x
n k
P k n
k
x
du
u
Note that in case
1
there are no results. The invariance principle used in the
study of estimate (3) in [15]
1
does not give any results, and as for estimate (5), it is
untenable in this case, which is easy to show.
Spatial autoregressive models have begun to be intensively studied relatively recently
and have not yet entered the monographic and educational literature. We will give some
overview of the results following the work [9].
P
mavzusidagi Respublika ilmiy-amaliy anjuman materiallari. Namangan 2025-yil.
381
The analysis of spatial patterns is of interest in many fields, such as geography,
geology, biology and agriculture. For a discussion of this, see [18]. These authors
considered the case of the so-called one-sided
(
( ))
AR p
model [6], which has the form [9]
1
2
,
0.
,
,
,
,
0,0
0
0
p
p
X
X
k l
i j k i l
j
k l
i
j
In [9] a special case of this model is considered, namely, when
1,
:
1
2
0,1
1,0
p
p
,
0
1,1
, and specific results on the asymptotic behavior of the
estimator
in the unstable case are obtained [6]. There are very few results of this type
for spatial models in the literature [6]. From a general point of view, it is desirable to deal
with models where
,
Xk l
is a linear combination of all neighbors on the lattice. In particular,
it would be interesting to consider a generalization of the Sandor Baran model when
1,
Xk
l
and
,
1
Xk l
have different weights
and
[6]. But even in this model with
complex mathematical problems arise with rather non-standard results [9].
In this paper, we propose an estimate of a simpler structure and using only a portion of
the observations located along the diagonal in a rectangle.
,
.
m n
R
This approach with a
significant reduction in the number of observations is relevant in geology problems and
some other areas of application of spatial autoregressive models. A simpler estimate
structure also allows us to reduce moment restrictions on noise.
SOLUTION OF THE PROBLEM AND RESULTS
First-order ordinary univariate autoregressive model. Near-critical case
1
n
.
In this paper, a first-order autoregressive model is considered. Process (2) is stable
in the case when
1,
it is unstable at
1
and is of the explosive type at
1.
Note that in the case
( )
1
n
of
n
there are no results. In this paper, we study a case
close to critical, when
( )
1
n
for
n
a particular type of estimate (5)
1 .
k
Construction of the estimate and its limit behavior
We will construct the estimation equation [28] by summing the relation (2)
t
from
1
to
n
1
1
1
1
n
n
n
k
k
k
k
k
k
X
X
or in abbreviated form
1
.
n
n
n
Y
Y
E
(6)
Solving equation (6) without taking into account
,
n
E
we obtain the estimate
,1
1
.
n
n
n
Y
Y
Now from (6) we find the deviation
P
mavzusidagi Respublika ilmiy-amaliy anjuman materiallari. Namangan 2025-yil.
382
*
,1
1
.
n
n
n
E
Y
(7)
Now let us note that from (2) we have
1
1
t
t
t
X
X
and therefore
1
1
.
n
n
n
Y
Y
E
(8)
Theorem 1.
If
k
i.i.d.r.v. with
2
1
1
0,
1
E
E
and
0
0,
X
1
.
n
When
1
,
c
n
c
,1
2
1
1
,
2
1
c
c
n
c
x
P n
x
arctg
Where
2
2
3
1
2
3 4
,
2
c
c
c
c
e
e
c
2
1
1 .
c
c
c
e
c
c
2
. If
( ) 1
,
n
c
n
n
,
n
o n
, when
n
the relation holds
1
,1
2
2
n
n
c
,
where is
1
2
,
a normal random vector with zero mean and covariance matrix
1
1
2 .
1
1
2
3.
If
( ) 1
,
n
c
n
n
,
0
n
n
, when
n
the statement holds
2
1
,1
3
2
n
n
,
where is
1
2
,
a normal random vector with zero mean and covariance matrix
3
1
2 2 .
3
1
2 2
For the first-order ordinary autoregressive model,
1
k
k
k
Y
Y
the limiting distribution
of non-standard parameter estimates was found
in the cases
1
c n
and
1
c n
[23].
The constructed estimates have, as a rule, simpler limit distributions in comparison
with traditional least squares estimates. At the same time, the simpler structure of the
proposed estimates also allows us to reduce the moment restrictions on the "noise" that
define the stochastic structure of the models.
P
mavzusidagi Respublika ilmiy-amaliy anjuman materiallari. Namangan 2025-yil.
383
References
1. Peter CB Phillips (2021). Estimation and Inference with Near Unit Roots. Cowles
foundation for research in economics yale university Box 208281 New Haven,
Connecticut 06520-8281.
2. Shi, S. and P. C. B. Phillips (2021). Diagnosing housing fever with an econometric
thermometer.
Journal of Economic Surveys, forthcoming
.
3. Yuhao Liu (2015). Finding moments of AR(k)-model parameter estimators. Brock
Reports in Mathematics and Statistics no. 150504.
4. Jan Vrbik (2015). Moments of AR(k) parameter estimators. Communications in Statistics
- Simulation and Computation 44 (2015) 1239-1252
5. Baran, S., Pap, G. (2011). Parameter estimation in a spatial unit root autoregressive model.
J. Multivariate Anal. 107. -Pp. 282 305.
6. Baran, S., Pap, G. and Zuijlen, M. v. Asymptotic inference for an unstable spatial AR
model. Statistics 38, 2004. -Pp.465 482.
7. Baran, S., Pap, G. and Zuijlen, M. v. Asymptotic inference for unit roots in spatial
triangular autoregression. Department of Mathematics, Radboud University Nijmegen,
The
Netherlands,
Report
No.
0506
(April
2005).
Url:
8. Baran, S., Pap, G. and Zuijlen, M. v. (2007). Asymptotic inference for unit roots in spatial
triangular autoregression. Acta Appl. Math. 96, -Pp. 17 42.
9. Baran. S., Pap. G., Martien C. A. Van Zuijlen (2004). Asymptotic inference for a nearly
unstable sequence of stationary spatial AR models. Statist. Probab. Lett, 2004. -V.69. -Pp.
53-61.
10.Handbook of Applied Statistics. V.2. -M.: Finance and Statistics, 1990. -526 p.
11. Mann H., Wald A. On the statistical treatment of linear stochastic difference equations.
Econometrics, 1943. -V.11. -Pp. 173-220.
12.Anderson TV On asymptotic distributions of estimates of parameters of stochastic
difference Equations. Ann. Math. Statist, 1959. -V.30. -Pp. 676-687.
13.Chan NH, Wei CZ Asymptotic inference for nearly nonstationary
(
( ))
AR p
processes.
Annals of Statistics, 1987. -V.15. -Pp. 1050-1063.
14.Chan NH, Wei CZ Limiting distributions of least squares estimates of unstable
autoregression processes. Annals of Statistics, 1988. -V.16. - No. 1. -Pp. 367-401.
15.White, JS The limiting distribution of the serial correlation coefficient in the explosive
case. Ann. Math. Statist, 1958. -V. 29. -Pp. 1188-1197.
16.Startsev AN A new approach to estimation of an autoregressive parameter. Proc. of Sixth
USSR-Japan Symp. World Scientific, 1991. -Pp.377-381.
17.Startsev A.N., Mirzaev T.S. On non-standard estimation methods in autoregressive
models in unstable cases. Journal of the Middle Volga Mathematical Society, 2011. -V.13.
-
-P. 25-35.
P
mavzusidagi Respublika ilmiy-amaliy anjuman materiallari. Namangan 2025-yil.
384
18.Basu. S., Reinsel. GC Properties of the spatial unilateral first-order ARMA model. Adv.
Appl. Probab, 1993. -V. 25. -Pp. 631-648.
19.V. Paulauskas. A note on self-normalization for a simple spatial autoregressive model.
Lithuanian Mathematical Journal, 2007. -V. 47. -Pp. 184-194.
20.Tjacco van der Meer, Gyula Pap, Martien C.A. van Zuijlen. Asymptotic inference for
nearly unstable AR(p) processes. Nijmegen: Catholic University, Department of
Mathematics, 1994. MATH QA3.R46 no.9413.
21.Jiang Long, Wei Wang, Jiangshuai Huang, Jing Zhou, Kexin Liu. Distributed Adaptive
Control for Asymptotically Consensus Tracking of Uncertain Nonlinear Systems With
Intermittent Actuator Faults and Directed Communication Topology. IEEE Transactions
on Cybernetics, 2021. -V. 51. -Pp. 4050-4061.
22.Badi H. Baltagi, Junjie Shu. A Survey of Spatial Unit Roots, Mathematics, 2024. 12, 1052.
(https://doi.org/10.3390/math12071052).
23.Sandar Baran, Gyula Pap. Parameter estimation in a spatial unilateral unit root
autoregressive model. Journal of Multivariative Analysis, 2012 may, -V. 107. -Pp. 282-
305.
24.
-dimensional simultaneous
autoregressive model. Metrika, 2009, October, -V. 74, -Pp 55-66.
