Deng Xin, Tian Chunyu, Ge Meimei, Ye Jing, Ding Yang, Wu Yi

1. School of Mathematical and Finance, Chuzhou University, Chuzhou 239000, China;2. China Electronics Technology Group Corporation No.58 Research Institute, Nanjing 210000, China;3. School of Big Data and Artificial Intelligence, Chizhou University, Chizhou 247000, China

Abstract: The simple linear errors-in-variables (EV) model with φ-mixing random errors was mainly studied. By using the central limit theorem and the Marcinkiewicz-type strong law of large numbers for the φ-mixing sequence, the asymptotic normality of the least square (LS) estimators for the unknown parameters were established under some mild conditions. In addition, based on the strong convergence for weighted sums of φ-mixing random variables, the strong consistency of the LS estimators were obtained. Finally, the simulation study was provided to verify the validity of the theoretical results.

Keywords: EV model; asymptotic normality; strong consistency; LS estimator; φ-mixing sequence

1 Introduction

Consider the following simple linear errors-in-variables (EV) model:

ηi=θ+βxi+εi,ξi=xi+δi, 1≤i≤n

(1)

whereθandβare unknown parameters； (ε1,δ1),(ε2,δ2),... are random errors with mean zero；x1,x2,... are unobservable；ξi,ηi,i=1,2,3,... are observable. From the formula (1), we have

ηi=θ+βξi+Xi,Xi=εi-βδi, 1≤i≤n

(2)

We consider formally (2) as a usual regression model ofηionξi, and get the least square (LS) estimators ofθandβ:

(3)

In this paper, we investigate a much wider dependent error structure: theφ-mixing random errors, and study the asymptotic normality and the strong consistency of the LS estimators (3) for the unknown parametersθandβin a simple linear EV model (1). Now, let us recall the concept of theφ-mixing random variables.

Definition 1A sequence {Xn,n≥1} of random variables is said to be aφ-mixing sequence, if

n→∞.

1 Main results

(4)

(5)

and there exists a constantc>0 such that:

|xi-xj|≤c|i-j|, ∀ 1≤i

(6)

(7)

and

(8)

Theorem 2Under the conditions of Theorem 1, assume that

(9)

and

(10)

Then,

Theorem 3Under the model (1), assume thatE|ε1|p<∞,E|δ1|p<∞ for somep>1/δ,0<δ≤1/2.Letτ>0.If

(11)

Then,

Theorem 4Under the assumptions of Theorem 3, if

(12)

for someν∈(0,1/2), then,

2 Simulation

In the subsection, we will carry out simulations to study the numerical performance of the asymptotic normality results and the strong consistency results.

Figure 1. Q-Q plots of (a) with n=300.

Figure 2. Q-Q plots of (a) with n=600.

Figure 3. Q-Q plots of (a) with n=900.

Figure 4. Q-Q plots of (a) with n=1200.

Figure 5.Boxplots of

figure 6.BOxplots of

3 The proof of main results

By simple calculation, we have

(13)

and

(14)

In order to prove the main results of this paper, we need the following lemmas, which are the central limit theorem, the Marcinkiewicz-type strong law of large numbers forφ-mixing sequence and the strong convergence for the weighted sums of theφ-mixing random variables, respectively.

Lemma 1[23]Let {Xn,n≥1} be a centered stochastic sequence ofφ-mixing random variables and {ani,1≤i≤n,n≥1} be a triangular array of real numbers such that:

|ani|=O(n-δ) for 1≤i≤n,

wheret=min(q,2).Then,

ProofThe proof of the lemma can be referred to Lemma 2 of Wu[26].

The proof of Theorem 1

Firstly, by Markov’s inequality and condition (4), we can get that

Hence,

(15)

By

(16)

and the proof of (15), we have that

Thus,

(17)

Secondly, note that

For anyγ>0, we have that

(18)

Thereby,

which implies by (15) that:

(19)

From (13), we have that

Combined with (15), (17) and (19), it is sufficient to prove

Next, we prove:

(20)

and

(21)

For (20), by conditions (6) and (8), we can obtain that

For (21), by condition (8), we can get that

The proof of Theorem 2

According to (14), we have that

(22)

Note by condition (9) that

From Lemma 2, it is followed by

It is observed that

and

By Kronecker’s Lemma and condition (10), it is easy to obtain that:

It is followed by conditions (7) and (10) that:

which is implied in Lemma 1 that

Finally, the desired result follows from the result of Theorem 1 and (22).

The proof of Theorem 3

(23)

(24)

by (16).

According to (11) and Lemma 3 (takingq=p>2), we can obtain:

(25)

From (18), (23) and (11), we have that

which implies in the arbitrariness ofγthat:

(Ⅱ) Ifp>4, with the similar proofs as the case 2

The proof of Theorem 4

Applying Lemma 3 (takingq=p>2), it is not difficult to show that

Furthermore,from condition (12), Theorem 3 and Lemma 3 (takingq=p>2), we have

Therefore, the desired result can be obtained from (14).

Acknowledgments

This work is supported by the Scientific Research Foundation Funded Project of Chuzhou University (2018qd01) and the Natural Science Foundation of Anhui Province (1908085QA01).

Conflictofinterest

The authors declare no conflict of interest.

Authorinformation

DengXinreceived her PhD degree in Statistics from Anhui University. She is currently a lecturer at the School of Mathematical and Finance of Chuzhou University. Her research mainly focuses on probability limit theory, parametric and non-parametric statistics.

WuYi(corresponding author) received his PhD degree in Statistics from Anhui University. He is currently a lecturer at the School of Big Data and Artificial Intelligence of Chizhou University. His research interests include probability limit theory，statistical models and sub-linear expectations.