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International Journal of Probability and Statistics
p-ISSN: 2168-4871 e-ISSN: 2168-4863
2021; 10(2): 27-45
doi:10.5923/j.ijps.20211002.01
Received: May 8, 2021; Accepted: Jun. 2, 2021; Published: Jun. 15, 2021
Berry-Esseen Type Bound in Partially Linear Regression Model under Mixing Sequences
Abstract
Reference
Full-Text PDF
Full-text HTMLSallieu Kabay Samura
Department of Mathematics and Statistics, Fourah Bay College, University of Sierra Leone, Sierra Leone
Correspondence to: Sallieu Kabay Samura , Department of Mathematics and Statistics, Fourah Bay College, University of Sierra Leone, Sierra Leone.
| Email: |  |
Copyright © 2021 The Author(s). Published by Scientific & Academic Publishing.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
It is well-known that the confidence intervals of 
and 
in partially linear regression model lie in the limit distributions of their estimators. However the accuracy of the confidence intervals depends on how fast the theoretical distributions of the estimators converge to their limits. As a results, Berry-Esseen type bounds can be used to assess the accuracy. The aim of this paper is to study the Barry-Esseen type bounds for the estimators of 
and 
in the partially linear regression model with 
satisfying 
with 
and 
being stationary 
-mixing random variables. By choosing suitable weighted functions, the Berry-Esseen type bounds for the estimators 
and 
can achieve 
and 
respectively. Simulation studies are conducted to demonstrate the performance of the proposed procedure.
Keywords:
Partially linear model, Berry-esseen bound, 
-mixing sequence
Cite this paper: Sallieu Kabay Samura , Berry-Esseen Type Bound in Partially Linear Regression Model under Mixing Sequences, International Journal of Probability and Statistics , Vol. 10 No. 2, 2021, pp. 27-45. doi: 10.5923/j.ijps.20211002.01.
Article Outline
1. Introduction
- Partially linear regression model is a combination of linear and nonparametric parts in which the relationship between the response and some explanatory variables are linear whereas the other predictors are emerged in the model in unspecified association form. Opsomer and Ruppert (1999) argued for the advantage of partially linear regression model, including that there is less worry of overfitting, that they are more easily interpretable, and that the estimator is more efficient for the parametric components. Also, various estimation and variable selection methods for the partially linear regression model have been developed which we refer to, Horowitz (2009), Liu et al. (2011), Roozbeh et al. (2012), Amini and Roozbeh (2016), Roozbeh and Arashi (2016), Roozbeh (2018) and Amini and Roozbeh (2019) to mention a few.Since its introduction by Engle et al. (1986), partially linear regression models have been studied by many authors. For example, Heckman (1986), Rice (1986), Chen (1988) and Speckman (1988) studied the consistency properties of the estimator of
under different assumptions. Schick (1996) and Liang and Härdle (1997) extended the root n consistency and asymptotic results for the case of heteroscedasticity. Härdle et al. (2000) provided a good comprehensive reference of the partially linear model. Chen et al. (1998) and Gao et al. (1994) established the strong consistency and asymptotic normality, respectively, for the least squares estimators and weighted least-squares estimator (WLSE, for short) of 
based on nonparametric estimates 
and 
You et al. (2007) further studied the model and developed an inferential procedure which includes a test of heteroscedasticity, a two-step estimator of 
, mean square errors of 
and 
and a bootstrap goodness of fit test. If 
then the model boils down to the heteroscedastic linear model, whose asymptotic properties of the WLSE of 
were studied by Carroll (1982), Robinson (1987) and Carroll and Härdle (1989), respectively.In this paper, we will further study the limit behaviors of the estimators in the partially linear regression model under 
-mixing random variables, the concept of which was first introduced by Bradley (1985) as follows.Let 
be a sequence of random variables defined on a fixed probability space 
. Denote 
and 
. Let 
and 
be positive integers. Write 
. Given 
-algebras 
and 
in 
, let
where 
and 
. Define the 
-mixing coefficients by
Definition 1.1. A sequence 
of random variable is said to be 
-mixing if 
as 
Since the concept of 
-mixing was introduced by Bradley (1985), many limit theorems were established. Bradley (1985) discussed central limit theorems under absolute regularity for 
-mixing sequences. Shao (1993) established limit theorems of 
-mixing sequences; Cai (1991) obtained strong consistency and rates for recursive nonparametric conditional probability density estimators under 
-mixing conditions; Lu and Lin (1997) gave the bounds of covariance of 
-mixing sequences; Shen and Zhang (2011) studied some convergence theorems for 
-mixing random variables, and obtained some new strong laws of large numbers for weighted sums of 
-mixing random variables; Gao (2016) investigated the 
-mixing sequences which are stochastically dominated, and presented some strong stability; Yu (2016) showed the Resenthal-type inequality of the 
-mixing sequences, and investigated the strong convergence theorems.The aim of this paper is to further study the Barry-Esseen type bounds for the estimators of 
and 
in the partially linear regression model (2.1) with 
satisfying 
with 
and 
being stationary 
-mixing random variables. By choosing suitable weighted functions, the Berry-Esseen type bounds for the estimators 
and 
can achieve 
and 
respectively.This work is organised as follows: In Section 2, we recall the partially linear regression model and construct the partial least squares estimator for both the parametric and non-parametric components. The main results and numerical analysis (simulations and real data) are presented in Section 3. The proofs of the main results are provided in Section 4.Throughout this paper, the symbols 
denote positive constants whose values may be different in different places. Let 
and 
be the indicator function of the set 
. Let 
2. Model and Estimation
- Consider the following partially linear regression model:
 | (2.1) |
is an unknown parameter of interest, 
are nonrandom design points, 
are the response variables, 
are random errors, 
is an unknown functions defined on closed interval [0,1].If 
is the true parameter, then model (2.1) is reduced to a nonparametric regression model 
Since 
we have 
. Using the least squares method, we obtain 
of 
by minimizing
The minimizer is found to be | (2.2) |
,
and 
Then based on 
we defined the nonparametric function 
by | (2.3) |
on [0,1] such that 
and(i) 
(ii) 
A2. 
and 
are defined on [0,1] and satisfy Lipschitz condition of order 1.A3. The probability weight function 
are defined on [0,1] and satisfy
A4. 
for some 
A5. There exist positive integers 
and 
such that 
A6. The spectral density 
of 
satisfies that 
Remark 2.1 Conditions (A1)-(A3) have been used frequently by many authors. For example, Gao et al. [1996], Sun et al. [2002], You et al. [2005], Liang et al. [2006], You and Chen [2007] and so on. (A4) is adopted in Sun et al. [2002], You et al. [2005], Liang et al. [2006], Liang and Fan [2009] and so forth. Moreover, if functions 
and 
satisfy a Lipschitz condition of order 1 on [0,1], then (A3) (iii) implies that 
and 
3. Main Results and Numerical Analysis
3.1. Main Results
- In this subsection, we present the Berry-Esseen type bounds for the estimators
, 
. We first introduce some notations which will be used in the theorem below.
Theorem 3.1. Let 
be a mean zero 
-mixing sequence, with 
, where 
and 
Suppose that conditions 
are satisfied, 
and 
for some 
. Let 
for some 
Assume that 
. Then, for 
and any 
we have | (3.1) |
and 
where 
and 
for some 
. Let 
, 
for some 
and 
for some 
, 
Then
Theorem 3.2. Suppose that the conditions in theorem 3.1 hold. Let 
and 
If 
and 
converge to zero, then | (3.2) |
for some 
and 
Suppose that 
hold with 
for each 
and 
for some 
Let 
for some 
and 
for some 
, then
3.2. Numerical Simulation
- In this subsection, We carry out a simulation to study the asymptotic normality of the estimators
and 
, respectively. The observations are generated for the following model:
where 
is an AR(1) type process 
with 
be an MA(1) process specified by 
and 
, for 
where 
and 
for 
Here, we choose the nearest neighbor weights to be the weight functions 
For any 
we rewrite 
as follows
if 
, then 
is permutated before 
when 
. Take 
and defined the nearest neighbor weight function as follows:
We generate the observed data with sample size 
as 
and 
respectively. We used R software to compute 
and 
and obtained the Q-Q plots of 
and 
respectively, based on 500 replications. | Figure 1. Q-Q plot of  with  =50, 100 and 150, respectively |
 | Figure 2. Q-Q plot of  with  =50, 100 and 150, respectively |
4. Proofs of the Main Results
- We first introduce several lemmas which will be used to prove the main results of the paper.Lemma 3.1. (cf. Yu, 2016) Let
be a sequence of 
-mixing random variables with 
, 
for some 
and 
, where 
and 
Assume that 
is an array of real numbers. Then there exists a positive constant 
depending only on 
and 
such that
Lemma 3.2. (Liang and Fan, 2009) Let 
be random variables. For positive numbers 
we have that
Lemma 3.3. (Lu and Lin (1997)) Let 
be a sequence of 
-mixing random variables. Suppose that 
and 
where 
and 
Then
Lemma 3.4. Let 
be a sequence of 
-mixing random variables. Suppose that p and q are two positive integers. Let 
for 
Then for any 
Proof. It is easily checked that | (4.1) |
we have | (4.2) |
that | (4.3) |
 | (4.4) |
and hence applying Lemma 3.3 and invoking again the inequality 
we find that | (4.5) |
 | (4.6) |
Proceeding in this manner, we obtain
This completes the proof of the lemma.Lemma 3.5. Let 
be an array of real numbers such that 
and 
where 
are some positive numbers. Suppose that 
for some 
and 
for some 
Then for any 
and
where 
are positive numbers and 
for some 
and any 
Proof. We only prove the first inequality, and the second one is completely analogous. According to the definition of 
we have that
By 
and Markov's inequality, we have
Note that
Applying Lemma 3.1 with 
Noting that 
and 
implies 
we have
Where the inequality in the last line above follows from 
and changing the order of simulation. The proof is completed.Lemma 3.6. Under the assumptions of Theorem 2.1, the following statements hold.
Proof. (i) Applying Lemma 3.2 with 
and 
inequality, we have by 
for 
(A1) (iii) and (1.2) that | (4.7) |
inequality again that
Moreover, we have by 
and (A1) (iii) again and 
that
Hence (i) has been proved.(ii) The inequalities (a)-(e) can be proved by applying Lemma 3.5. Now we will verify them one by one.(a) Noting by (A1) (iii) that 
and 
the result follows immediately form Lemma 3.5.(b) Similarly, noting additionally that 
we have by Lemma 3.5 again that the result follows.(c) Note that 
Therefore, we have 
Let 
we have Lemma 3.5 again that
(d) Observe that 
Utilizing the Abel Inequality (see Mitrinovic, 1970, Theorem 1, p.32), we have by (A1) and (A3) that
and
Thus the result follow from Lemma 3.5 immediately.(e) It follows 
that
Similar to the proofs of (c) and (d), we have
Choosing 
we have by Lemma 3.5 again that
(f) The inequality (f) can be derived immediately by (i) and Markov's inequality.(g) It follows from the Abel Inequality and Remark 2.1 again that
The prove is completed.Lemma 3.7 Let 
Then under the assumption of Theorem 3.1, we have
Proof. Similar to Lemma 3.6 in Liang and Fan (2009), we can complete the proof of the lemma. The details are omitted.Assume that 
are independent random variables and 
has the same distribution as that of 
for each 
Let 
then 
Lemma 3.7 Under the assumptions of Theorem 3.1, we have
Proof. We have by Berry-Esseen inequality (Petrov, 1995, p.154, Theorem 5.7) that
Applying Lemma 3.3 with 
and noting that 
we have by choosing 
that
which together with 
from Lemma 3.7 yields that
The prove is completed.Lemma 3.8 Under the assumptions of Theorem 3.1, we have
Proof. Suppose that 
are characteristic functions of 
respectively.We have by Esseen inequality (Petrov, 1995, p.146, Theorem 5.3) that for any 
Applying Lemma 3.4 with 
and Lemma 3.2 with 
and similar to the proof of (3.2), we have by 
that
which implies that 
On the other hand, from Lemma 3.7 and we have
which derives that 
By choosing 
we can obtain that
This completes the proof of the lemma.Proof of Theorem 3.1 We can observe that
It is easy to show that
and
By changing the order of summation, we obtain that
Let 
Denote
The we have that
It follows from Lemma 3.7-3.9 that
Consequently, from (3.1), Lemma 3.1 and Lemma 3.6, we have
The proof of the theorem is completed.To prove Theorem 2.2, we need the following lemmas.Lemma 3.8 Under the assumption of Theorem 2.2, the following statements hold.
Furthermore, if 
for some positive numbers 
then
Proof. (i) Similar to the proof of (3.2), we have by 
that
Similarly,
Noting that 
analogous to the proof of (3.3) we have 
(ii) Noting that
and
the first inequality follows immediately form Lemma 3.5.Similarly, noting that
we can also get the second one from Lemma 3.5.(iii) Observe that 
in Theorem 2.1, the proof of (iii) can be easily obtained by following the proof of Lemma 3.9 in Liang and Fan (2009). The details are omitted here.Lemma 3.11. Let 
Then under the assumptions of Theorem 2.2, we have
Proof. Similar to the proof of Lemma 3.10 in Liang and Fan (2009), we can prove the lemma.The details are omitted.Assume that 
are independent random variables and 
has the same distribution as that of 
for each 
Let 
.Lemma 3.12 Under the assumptions of Theorem 2.2, we have
Proof. Note that 
Similar to the proof of Lemma 3.8, we have by Lemma 3.3, 
and changing the order of summation that
which together with 
from Lemma 3.11 yields that
The proof is completed.Lemma 3.13 Under the assumptions of Theorem 2.2, we have
Proof. Suppose that 
are the characteristic functions of 
respectively. Similar to the proof of Lemma 3.9, we have
together with Lemma 3.11 and 3.12 yield that
This completes the proof of the lemma.Proof of Theorem 3.1 We have that
Note that
and
Similar to the decomposition for 
we denote
Then we have that
we have by Lemma 3.11-3.13 that
which together with (3.4), Lemma 3.1 and Lemma 3.10 yields that
The proof is completed.