Mathematical Problems in Engineering
Volume 2019 (2019), Article ID 6593821, 11 pages
https://doi.org/10.1155/2019/6593821

Research Article

Locally Most Powerful Test for the Random Coefficient Autoregressive Model

Li Bi,¹ Feilong Lu,¹ Kai Yang,² and Dehui Wang¹

¹School of Mathematics, Jilin University, Changchun 130012, China
²School of Mathematics and Statistics, Changchun University of Technology, Changchun 130012, China

Correspondence should be addressed to Dehui Wang; wangdh@jlu.edu.cn

Received 3 April 2019; Accepted 17 June 2019; Published 27 June 2019

Academic Editor: Jixiang Yang

Copyright © 2019 Li Bi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this article, we study the problem of testing the constancy of the coefficient in a class of stationary first-order random coefficient autoregressive (RCAR(1)) model. We construct a new test statistic based on the locally most powerful-type (LMP) test. Under the null hypothesis, we derive the limiting distribution of the proposed test statistic. In the simulation, we compare the power between LMP test and empirical likelihood (EL) test and find that the accuracy of using LMP is 6.7%, 28.8%, and 26.1% higher than that of EL test under normal, student’s , and symmetric contamination errors, respectively. A real life data is given to illustrate the practical effectiveness of our test.

1. Introduction

In time series analysis, autoregressive and linear processes are widely used due to their mathematical tractability. In fact, the autoregressive model has two prominently advantages: their estimation procedures are well established and the existence of stationary solutions are easily derived. Therefore, this model has been investigated in the field of signal detection and classification, psychometry, and biomedical engineering. See, for example, Hoque [1], Ogawa et al. [2], Subasi et al. [3], and Maleki et al. [4]. The first-order autoregressive model (AR(1)) is defined as

where is an independent and identically distributed (i.i.d.) random error sequence with probability density function , , , and independent of for all However, it has been found that a variety of data sets cannot be modelled precisely by assuming linearity. Financial data, for instance, present heteroscedasticity, and biological data suffer from random perturbations. To address this problem, Conlisk [5] considered a random coefficient autoregressive model defined as follows:

where is a sequence of i.i.d random variables with and . is an i.i.d. random error sequence with mean zero and variance . The random variable is assumed to be independent of , which is independent of . Note that when , the RCAR(1) model reduces to the standard AR(1) model.

The RCAR(1) models have been widely applied in mathematical literature due to its strong application value in practice. For instance, Nicholls and Quinn [6–8] derived the least square (LS) estimates of the model parameter and showed that the LS estimates are strongly consistent and obey the central limit theorem. Brandt [9] restated the necessary and sufficient conditions for the existence of a strict stationarity and ergodicity solution in the RCAR(1) model. Wang and Ghosh [10] considered the Basyesian estimation method to estimate RCAR model parameter and explored the frequentist properties of the Bayes estimator. Wang et al. [11] obtained the asymptotic properties of the maximum likelihood estimators of these parameters in RCAR(1) model under the unit root assumption.

More recently, the problem of testing traditional AR(1) model against RCAR(1) model is thus of essential issue in this settings. This detection problem was firstly investigated by Nicholls and Quinn [8], where a Gaussian Lagrange multiplier test is derived for the problem. However, this likelihood ratio approach has several weaknesses, as the true value of the parameter under the null hypothesis lies on the boundary of the parameter space, the asymptotics will not be easily established. Ramanathan and Rajarshi [12] considered a non-Gaussian test method to handle the change problem for parameters in RCAR(1) case. Their signed rank test, however, require a symmetry assumption on the innovation density, which is highly unnatural in this context. Still for the RCAR(1) model, Lee [13] proposed the cumulative sum test for parameter change in RCAR(1) model. But, it detects a change only in a function of the parameter rather than the parameter themselves. Recently, Moreno and Romo [14] studied robust unit root test with autoregressive errors. The unit root test shows some size distortions and requires many estimates in the construction of test.

To deal with these drawbacks, we construct a locally most powerful-type (LMP) test for testing the constancy of parameters in the RCAR(1) model. The LMP test has been investigated by several authors. Rohatgi et al. [15], Chikkagoudar and Biradar [16], and Manik et al. [17] considered the LMP test for parameter constancy. The test has merit in that its calculation process is less cumbersome and it produces stable sizes especially when parameter nearby the true value. In contrast, the traditional tests mentioned above show some size distortions and require many estimates in the construction of tests, see Ramanathan and Rajarshi [12]. In a general way, our test can conventionally discard correlation effects and enhance the performance of the test. In this paper, we illustrate how our proposed method can be implemented for finite samples under normal, student’s t, and symmetric contamination errors (see Huber [18] and Hettmansperger [19]). Through numerical simulation studies, we can see that our test has a stronger power in terms of maintaining the empirical level and power than the empirical likelihood (EL) test suggested by Zhao et al. [20].

The outline of this paper is organized as follows. In Section 2, we introduce our test statistic and derive its limiting distribution under the null hypothesis. Numerical simulations to evaluate the empirical size and power of our test technique are discussed in Section 3. Section 4 presents a real life data example to illustrate the superior of our methods. We provide brief concluding remarks in Conclusions. Appendix provides the proofs of the main results.

2. Methodology and Main Results

In this section, we will construct a test statistic to test whether is a constant. To achieve this task, we set up the null and alternative hypotheses

In what follows, we give our main results. Suppose that the time series data are generated from (2). Let be an i.i.d. sequence of random variables with common probability distribution ; is an i.i.d sequence of random variables with density function In this article, we regard as a given number; alternatively, if is a random variable, we shall consider only inferences conditionally on are fixed. Note that , , and are independent.

Next, we assume the following conditions to establish the asymptotic properties of the test statistic:

(C1)

(C2) The distribution of is such that

(C3) The third and mixed derivatives of with respect to and are uniformly bounded on .

(C4) Differentiation thrice with respect to of under the integration is bounded.

It is easy to derive that is a Markov chain on with the following transition probabilities:

The Markovity follows from (2) and the fact that is an i.i.d sequence. The conditional density is obtained by noting that conditional on Conditional on and is an i.i.d sequence with the density , hence, the probability density function of is given by . So, we get the result in the above equation.

Therefore, we can write down the likelihood function for RCAR(1) model:

Furthermore, employing this spirit of LMP approach introduced by Manik et al. [17], we can obtain our test statistic

where

And the detailed procedures can be found in Appendix A.

Let denote vector of all unknown parameters of the RCAR(1) process. Hereafter, we use the notation to represent the maximum likelihood estimator (MLE) of . In addition, the actual test statistic can be obtained by replacing with their MLE It can be easily derived that is a zero mean martingale.

Before we state our main results, the following assumptions will be made:

(A.1) , for some

Then about the asymptotic distribution of , we have the following theorem.

Theorem 1. Let be a sequence of strictly stationary, ergodic, and -measurable solutions to equation (2) under conditions of and above . Then under , we have asymptotic normality

here

(The expressions for , , and are derived in Appendix B).

3. Simulation

In this section, we carry out some simulation studies to compare performances of the locally most powerful-type test and the empirical likelihood test in terms of empirical size and power. The empirical size and power for the two tests in Tables 1–6 are based on 1000 repetitions with the help of R software. Within each study, we set the initial values and employ the significance level at 0.05. Throughout this simulation, we used notation LMP for locally most powerful-type test by our algorithm, EL for empirical likelihood method.

Table 1: Empirical sizes of LMP and EL tests at nominal level 0.05 for model N1.

100
MP
0.095
0.105
0.109
0.101
0.116
0.103
0.089
0.098

EL
0.008
0.004
0.002
0.003
0.003
0.002
0.003
0.002

200
LMP
0.089
0.082
0.085
0.086
0.085
0.082
0.083
0.087

EL
0.006
0.007
0.004
0.005
0.004
0.004
0.003
0.003

300
LMP
0.074
0.077
0.082
0.069
0.076
0.071
0.076
0.074

EL
0.006
0.009
0.007
0.006
0.005
0.007
0.008
0.009

400
LMP
0.063
0.070
0.075
0.068
0.067
0.063
0.070
0.072

EL
0.007
0.009
0.006
0.004
0.007
0.008
0.005
0.004

500
LMP
0.066
0.060
0.077
0.060
0.065
0.062
0.060
0.055

EL
0.009
0.005
0.006
0.005
0.006
0.009
0.014
0.008

600
LMP
0.062
0.057
0.067
0.053
0.065
0.060
0.059
0.068

EL
0.009
0.005
0.011
0.007
0.009
0.007
0.008
0.007

700
LMP
0.064
0.059
0.068
0.056
0.057
0.062
0.058
0.063

EL
0.005
0.010
0.008
0.007
0.007
0.008
0.005
0.003

800
LMP
0.060
0.061
0.060
0.056
0.058
0.055
0.055
0.063

EL
0.009
0.007
0.007
0.010
0.011
0.012
0.008
0.009

900
LMP
0.055
0.062
0.054
0.048
0.053
0.053
0.062
0.062

EL
0.015
0.015
0.008
0.010
0.011
0.007
0.010
0.009

1000
LMP
0.063
0.051
0.057
0.052
0.058
0.062
0.057
0.054

EL
0.019
0.009
0.012
0.009
0.011
0.012
0.007
0.009

5000
LMP
0.051
0.051
0.051
0.053
0.050
0.048
0.049
0.053

EL
0.020
0.022
0.018
0.020
0.016
0.021
0.024
0.022

Table 2: Empirical sizes of LMP and EL tests at nominal level 0.05 for model N2.

100
MP
0.060
0.065
0.066
0.061
0.065
0.068
0.075
0.067

EL
0.003
0.003
0.002
0.003
0.004
0.003
0.004
0.001

200
LMP
0.059
0.061
0.065
0.059
0.066
0.063
0.068
0.061

EL
0.002
0.004
0.002
0.002
0.004
0.004
0.002
0.004

300
LMP
0.058
0.057
0.059
0.055
0.063
0.058
0.063
0.055

EL
0.005
0.004
0.003
0.009
0.006
0.006
0.004
0.002

400
LMP
0.058
0.055
0.058
0.055
0.062
0.054
0.061
0.058

EL
0.004
0.003
0.005
0.003
0.004
0.003
0.005
0.006

500
LMP
0.056
0.055
0.056
0.052
0.057
0.059
0.055
0.054

EL
0.006
0.004
0.005
0.005
0.007
0.005
0.006
0.007

600
LMP
0.060
0.057
0.058
0.060
0.062
0.059
0.056
0.053

EL
0.003
0.005
0.005
0.005
0.008
0.007
0.009
0.006

700
LMP
0.055
0.056
0.058
0.057
0.060
0.060
0.056
0.058

EL
0.006
0.005
0.005
0.008
0.006
0.006
0.004
0.008

800
LMP
0.050
0.054
0.054
0.052
0.056
0.054
0.054
0.054

EL
0.005
0.007
0.008
0.006
0.006
0.008
0.008
0.008

900
LMP
0.051
0.052
0.058
0.056
0.056
0.060
0.049
0.051

EL
0.005
0.006
0.006
0.009
0.006
0.008
0.010
0.005

1000
LMP
0.052
0.056
0.057
0.055
0.050
0.053
0.050
0.054

EL
0.007
0.008
0.005
0.006
0.012
0.009
0.009
0.007

5000
LMP
0.052
0.050
0.050
0.050
0.051
0.049
0.054
0.052

EL
0.015
0.014
0.012
0.016
0.017
0.014
0.012
0.013

Table 3: Empirical sizes of LMP and EL tests at nominal level 0.05 for model N3.

100
LMP
0.064
0.068
0.065
0.072
0.053
0.078
0.071
0.053
0.047

EL
0.000
0.000
0.001
0.000
0.000
0.001
0.000
0.000
0.001

300
LMP
0.045
0.066
0.053
0.038
0.054
0.081
0.051
0.043
0.040

EL
0.000
0.001
0.000
0.001
0.001
0.002
0.000
0.001
0.000

500
LMP
0.058
0.057
0.057
0.049
0.050
0.060
0.043
0.049
0.044

EL
0.001
0.000
0.001
0.000
0.000
0.001
0.000
0.000
0.001

1000
LMP
0.050
0.059
0.052
0.051
0.045
0.054
0.036
0.043
0.036

EL
0.002
0.001
0.001
0.001
0.001
0.001
0.003
0.001
0.001

5000
LMP
0.053
0.050
0.054
0.050
0.048
0.050
0.047
0.050
0.046

EL
0.003
0.004
0.003
0.007
0.007
0.006
0.008
0.006
0.011

100
LMP
0.078
0.081
0.089
0.069
0.089
0.095
0.083
0.091
0.086

EL
0.000
0.002
0.001
0.000
0.000
0.000
0.000
0.000
0.001

300
LMP
0.080
0.070
0.074
0.070
0.070
0.056
0.063
0.057
0.055

EL
0.000
0.000
0.001
0.000
0.000
0.001
0.002
0.000
0.001

500
LMP
0.072
0.081
0.068
0.067
0.058
0.059
0.060
0.046
0.053

EL
0.001
0.000
0.001
0.000
0.000
0.001
0.001
0.000
0.004

1000
LMP
0.064
0.061
0.067
0.067
0.055
0.055
0.052
0.051
0.045

EL
0.000
0.000
0.000
0.001
0.002
0.003
0.003
0.000
0.001

5000
LMP
0.055
0.053
0.050
0.050
0.055
0.046
0.050
0.047
0.050

EL
0.004
0.002
0.003
0.002
0.002
0.006
0.003
0.003
0.009

Table 4: Empirical powers of LMP and EL tests at nominal level 0.05 for model A1.

100
LMP
0.165
0.102
0.171
0.110
0.171
0.107
0.174
0.090

EL
0.055
0.024
0.040
0.023
0.050
0.022
0.041
0.025

200
LMP
0.318
0.143
0.294
0.147
0.320
0.140
0.311
0.160

EL
0.191
0.070
0.156
0.073
0.202
0.049
0.155
0.067

300
LMP
0.513
0.209
0.494
0.253
0.511
0.175
0.449
0.241

EL
0.405
0.144
0.332
0.173
0.397
0.124
0.294
0.168

400
LMP
0.674
0.232
0.561
0.330
0.663
0.264
0.590
0.338

EL
0.578
0.176
0.456
0.272
0.559
0.189
0.450
0.260

500
LMP
0.791
0.377
0.720
0.427
0.770
0.374
0.725
0.438

EL
0.722
0.298
0.629
0.366
0.705
0.303
0.625
0.374

600
LMP
0.842
0.415
0.792
0.521
0.832
0.424
0.790
0.508

EL
0.798
0.352
0.717
0.460
0.793
0.365
0.712
0.458

700
LMP
0.912
0.483
0.862
0.598
0.919
0.478
0.855
0.598

EL
0.882
0.429
0.809
0.541
0.885
0.423
0.794
0.536

800
LMP
0.964
0.560
0.915
0.645
0.947
0.568
0.918
0.643

EL
0.938
0.512
0.877
0.604
0.928
0.516
0.872
0.599

900
LMP
0.967
0.633
0.936
0.727
0.971
0.623
0.955
0.703

EL
0.952
0.586
0.910
0.696
0.959
0.586
0.933
0.687

1000
LMP
0.988
0.712
0.963
0.799
0.978
0.702
0.962
0.776

EL
0.981
0.678
0.942
0.774
0.969
0.676
0.938
0.749

5000
LMP
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000

EL
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000

Table 5: Empirical powers of LMP and EL tests at nominal level 0.05 for model A2.

100
LMP
0.289
0.110
0.220
0.149
0.241
0.096
0.183
0.131

EL
0.045
0.016
0.027
0.019
0.047
0.018
0.027
0.029

200
LMP
0.596
0.252
0.519
0.310
0.525
0.199
0.452
0.272

EL
0.161
0.052
0.118
0.055
0.186
0.057
0.146
0.063

300
LMP
0.790
0.400
0.722
0.475
0.762
0.351
0.686
0.429

EL
0.336
0.111
0.276
0.124
0.373
0.124
0.293
0.152

400
LMP
0.899
0.538
0.861
0.644
0.862
0.462
0.815
0.564

EL
0.486
0.147
0.399
0.196
0.520
0.173
0.440
0.223

500
LMP
0.964
0.624
0.933
0.757
0.957
0.596
0.906
0.699

EL
0.622
0.218
0.529
0.285
0.695
0.262
0.590
0.345

600
LMP
0.979
0.701
0.951
0.819
0.965
0.657
0.947
0.754

EL
0.708
0.237
0.606
0.351
0.767
0.301
0.650
0.395

700
LMP
0.995
0.791
0.974
0.862
0.983
0.727
0.978
0.824

EL
0.814
0.351
0.669
0.404
0.839
0.389
0.773
0.483

800
LMP
0.996
0.837
0.990
0.927
0.994
0.809
0.984
0.882

EL
0.845
0.404
0.767
0.512
0.895
0.488
0.839
0.585

900
LMP
0.999
0.874
0.995
0.934
0.996
0.831
0.993
0.924

EL
0.879
0.430
0.801
0.548
0.929
0.514
0.877
0.622

1000
LMP
1.000
0.936
1.000
0.971
1.000
0.889
0.996
0.951

EL
0.913
0.515
0.887
0.626
0.946
0.601
0.930
0.706

5000
LMP
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000

EL
0.992
0.967
0.994
0.985
1.000
1.000
1.000
0.999

Table 6: Empirical powers of LMP and EL tests at nominal level 0.05 for model A3 with .

100
LMP
0.141
0.054
0.034
0.120
0.073
0.038
0.084
0.082
0.046

EL
0.039
0.009
0.006
0.026
0.012
0.008
0.018
0.012
0.009

300
LMP
0.775
0.388
0.183
0.628
0.493
0.299
0.475
0.448
0.331

EL
0.325
0.102
0.030
0.231
0.097
0.052
0.143
0.114
0.061

500
LMP
0.959
0.675
0.381
0.880
0.791
0.594
0.754
0.729
0.583

EL
0.656
0.236
0.070
0.500
0.268
0.143
0.347
0.222
0.131

1000
LMP
1.000
0.967
0.753
0.996
0.992
0.934
0.982
0.982
0.939

EL
0.945
0.599
0.255
0.891
0.673
0.399
0.720
0.609
0.402

5000
LMP
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000

EL
1.000
1.000
0.981
1.000
1.000
0.997
0.999
1.000
0.997

100
LMP
0.118
0.043
0.035
0.071
0.041
0.025
0.049
0.052
0.037

EL
0.042
0.013
0.015
0.032
0.021
0.014
0.020
0.013
0.011

300
LMP
0.673
0.278
0.101
0.545
0.372
0.190
0.388
0.313
0.220

EL
0.301
0.077
0.021
0.199
0.101
0.043
0.128
0.093
0.055

500
LMP
0.914
0.558
0.277
0.868
0.681
0.423
0.700
0.622
0.467

EL
0.571
0.183
0.054
0.486
0.239
0.128
0.353
0.212
0.113

1000
LMP
1.000
0.929
0.627
0.995
0.965
0.849
0.970
0.954
0.865

EL
0.927
0.503
0.204
0.852
0.637
0.375
0.702
0.548
0.368

5000
LMP
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000

EL
1.000
1.000
0.963
1.000
1.000
0.996
1.000
1.000
0.994

3.1. Empirical Size

To calculate empirical sizes, we pay our attention to three kinds of model (1) defined as follows:

(N1) ,

(N2) ,

(N3) , The distribution function of -contamination is

where is a fixed constant satisfying and is the distribution function of standard normal random variable,

Let us now describe how the simulated results are obtained. First of all, we use the model N1 to generate data when the 0.5, 1 and 0.3, 0.5, 0.7, 0.9. Secondly, we generate data from model N2 with the degrees of freedom 8, 10 and 0.3, 0.5, 0.7, 0.9. For model N1 and N2, we take the sample size 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, and 5000. Thirdly, we simulate samples from model N3 with the , , and 0.3, 0.5, 0.7, 0.9. For model N3, we set the sample size and . The empirical levels of three models are presented in Tables 1–3, respectively. As seen from Tables 1–3, the LMP test give similar results under normal, student’s t, and symmetric contamination errors. In addition, the LMP test produces sizes closer to the nominal significance level 0.05 quite satisfactorily, especially for larger sample sizes. However, the EL method has lower levels for each sample size, so one may make wrong decisions in tests based on them. Looking at three models results, we can conclude that our method has a greater effect than the EL test in terms of the empirical level.

3.2. Empirical Power

In order to investigate the empirical powers, we consider the alternatives under three versions of model (2):

(A1) ,

(A2) ,

(A3) , The distribution function of -contamination is

where is a fixed constant satisfying and is the distribution function of standard normal random variable, ,

To calculate the empirical powers of the two tests, in the first, we generate samples from the model A1 in which the parameters 0.5, 1, 0.5, 1, and 0.5, 1. In the second setting, we simulate samples from model A2 with the degrees of freedom the parameters and For model A1 and A2, we take the sample size 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, and 5000. In the last case, we generate data from the model A3 in which the , , and the parameters , For model A3, we employ the sample size 100, 300, 500, 1000, and 5000. The empirical power of the above three models are summed up in Tables 4–6. In three cases, the power of two test statistics increase with the sample size, while our test produces relatively better powers than the EL test. Furthermore, both tests powers are close to 1 at the sample size . Overall, from these simulations, we conclude that the accuracy of using LMP is 6.7%, 28.8%, 26.1% higher than that of EL test under normal, student’s , and symmetric contamination errors, respectively. As anticipated, our finding shows that the LMP test is a functional tool to detect a parameter change for RCAR(1) model. Therefore, we recommend the LMP for practical use because its computation is not difficult and the overall performance is better than that EL under normal, student’s t, and symmetric contamination errors.

4. A Real Life Data Analysis

In this section, we illustrate how the LMP method can be applied to a practical application. This data consist of 78 monthly number of annulus growth rate of the import bill in Australia, starting in December 2010 and ending in May 2017. The data are available online at the CEInet Statistics Database site http://db.cei.cn/page/Default.aspx. The mean and variance of the data are found to be 0.2069 and 15.3347. The data are denoted as Figure 1 is the sample path plot for the real data ,

Figure 1: The left one is the sample path of original series and the right one is the sample plot of centering series for the real data.

For such a process, the mean function of is constant and we may assume that the process mean is subtracted out to produce a process with zero mean. The plot of the sample path, autocorrelation function (ACF), and the partial autocorrelation function (PACF) for series are given in Figures 1 and 2, respectively. From Figure 1, we can see may come from a stationary autoregressive time series process. From Figure 2, we conclude that is from first-order autoregressive process. Moreover, we test the normality of the residual data according to the Normal Q-Q plot introduced by Henry C Thode [21]. From Figure 3, we notice that the scatter points on the Normal Q-Q plot are close to the reference line, so the residual data can basically be seen as following the normal distribution. Hence, we derive that the model N1 would be more suitable to model these data.

Figure 2: The sample autocorrelation function (ACF) and partial autocorrelation function (PACF) plots of centering series for the real data.

Figure 3: The normal Q-Q plot of centering series for the real data.

From the time series plot given in Figure 1, the constancy of the parameter may be suspected, and, therefore, we are interested in testing the hypothesis of constancy of the coefficient parameter. We carry out the test for against The value of the test statistic in (8) turned out to be 2.1151, with value 0.0344, which indicates the rejection of the null hypothesis at 5% level of significance. Thus, in this case the RCAR(1) model would be much more appropriate as opposed to the AR(1) model.

5. Conclusions

In this article, we propose a locally most powerful-type test for testing the constancy of the random coefficient parameter in autoregressive model and derive their limiting null distributions under regularity conditions. It is clear from the applications that the coefficient need not remain constant throughout the time. Therefore, it is essential to have such a test conducted whenever the random variation is suspected. Through our simulation study and a real-data analysis, we demonstrate that our test succeeds and it performs better than the competitor. Finally, we anticipate that our locally most powerful-type test can be extended to other types of time series model.

Appendix

A. Process of Establish the Test Statistic

In this Appendix A, we present the detailed steps to obtain the test statistics.

Proof. Therefore, we can get the likelihood function

Now, the Taylor series expansion of around which is the mean of gives that

Obviously, we can conclude that if That is to say, the distribution function of random variable is degenerated. Furthermore, we derive that

By taking the derivative of with respect to , at , we have the following equation:

Thus, the LMP test statistic has the form

This completes the proof.

B. Proof of Theorem 1

Proof of Theorem 1. In the following, we can write down the log-likelihood function under the null hypothesis:

According to the maximum likelihood principle, the maximum likelihood estimate can be obtained by maximizing the log-likelihood function with respect to . Applying the Taylor series expansion to gives

where is on the line segment between and , so . Under the null hypothesis, we can write

Using the similar method as in the above, expanding around gives that

with . The remainder term is since Under and , we conclude that . Therefore,

Now the first term of (B.5) converges to

with This is because

Next, we can obtain the elements of Fisher information matrix

Thus (B.5) is asymptotically equivalent to

We present the following lemma recommended by Billingsley [22] to establish the asymptotic normality of the test statistic.
Lemma B.1. Under assumption of , we have as ,
(i) ; (ii) , where .
Proof. By the ergodic theorem, the first conclusion holds.
Now, let and is a sigma field. Note that It is easy to see that . Then, we have

Thus is a martingale. We have shown , which means is uniformly integrable. By Theorem 1.1 of Billingsley [22], we have that as

Hence, using the following version of Martingale Central Limit Theorem from Hall and Heyde [23], we have . The proof of Lemma B.1 is thus completed.                                                                 □
Similarly, we can verify that is a martingale. By ergodic and stationary properties, we obtain that as ,

Therefore, .
In the same way, for any vector we have

By the Cramer-Wold device, we obtain

where Then we can make the conclusion that

where The proof of the theorem is completed.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Nos. 11871028, 11731015, 11571051, and 11501241), Natural Science Foundation of Jilin Province (Nos. 20180101216JC, 20170101057JC, and 20150520053JH), Program for Changbaishan Scholars of Jilin Province (2015010), and Science and Technology Program of Jilin Educational Department during the “13th Five-Year” Plan Period (No. 2016316).

References

A. Hoque, “Finite sample analysis of the first order autoregressive model,” Calcutta Statistical Association Bulletin, vol. 34, no. 1-2, pp. 51–63, 1985.

T. Ogawa, H. Sonoda, S. Ishiwa, and Y. Shigeta, “An application of autoregressive model to pattern discrimination of brain electrical activity mapping,” Brain Topography, vol. 6, no. 1, pp. 3–11, 1992.

A. Subasi, A. Alkan, E. Koklukaya, and M. K. Kiymik, “Wavelet neural network classification of EEG signals by using AR model with MLE preprocessing,” Journal of the International Neural Network Society, vol. 18, pp. 985–997, 2005.

M. Maleki and A. R. Nematollahi, “Autoregressive models with mixture of scale mixtures of gaussian innovations,” Iranian Journal of Science & Technology, Transactions A: Science, vol. 41, no. 4, pp. 1099–1107, 2017.

J. Conlisk, “Stability in a random coefficient model,” International Economic Review, vol. 15, no. 2, pp. 529–533, 1974.

D. F. Nicholls and B. G. Quinn, “The estimation of random coefficient autoregressive models I,” Journal of Time Series Analysis, vol. 1, no. 1, pp. 37–46, 1980.

D. F. Nicholls and B. G. Quinn, “The estimation of random coefficient autoregressive models II,” Journal of Time Series Analysis, vol. 2, no. 3, pp. 185–203, 1981.

D. F. Nicholls and B. G. Quinn, Random Coefficient Autoregressive Models: An Introduction, Lecture Notes in Statistics, Springer-Verlag, New York, NY, USA, 1982.

A. Brandt, “The stochastic equation Yn+1 = AnYn + Bn with stationary coefficients,” Advances in Applied Probability, vol. 18, no. 1, pp. 211–220, 1986.

D. Wang and S. K. Ghosh, “Bayesian estimation and unit root tests for random coefficient autoregressive models,” Model Assisted Statistics and Applications, vol. 3, no. 4, pp. 281–295, 2008.

D. Wang, S. K. Ghosh, and S. G. Pantula, “Maximum likelihood estimation and unit root test for first order random coefficient autoregressive models,” Journal of Statistical Theory and Practice, vol. 4, no. 2, pp. 261–278, 2010.

T. V. Ramanathan and M. B. Rajarshi, “Rank tests for testing the randomness of autoregressive coefficients,” Statistics & Probability Letters, vol. 21, no. 2, pp. 115–120, 1994.

S. Lee, J. Ha, O. Na, and S. Na, “The cusum test for parameter change in time series models,” Scandinavian Journal of Statistics, vol. 30, no. 4, pp. 781–796, 2003.

M. Moreno and J. Romo, “Robust unit root tests with autoregressive errors,” Communications in Statistics—Theory and Methods, vol. 45, no. 20, pp. 5997–6021, 2016.

V. K. Rohatgi, A. K. Md Ehsanes Sleh, R. Ahluwalia, and P. Ji, “Null distribution of locally most powerful tests for the two sample problem when the combined sample is type II censored,” Communications in Statistics - Theory and Methods, vol. 19, pp. 2337–2355, 1992.

M. S. Chikkagoudar and B. S. Biradar, “Locally most powerful rank tests for comparison of two failure rates based on multiple type-ii censored data,” Communications in Statistics: Theory and Methods, vol. 41, no. 23, pp. 4315–4331, 2012.

A. Manik, N. Balakrishna, and T. V. Ramanathan, “Testing the constancy of the thinning parameter in a random coefficient integer autoregressive model,” Statistical Papers, pp. 1–25, 2017.

P. J. Huber, Robust Statistics, John Wiley & Sons, New York, NY, USA, 1981.

T. P. Hettmansperger, Statistical Inference Based on Ranks, John Wiley & Sons, Inc., New York, NY, USA, 1984.

Z.-W. Zhao, D.-H. Wang, and C.-X. Peng, “Coefficient constancy test in generalized random coefficient autoregressive model,” Applied Mathematics and Computation, vol. 219, no. 20, pp. 10283–10292, 2013.

H. C. Thode, Testing for Normality, CRC Press, New York, NY, USA, 2002.

P. Billingsley, “The lindeberg-Lvy theorem for martingales,” Proceedings of the American Mathematical Society, vol. 12, no. 1, pp. 788–792, 1961.

P. Hall and C. C. Heyde, Martingale Limit Theory and Its Application, Academic Press, New York, NY, USA, 1980.