MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi 10.1155/2019/6593821 6593821 Research Article Locally Most Powerful Test for the Random Coefficient Autoregressive Model Bi Li 1 Lu Feilong 1 Yang Kai 2 https://orcid.org/0000-0002-9185-9034 Wang Dehui 1 Yang Jixiang 1 School of Mathematics Jilin University Changchun 130012 China jlu.edu.cn 2 School of Mathematics and Statistics Changchun University of Technology Changchun 130012 China ccut.edu.cn 2019 2762019 2019 03 04 2019 17 06 2019 2762019 2019 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.

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 t, and symmetric contamination errors, respectively. A real life data is given to illustrate the practical effectiveness of our test.

National Natural Science Foundation of China 11871028 11731015 11571051 11501241 Natural Science Foundation of Jilin Province 20180101216JC 20170101057JC 20150520053JH Program for Changbaishan Scholars of Jilin Province 2015010 Science and Technology Program of Jilin Educational Department during the “13th Five-Year” Plan Period 2016316
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 , Ogawa et al. , Subasi et al. , and Maleki et al. . The first-order autoregressive model (AR(1)) is defined as(1)Xt=αXt-1+εt,t1,where {εt} is an independent and identically distributed (i.i.d.) random error sequence with probability density function fε, E(εt)=0, Var(εt)=σε2, and {εt} independent of X0 for all t. 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  considered a random coefficient autoregressive model defined as follows:(2)Xt=αtXt-1+εt,t1,where {αt} is a sequence of i.i.d random variables with E(αt)=α and Var(αt)=σα2. {εt} is an i.i.d. random error sequence with mean zero and variance σε2. The random variable X0 is assumed to be independent of {αt}, which is independent of {εt}. Note that when σα2=0, 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  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  restated the necessary and sufficient conditions for the existence of a strict stationarity and ergodicity solution in the RCAR(1) model. Wang and Ghosh  considered the Basyesian estimation method to estimate RCAR model parameter and explored the frequentist properties of the Bayes estimator. Wang et al.  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 , 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  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  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  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. , Chikkagoudar and Biradar , and Manik et al.  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 . 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  and Hettmansperger ). 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. .

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 αt is a constant. To achieve this task, we set up the null and alternative hypotheses(3)H0:σα2=0vsH1:σα2>0.In what follows, we give our main results. Suppose that the time series data x1,x2,,xn are generated from (2). Let {αt} be an i.i.d. sequence of random variables with common probability distribution Fα; {εt} is an i.i.d sequence of random variables with density function fε. In this article, we regard X0 as a given number; alternatively, if X0 is a random variable, we shall consider only inferences conditionally on X0 are fixed. Note that X0, {αt}, and {εt} are independent.

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

(C1) α2+σα2<1.

(C2) The distribution Fα of αt is such that Eαt3<.

(C3) The third and mixed derivatives of logfε with respect to α and σε2 are uniformly bounded on (α,σε2).

(C4) Differentiation thrice with respect to (α,σε2) of fε under the integration is bounded.

It is easy to derive that Xt is a Markov chain on {0,1,2,} with the following transition probabilities: (4)fXtXt-1xtxt-1=fXtXt-1xtxt-1,αtdFα=fεxt-αtxt-1dFα.The Markovity follows from (2) and the fact that {αt} is an i.i.d sequence. The conditional density fXtXt-1(xtxt-1) is obtained by noting that conditional on αt. Conditional on (Xt-1,αt) and {εt} is an i.i.d sequence with the density fε, hence, the probability density function of Xt is given by fε(xt-αtxt-1). So, we get the result in the above equation.

Therefore, we can write down the likelihood function LH1 for RCAR(1) model:(5)LH1x1,,xt=fX0x0t=1nfεxt-αtxt-1dFα.Furthermore, employing this spirit of LMP approach introduced by Manik et al. , we can obtain our test statistic(6)Qnβ=t=1n12fεxt-αxt-1fεxt-αxt-1=t=1nRtβ,where(7)Rtβ=12fεxt-αxt-1fεxt-αxt-1.And the detailed procedures can be found in Appendix A.

Let β=(α,σε2)T denote vector of all unknown parameters of the RCAR(1) process. Hereafter, we use the notation β^=(α^,σ^ε2) to represent the maximum likelihood estimator (MLE) of β. In addition, the actual test statistic can be obtained by replacing β with their MLE β^. It can be easily derived that {Rt(β),FtR} is a zero mean martingale.

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

(A.1) suptERtβ2+η<, for some η>0.

Then about the asymptotic distribution of Qn(β^), we have the following theorem.

Theorem 1.

Let {xt} be a sequence of strictly stationary, ergodic, and Ft-measurable solutions to equation (2) under conditions of (C1)-(C4) and above (A.1). Then under H0, we have asymptotic normality(8)n-1Qnβ^dN0,ω^2,here ω^2=σ2-WTΣ-1W.

(The expressions for , σ2, and Σ-1 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 16 are based on 1000 repetitions with the help of R software. Within each study, we set the initial values X01 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.

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

N ( σ N , α )
( 0.5,0.3 ) ( 0.5,0.5 ) ( 0.5,0.7 ) ( 0.5,0.9 ) ( 1,0.3 ) ( 1,0.5 ) ( 1,0.7 ) ( 1,0.9 )
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

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

N ( m , α )
( 8,0.3 ) ( 8,0.5 ) ( 8,0.7 ) ( 8,0.9 ) ( 10,0.3 ) ( 10,0.5 ) ( 10,0.7 ) ( 10,0.9 )
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

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

N ( σ 1 , σ 2 , α )
( 1,3 , 0.1 ) ( 1,3 , 0.2 ) ( 1,3 , 0.3 ) ( 1,3 , 0.4 ) ( 1,3 , 0.5 ) ( 1,3 , 0.6 ) ( 1,3 , 0.7 ) ( 1,3 , 0.8 ) ( 1,3 , 0.9 )
ξ = 0.8
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
ξ = 0.9
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

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

N ( σ N , a , b )
( 0.5,0.5,0.5 ) ( 0.5,0.5,1 ) ( 0.5,1 , 0.5 ) ( 0.5,1 , 1 ) ( 1,0.5,0.5 ) ( 1,0.5,1 ) ( 1,1 , 0.5 ) ( 1,1 , 1 )
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

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

N ( m , a , b )
( 6,0.5,0.5 ) ( 6,0.5,1 ) ( 6,1 , 0.5 ) ( 6,1 , 1 ) ( 10,0.5,0.5 ) ( 10,0.5,1 ) ( 10,1 , 0.5 ) ( 10,1 , 1 )
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

Empirical powers of LMP and EL tests at nominal level 0.05 for model A3 with (σ1,σ2)=(1,3).

N ( a , b )
( 0.5,0.5 ) ( 0.5,1 ) ( 0.5,1.5 ) ( 1,0.5 ) ( 1,1 ) ( 1,1.5 ) ( 1.5,0.5 ) ( 1.5,1 ) ( 1.5,1.5 )
ξ = 0.8
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
ξ = 0.9
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) αt=α, εt~N(0,σN2).

(N2) αt=α, εt~t(m).

(N3) αt=α, εt~Fε(x). The distribution function of ξ-contamination is(9)Fεx=ξΦxσ1+1-ξΦxσ2,where ξ is a fixed constant satisfying 0<ξ<1 and Φ(x) is the distribution function of standard normal random variable, σi>0,i=1,2.

Let us now describe how the simulated results are obtained. First of all, we use the model N1 to generate data when the σN= 0.5, 1 and α= 0.3, 0.5, 0.7, 0.9. Secondly, we generate data from model N2 with the degrees of freedom m= 8, 10 and α= 0.3, 0.5, 0.7, 0.9. For model N1 and N2, we take the sample size N= 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, and 5000. Thirdly, we simulate samples from model N3 with the (ξ,σ1,σ2,)=(0.8,1,3), (ξ,σ1,σ2,)=(0.9,1,3), and α= 0.3, 0.5, 0.7, 0.9. For model N3, we set the sample size N=100,300,500,1000 and 5000. The empirical levels of three models are presented in Tables 13, respectively. As seen from Tables 13, 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) αt~Beta(a,b), εt~N(0,σN2).

(A2) αt~Beta(a,b), εt~t(m).

(A3) αt~Beta(a,b), εt~Fε(x). The distribution function of ξ-contamination is(10)Fεx=ξΦxσ1+1-ξΦxσ2,where ξ is a fixed constant satisfying 0<ξ<1 and Φ(x) is the distribution function of standard normal random variable, σi>0, i=1,2.

To calculate the empirical powers of the two tests, in the first, we generate samples from the model A1 in which the parameters a= 0.5, 1, b= 0.5, 1, and σN= 0.5, 1. In the second setting, we simulate samples from model A2 with the degrees of freedom m=6,10 the parameters a=0.5,1 and b=0.5,1. For model A1 and A2, we take the sample size N= 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 (ξ,σ1,σ2,)=(0.8,1,3), (ξ,σ1,σ2,)=(0.9,1,3), and the parameters a=0.5,1,1.5, b=0.5,1,1.5. For model A3, we employ the sample size N= 100, 300, 500, 1000, and 5000. The empirical power of the above three models are summed up in Tables 46. 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 N=5000. 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 t, 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 y1,y2,y78. Figure 1 is the sample path plot for the real data yt, t=1,2,,78.

The left one is the sample path of original series yt and the right one is the sample plot of centering series Xt for the real data.

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

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

The normal Q-Q plot of centering series Xt 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 H0:σα2=0 against H1:σα2>0. The value of the test statistic in (8) turned out to be 2.1151, with p 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 (A.1)LH1x1,,xt=fX0x0t=1nfXtXt-1xtxt-1=fX0x0t=1nfεxt-αtxt-1dFα. Now, the Taylor series expansion of LH1 around α which is the mean of αt gives that (A.2)LH1x1,,xt=fX0x0t=1nfεxt-αtxt-1dFα.=fX0x0t=1nfεxt-αxt-1+αt-αfεxt-αxt-1+αt-α22!fεxt-αxt-1+αt-α33!fεxt-αxt-1+dFα=fX0x0t=1nfεxt-αxt-1+12σα2fεxt-αxt-1+fεxt-αxt-1αt-α33!dFα+.Obviously, we can conclude that P(αt-α=0)=1 if σα2=0. That is to say, the distribution function of random variable αt is degenerated. Furthermore, we derive that (A.3)αt-αkk!dFα=0,k=1,2,3.By taking the derivative of logLH1 with respect to β, at σα2=0, we have the following equation:(A.4)logLH1σα2σα2=0=t=1nlogfεxt-αxt-1+1/2σα2fεxt-αxt-1+fεxt-αxt-1αt-α3/3!dFα+σα2σα2=0=t=1nlogfεxt-αxt-1+1/2σα2fεxt-αxt-1σα2σα2=0=12t=1nfεxt-αxt-1fεxt-αxt-1.Thus, the LMP test statistic has the form(A.5)Qnβ=t=1n12fεxt-αxt-1fεxt-αxt-1.This completes the proof.

B. Proof of Theorem <xref ref-type="statement" rid="thm2.1">1</xref> Proof of Theorem <xref ref-type="statement" rid="thm2.1">1</xref>.

In the following, we can write down the log-likelihood function under the null hypothesis: (B.1)logL=t=1nlogfεxt-αxt-1.According to the maximum likelihood principle, the maximum likelihood estimate β^ can be obtained by maximizing the log-likelihood function logL with respect to β. Applying the Taylor series expansion to logfε/β gives(B.2)0=logfεβ=logfεβ+β^-β2logfεβββ,where β is on the line segment between β and β^, so βpβ(n). Under the null hypothesis, we can write (B.3)β^-β=-t=1n2logfεββ+opn-1t=1nlogfεβ.Using the similar method as in the above, expanding (1/n)t=1nRt(β^) around β gives that(B.4)1nt=1nRtβ^=1nt=1nRtβ+β^-βT1nt=1nRtββ+op1, with Rt(β)/β=Rt/α,Rt/σ2T. The remainder term is op(1) since β^pβ. Under (C3) and (C4), we conclude that 2Rt(β)/ββ=Op(n). Therefore,(B.5)β^-βT1nt=1nRtββ=-1nt=1nRtββT×1nt=1n2logfεββ+op1-1×1nt=1nlogfεβ, Now the first term of (B.5) converges to (B.6) 1 n t = 1 n R t β β = 1 n t = 1 n β log f ε σ α 2 σ α 2 = 0 = 1 n t = 1 n 2 log f ε α σ α 2 1 n t = 1 n 2 log f ε σ ε 2 σ α 2 σ α 2 = 0 W a s n , with W=(W11,W12)T. This is because (B.7)W11=Elogfεασα2=E-logfεαlogfεσα2=-12Elogfεα2logfεα2+logfεα2,atσα2=0,W12=Elogfεσε2σα2=E-logfεσε2logfεσα2=-12Elogfεσε22logfεα2+logfεα2,atσα2=0. Next, we can obtain the elements of Fisher information matrix (B.8) Σ = 1 n t = 1 n E log f ε β log f ε β T F t - 1 y = - E 2 log f ε α 2 - E 2 log f ε α σ ε 2 - E 2 log f ε α σ ε 2 - E 2 log f ε σ ε 2 σ ε 2 . Thus (B.5) is asymptotically equivalent to(B.9)-WΣ-11nt=1nlogfεβ. We present the following lemma recommended by Billingsley  to establish the asymptotic normality of the test statistic.

Lemma B.1. Under assumption of ( A . 1 ) , we have as n ,

(i) l i m n 1 / n t = 1 n E ( R t 2 ( β ) F t - 1 R ) = σ 2 , a . s . ; (ii) 1/nt=1nRt(β)dN0,σ2, where σ2=1/4E2logfε/α2+logfε/α2.

Proof. By the ergodic theorem, the first conclusion holds.

Now, let Ft=σ{x0,x1,,xt},t1 and F0 is a sigma field. Note that Qn(β)=t=1nRt(β). It is easy to see that [Rn(β)|Fn-1]=0. Then, we have (B.10)EQnβFn-1=EQn-1β+RnβFn-1=Qn-1β, Thus {Qn,Fn,n0} is a martingale. We have shown suptE|Rt(β)|2+η<, which means E[Rt2(β)] is uniformly integrable. By Theorem 1.1 of Billingsley , we have that as n(B.11)1nt=1nRt2βa.s.ERn2βFn-1=σ2. Hence, using the following version of Martingale Central Limit Theorem from Hall and Heyde , we have 1/nt=1nRt(β)dN(0,σ2). The proof of Lemma B.1 is thus completed.                 □

Similarly, we can verify that Sn=t=1nlogfε/β is a martingale. By ergodic and stationary properties, we obtain that as n, (B.12)1nt=1n2logfεββTa.s.ElogfεβlogfεβT=Σ.Therefore, (1/n)SndN(0,Σ).

In the same way, for any vector c=(c1,c2,c3)TR3(0,0,0) we have(B.13)1ncTt=1nRtβt=1nlogfεβ=1nt=1nc1Rtβ+c2logfεα+c3logfεσε2,dN0,Ec1Rtβ+c2logfεα+c3logfεσε22By the Cramer-Wold device, we obtain (B.14)1nt=1nRtβt=1nlogfεβdN00,σ2WTWΣ.where W=(W11,W12)T. Then we can make the conclusion that(B.15)1nt=1nRtβ^dN0,ω2,where ω2=σ2-WTΣ-1W. 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).

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