Qin and Liu Journal of Inequalities and Applications (2015) 2015:48 DOI 10.1186/s13660-015-0569-8

O Journal of Inequalities and Applications

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RESEARCH

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A robust test for mean change in dependent observations

Ruibing Qin1* and Weiqi Liu2

"Correspondence: rbqin@hotmail.com School of Mathematical Science, Shanxi University,Taiyuan, Shanxi 030006, P.R. China Full list of author information is available at the end of the article

Abstract

A robust test based on the indicators of the data minus the sample median is proposed to detect the change in the mean of a-mixing stochastic sequences. The asymptotic distribution of the test is established under the null hypothesis that the mean ¡l remains as a constant. The consistency of the proposed test is also obtained under the alternative hypothesis that ¡l changes at some unknown time. Simulations demonstrate that the test behaves well for heavy-tailed sequences. MSC: Primary 62G08;62M10

Keywords: change point; median; robust test; consistency

ft Spri

ringer

1 Introduction

The problem of a mean change at an unknown location in a sequence of observations has received considerable attention in the literature. For example, Sen and Srivastava [1], Hawkins [2], Worsley [3] proposed tests for a change in the mean of normal series. Yao [4] proposed some estimators of the change point in a sequence of independent variables. For serially correlated data, Bai [5] considered the estimation of the change point in linear processes. Horvath and Kokoszka [6] gave an estimator of the change point in a long-range dependent series.

Most of the existing results in the statistic and econometric literature have concentrated on the case that the innovations are Gaussian. In fact, many economic and financial time series can be very heavy-tailed with infinite variances; see e.g. Mittnik and Rachev [7]. Therefore, the series with infinite-variance innovations aroused a great deal of interest of researchers in statistics, such as Phillips [8], Horvath and Kokoskza [9], Han and Tian [10,11]. It is more efficient to construct robust procedures for heavy-tailed innovations, such as the M procedures in Huskova [12,13] and the references therein. De Jong et al. [14] proposed a robust KPSS test based on the 'sign' of the data minus the sample median, which behaves rather well for heavy-tailed series. In this paper, we shall construct a robust test for the mean change in a sequence.

The rest of this paper is organized as follows: Section 2 introduces the models and necessary assumptions for the asymptotic properties. Section 3 gives the asymptotic distribution and the consistency of the test proposed in the paper. In Section 4, we shall show the statistical behaviors through simulations. All mathematical proofs are collected in the Appendix.

© 2015 Qin and Liu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalworkis properly credited.

2 Model and assumptions

In the following, we concentrate ourselves on the model as follows:

Yt = f(t)+Xt, f(t)

f 1, t < k0, f 2, t > ko,

where ko is the change point.

In order to obtain the weak convergence and the convergence rate, X(t) satisfies the following.

Assumption 1

1. The Xj are strictly stationary random variables, and f is the unique population median of {Xt,1 < t < T}.

2. The Xj are strong (a-) mixing, and for some finite r >2 and C > 0, and for some n >0, a(m) < Cm-r/(r-2)-n.

3. Xj - f has a continuous density/(x) in a neighborhood [-n, n] of 0 for some n >0, and Mxe[-n,n]/(x) > 0.

4. a2 e (0, to), where a2 is defined as follows:

To derive the CLT of sign-transformed data, we need a kernel estimator, so we make the following assumption on the kernel function.

Assumption 2

1. k(-) satisfies /-TO (%)l d% < to, where

2. k(x) is continuous at all but a finite number of points, k(x) = k(-x), \k(x) \ < l(x) where l(x) is nondecreasing and /0to \l(x)\ dx < to, and k(0) = 1.

3. yT/T ^ 0, and yT —^ to as T ^to.

Remark 1 De Jong et al. [14] test the stationarity of a sequence under Assumption 1. We detect change in the mean of a sequence, so Assumption 1 holds under the null hypothesis and the alternative one. Since there is no moment condition for Xt in Assumption 1, even Cauchy series are allowed. The a-mixing sequences can include many time series, such as autoregressive or heteroscedastic series under some conditions. Assumption 2 allows some choices such as the Bartlett, quadratic spectral, and Parzen kernel functions.

3 Main results

Let mT = med{ Y1,..., YT}. Then we transform the data Y1,..., YT into the indicator data sgn(Yt - mT), where sgn(x) = 1 if x >0, sgn(x) = -1 if x <0, sgn(x) = 0 if x = 0. Based on these indicator data, De Jong et al. [14] replace et = Yt - YT with sgn(Yt - mT) in the usual KPSS test and their simulations show that the new KPSS test exhibits some robustness for the heavy-tailed series.

lim El T-1/2 Vsgn(Xt - fx)

OO I ' J

The popularly used test to detect a mean change is based on the CUSUM type as follows:

T T ] 7=1 T (1- T)] t=iTtl+1 J

We rewrite ET(t) under H0 as

E [TT][T(1- z)] I 1 H 1 ^ J

ET"~ N irY-T^c1-^ ,=^.1r-[

W-Z^kk E <Y- - Yt)-T^ E (Yt - Yt (3)

I L J t=1 L v " t=[Tt]+1 J

According to the idea of De Jong et al. [14], replace et = Yt - YT with sgn( Yt - mT)in (3); then we get a robust version of CUSUM as follows:

[Tx][T(1 - x)] [ 1 [tt] 1 T ]

ET=—T2— (TJ gsgn(Yt -mT) - [T^ s^{Yt -mT)\. (4)

Then the test statistic proposed in this paper is

Tt = T1/2a-1 max IEt(t)|. (5)

t e(0,1) 1

Under Assumptions 1, 2, we can obtain two asymptotic results as follows. Theorem 1 I/Assumptions 1, 2 hold, then under the null hypothesis H0, we have

T1/2 a-1 max\Et \ sup|W(t)-t W(1)|, as T —to, (6)

t e(0,1) t e(0,1)

where '=}■' stands/or the weak convergence.

Under the alternative hypothesis H1, a change in the mean happens at some time, we denote the time as [Tt0]. Let F(•) be the common distribution function of Xt and f * be the median of

F*(•) = T0F(• - f 1) + (1 - T0)F(• - f 2). (7)

Then we have the following. Theorem 2 I/Assumptions 1, 2 hold, then under the alternative hypothesis H1, we have max 1 Et(t)| —— T0(1- T0)\A\, (8)

t e(0,1) 1

where A = F(f * - f 1) - F(f * - f 2).

Remark 2 By Theorem 1, we reject H0 if TT > cp, where the critic value cp is the (1 -p) quantile of the Kolmogorov-Smirnov distribution. By Theorem 2, TT is consistent asymp-

totically as the sample size T — to.

In order to apply the test in (5), we employ the HAC estimator instead of the unknown a2 as

&T = TK(i -Wyt) sgn(Yi - mT) sgn(Y - mT), (9)

1=1 j=i

then the following theorem proves two results of the estimator &T under H0 and HA, respectively.

Theorem 3 (i) Assuming that the conditions of Theorem 1 hold, then we have, asT — <x,

&T - a2. (10)

(ii) Assuming that the conditions of Theorem 2 hold, then we have, asT — <x,

oT - aT, (1)

where a22 is defined as follows:

a2 = Um ^T-1/2 sgn(Yt - fi*)J .

4 Simulation and empirical application 4.1 Simulation

In this section, we present Monte Carlo simulations to investigate the size and the power of the robust CUSUM and the ordinary CUSUM tests. Since a lot of information has been lost during the inference by using the indicator data instead of the original data, so we are concerned whether the indicator CUSUM test is robust to the heavy-tailed sequences; moreover, we may ask: how large is the loss in power in using indicators when the data has a nearly normal distribution? The HAC estimator a2 in the robust CUSUM test is a kernel estimator, so it is important to analyze whether the performance is affected by the choice of the kernel function k(-) and the bandwidth yT.

We consider the model as follows:

« = l0^ t T (12)

[¡2+ Xt, t > Tto,

Xt is an autoregressive process Xt = 0.5Xt_i + et, where the {et} are independent noise generated by the program from JP Nolan. We vary the tail thickness of {et} by the different characteristic indices a = 1.97,1.83,1.41,1.14, respectively. Accordingly the break times are to = 0.3,0.5, respectively. During the simulations, we adopt 1.358 as the asymptotic critical value of supT e(01) \W(r )-t W (1)| at 95% for the various sample sizes T = 300,500,1,000.

First, we consider the size of the tests. Tables 1 and 2 report the results when a2 are estimated by the Bartlett kernel and the quadratic spectral kernel with the bandwidth Yt = [4(T/100)1/4] and yt = [8(T/100)1/4], respectively, in 1,000 repetitions. From Tables 1 and 2, the ordinary CUSUM test based on the Bartlett kernel has better sizes, however,

Table 1 The empirical levels of the robust CUSUM test and the CUSUM test for dependent innovations

CUSUM RCUSUM

T = 300 T =500 T = 1,000 T = 300 T =500 T =1,

The tests based on the Bartlett kernel function

a =1.97 0.045 0.026 0.036 0.042 0.046 0.059

a =1.83 0.028 0.028 0.033 0.037 0.032 0.043

a =1.41 0.010 0.010 0.025 0.030 0.036 0.044

a = 1.14 0.005 0.010 0.008 0.045 0.049 0.048

The tests based on the quadratic spectral kernel function

a =1.97 0.471 0.491 0.489 0.068 0.048 0.050

a =1.83 0.428 0.462 0.478 0.062 0.077 0.063

a =1.41 0.458 0.449 0.486 0.066 0.072 0.053

a =1.14 0.474 0.476 0.507 0.083 0.073 0.055

The values in Table 1 are based on the bandwidth yt = [4(T/100)1/4].

Table 2 The empirical levels of the robust CUSUM test and the CUSUM test for dependent innovations

CUSUM RCUSUM

T = 300 T =500 T = 1,000 T = 300 T =500 T =1,

The tests based on the Bartlett kernel function

a = 1.97 0.028 0.032 0.034 0.034 0.033 0.046

a = 1.83 0.019 0.032 0.023 0.034 0.037 0.037

a = 1.41 0.009 0.013 0.021 0.035 0.038 0.048

a = 1.14 0.004 0.008 0.01 0.038 0.036 0.047

The tests based on the quadratic spectral kernel function

a = 1.97 0.425 0.447 0.470 0.037 0.043 0.040

a = 1.83 0.414 0.444 0.456 0.026 0.043 0.048

a = 1.41 0.484 0.463 0.483 0.040 0.035 0.041

a =1.14 0.459 0.490 0.454 0.028 0.048 0.042

The values in Table 2 are based on the bandwidth yt = [8(T/100)1/4].

the one based on the quadratic spectral kernel has a severe problem of overrejection, so we can conclude that the choice of the kernel function has higher impact on the sizes of the two CUSUM tests than the selection of the bandwidth. Comparing the two tests based on the Bartlett kernel, the ordinary CUSUM test becomes underrejecting as the tail index a changes from 2 to 1, and the sizes of the robust test are closer to the nominal size 0.05. Furthermore, the size is closer to 0.05 as the sample size T increases, which is consistent with Theorem 1.

Now we shall show the power of the two tests through empirical powers. The empirical powers are calculated based on the rejection numbers of the null hypothesis H0 in 1,000 repetitions when the alternative hypothesis H1 holds. The results are included in Tables 3, 4,5,6. On the basis ofTables 3,4,5,6, we can draw some conclusions. (i) The two CUSUM tests based on the Bartlett kernel and the quadratic spectral kernel become more powerful as the sample size T becomes larger. (ii) As the tail of the innovations gets heavier, the ordinary CUSUM test becomes less powerful, especially, the test hardly works, while the CUSUM test based on indicators is rather robust to the heavy-tailed innovations. (iii) The selection of the bandwidth has lower impact on the powers of the two CUSUM tests.

Finally, we consider the effects of the skewness in the innovations {et} on the power of the proposed test through simulations. In order to obtain the results reported in Table 7, we take the e(t) in the model (12) as chi square distributions with a freedom degree n =

Table 3 The empirical powers of the robust CUSUM test and the CUSUM test for dependent innovations

CUSUM RCUSUM

T = 300 T =500 T = 1,000 T = 300 T =500 T =1

The change point r0 = a =1.97 0.849 0.3 0.991 0.998 0.951 0.999 1.000

a =1.83 0.692 0.919 0.977 0.964 1.000 1.000

a =1.41 0.222 0.361 0.530 0.957 0.995 1.000

a =1.14 0.047 0.065 0.076 0.964 0.998 1.000

The change point r0 = a =1.97 0.988 0.5 0.997 0.997 0.991 1.000 1.000

a =1.83 0.913 0.966 0.979 0.985 1.000 1.000

a =1.41 0.360 0.531 0.651 0.994 1.000 1.000

a =1.14 0.097 0.108 0.133 0.996 1.000 1.000

The change point r0 = a =1.97 0.972 0.7 0.995 0.999 0.958 0.999 1.000

a =1.83 0.875 0.944 0.978 0.962 0.997 1.000

a =1.41 0.300 0.446 0.542 0.964 0.999 1.000

a =1.14 0.063 0.080 0.104 0.972 1.000 1.000

The values in Table 3 are based on the Bartlett kernel and the bandwidth yt = [4(T/100)1/4].

Table 4 The empirical powers of the robust CUSUM test and the CUSUM test for dependent innovations

CUSUM RCUSUM

T = 300 T =500 T = 1,000 T = 300 T =500 T = 1,000

The change point to = 0.3

a =1.97 0.348 0.848 0.995 0.921 1.000 1.000

a =1.83 0.241 0.676 0.953 0.931 0.993 1.000

a =1.41 0.111 0.242 0.409 0.944 0.997 0.997

a =1.14 0.029 0.056 0.080 0.943 1.000 1.000

The change point t0 = 0.5

a =1.97 0.931 0.995 0.997 0.993 1.000 1.000

a =1.83 0.796 0.954 0.985 0.989 1.000 1.000

a =1.41 0.285 0.456 0.605 0.990 1.000 1.000

a =1.14 0.057 0.088 0.106 0.989 1.000 1.000

The change point t0 = 0.7

a =1.97 0.937 0.997 0.997 0.949 1.000 1.000

a =1.83 0.783 0.926 0.969 0.934 1.000 1.000

a =1.41 0.238 0.373 0.553 0.938 0.997 1.000

a =1.14 0.046 0.068 0.094 0.948 0.997 1.000

The values in Table 4 are based on the Bartlett kernel and the bandwidth yt = [8(T/100)1/4].

1,2 and 10, respectively. On the basis of the simulations, the skewness of the innovations affects the powers the two CUSUM test significantly.

4.2 Empirical application

In this section, we take an empirical application on a series of daily stock price of LBC (SHANDONG LUBEI CHEMICAL Co., LTD) in the Shanghai Stocks Exchange. The stock prices in the group are observed from July 1st, 2004 to December 30th, 2005 with samples of 367 observations (as shown in Figure 1) and can be found in http://stock.business.sohu.com. As in Figure 2, the logarithm sequence is seen to exhibit a number of 'outliers', which are a manifestation of their heavy-tailed distributions, see Wang et al. [15]; the data can be well fitted by stable sequences.

Table 5 The empirical powers of the robust CUSUM test and the CUSUM test for dependent innovations

CUSUM RCUSUM

T = 300 T =500 T = 1,000 T = 300 T =500 T = 1,000

The change a =1.97 point r0 = 0.979 0.3 1.000 1.000 0.869 0.964 0.999

a =1.83 0.957 0.995 0.996 0.847 0.957 0.994

a =1.41 0.824 0.882 0.917 0.729 0.855 0.963

a =1.14 0.644 0.672 0.652 0.574 0.753 0.895

The change a = 1.97 point r0 = 0.998 0.5 0.999 1.000 0.939 0.983 1.000

a = 1.83 0.982 0.994 0.992 0.915 0.979 0.998

a = 1.41 0.802 0.826 0.889 0.805 0.929 0.996

a = 1.14 0.604 0.593 0.646 0.670 0.819 0.943

The change a = 1.97 point r0 = 0.993 0.7 1.000 1.000 0.873 0.961 0.996

a = 1.83 0.736 0.773 0.845 0.820 0.947 0.999

a = 1.41 0.736 0.773 0.845 0.717 0.867 0.972

a = 1.14 0.570 0.556 0.594 0.577 0.731 0.878

The values in Table 5 are based on the quadratic spectral kernel and the bandwidth yj = [4(J/100)1/4].

Table 6 The empirical powers of the robust CUSUM test and the CUSUM test for dependent innovations

CUSUM RCUSUM

T = 300 T =500 T = 1,000 T = 300 T =500 T = 1,000

The change point r0 = 0.3

a = 1.97 0.467 0.881 1.000 0.808 0.941 0.999

a = 1.83 0.521 0.874 0.993 0.764 0.920 0.995

a = 1.41 0.658 0.770 0.893 0.582 0.788 0.961

a = 1.14 0.565 0.629 0.668 0.440 0.642 0.847

The change point r0 = 0.5

a = 1.97 0.974 0.999 1.000 0.891 0.967 0.997

a = 1.83 0.958 0.987 0.994 0.866 0.969 0.999

a = 1.41 0.792 0.860 0.897 0.726 0.876 0.992

a = 1.14 0.594 0.640 0.631 0.568 0.720 0.921

The change point r0 = 0.7

a = 1.97 0.992 1.000 1.000 0.782 0.924 0.997

a = 1.83 0.974 0.981 0.992 0.802 0.924 0.990

a = 1.41 0.749 0.800 0.881 0.604 0.756 0.942

a = 1.14 0.544 0.580 0.590 0.448 0.598 0.838

The values in Table 6 are based on the quadratic spectral kernel and the bandwidth yj = [8(j/100)1/4].

Table 7 The empirical powers of the two CUSUM test for the skewed dependent innovations

CUSUM RCUSUM

X 2(1) X 2 (2) X 2(10) X2(1) X 2 (2) X2(10)

T0 = 0.3 0.9400 0.6690 0.3550 0.0 0.6760 0.2090

T0 = 0.5 0.9940 0.8130 0.4270 0.0350 0.8280 0.2880

T0 = 0.7 0.9900 0.7140 0.3480 0.0150 0.7530 0.2250

The values in Table 7 are based on the Bartlett kernel and the bandwidth yj = [4(j/100)1/4].

4 ViAT ~

3 V» \J V

50 100 150 200 250 300 350 400 the original stocks prices of LBHG

Figure 1 Stock prices of LBC in Shanghai Stock Exchange.

Figure 2 The logarithm return rates of LBC in Shanghai Stock Exchange.

Fitting a mean and computing the test proposed in this paper r1 = 4.2123 > 1.358, which indicates that a change in mean occurred, and ET(k) attains its maximum at k0 = 175 (21st, March, 2004) (as shown in Figure 3). Recall that LBC issued an announcement that its net profits in 2005 would decrease to 50% of that in 2004, in the 3rd Session Board of Directors' 17th Meeting on March 8th, 2005 (k1 = 166). The influence of the bad news was so strong that the stock price fell immediately in the following nine days, the mean of the logarithm return rate has a significant change after k0 = 175.

5 Concluding remarks

In this paper, we construct a nonparametric test based on the indicators of the data minus the sample median. When there exists no change in the mean of the data, the test has the usual distribution of the sup of the absolute value of a Brownian bridge. As Bai [5] pointed out, it is a difficult task in applications of autoregressive models. First, the order

180 N

i ' ' '

160 - / \

140 - / \

120 - / \

100 - / \

80 - / \

60 - / \

40 - / \

20 - / \

50 100 150 200 250 300 350 400 the robust CUSUM values of the LBHG

Figure 3 The robust CUSUM values of LBC in Shanghai Stock Exchange.

of an autoregressive model is not assumed to be known a priori and has to be estimated. Second, the often-used way to determine the order via the Akaike information criterion (AIC) and the Bayes information criterion (BIC) tends to overestimate its order if a change exists. However, the proposed test does not rely on the precise autoregressive models and the prior knowledge on the tail index a, so the proposed test is more applicable, although there exists a little distortion in its size for dependent sequences.

Appendix: Proofs of main results

The proof of Theorem 1 is based on the following four lemmas.

Lemma 1 For Lr-bounded strong (a-) mixing random variables yTt e R, for which the mixing coefficients satisfy a(m) < Cm-r/(r-2)-n for some n > 0,

EZ(yTt - EyTt) < C'J^WyTt II2 (13)

, , yrt N ' ^ — \ t=i / t=i

for constants C and C, where ||X|| = (E|X|r)1/r.

This lemma is Lemma 1 in De Jong et al. [14]; it is crucial for the proof of the following lemmas and theorems.

Lemma 2 Let

yj(0) = sgn(Yj - f 1 - fx - 0T-1/2) - sgn(Y - f 1 - f). (14)

If the null hypothesis H0 holds, then under Assumption 1, for all K, e > 0,

lim lim sup P[ sup T-1/2 T|yj (0)-yfa') - Ey,(0) + Ey;(</>') | > e J =0.

T^rn y0,0'e[-K,K]:|0-0'|<ä t=1 )

Proof Since the proof is similar to Lemma 2 of De Jong et al. [14], we omit it. □

Lemma 3 Letyj(4>) be as in (14), and let

[T t ]

Gt (t , 4>) = T-1/2£ (16)

If the null hypothesis H0 holds, then under Assumption 1, for any K > 0,

sup sup |GT(t,$)-EGT(t,^ 0. (17)

t e[0,1] 0e[-K,K]

Proof The proof is similar to Lemma 3 ofDe Jong etal. [14], so we omit it. □

Lemma 4 If the null hypothesis H0 holds, then under Assumption 1,

T1/2(mT - /1 - ß) = 2-1f (0)-1a Wt(1) + op(1). (18)

Proof The proof is similar to Lemma 4 of De Jong et al. [14], so we omit it. □

Proof of Theorem 1 According to Lemma 4, we can find a large K so that -K < T1/2 (mT -ii -A) < K. Then

[Tt] [Tt]

T-1/2St,[tt] = T-1/2£ sgn( Y; - mT) = T-1/^ sgn((Yj - / - ß)- (mT - / - AO)

j=1 j=1

= Gt(r, T1/2(mT - /1 -A)) -EGT(r, T1/2(mT - /1 -A))

+ T-1/2 ^ sgn(Yj - i1 - A)- 2T-1/2[Tr](mT - /1- A )f (fnT - i1 - A) j=1

= I1+12-13, (19)

where mmT is on the line between mT and / + A and mmT - i1 - A = oP(1) by Lemma 4. Then I1 = oP(1) holds uniformly for all r e [0,1] by Lemmas 3, 4. By definition, I2 = a WT(r). I3 = ra WT(1) + oP(1) by Lemma 4. So we have

T-1/2St,[tt ] = a (Wt (r) - r Wt (1)) + op(1). (20)

Noting that | T-1/2 Em sgn( Y - mT) |< T-1/2, we have

~~7TST,[T(1-r)] = T 1/2 J2 sgn(Yj - mT) j=[T t ]+1

T [Tr ]

= T-1I2J2 sgn(Yj - mT) - T"1/2 £ sgn( Y - mT)

j=1 j=1

= O(T-1/2) - \ Gt(r, T1/2(mT - i1 - A)) - EGt(r, T 1,2(mr - / - A))

+ T-1/2£] sgn(Yj - fn- ¡l) j=i

- 2T-1/2 [Tt](fflT - fi - l)f (mt - fi - l) J = 0(T-1/2) - op(1)- {oWt(t)- TOWT(1)}. (21)

Based on (20), (21), by the functional central limit theorem,

T1/2 a-1 max |St | sup|W(t)-t W(1)|, as T -to. (22)

t e(0,1) t e(0,1)

If we can show a 2 -— a2, the proof of Theorem 1 is completed. Under the null hypothesis H0, ¡1 remains as a constant, so we can prove the consistency of a2 just as De Jong etal. [14]. □

The proof of Theorem 2 is based on Lemmas 1, 5, 6, 7 as follows.

Lemma 5 Ifthe alternative hypothesis H1 holds and k0 = [T t0] is the change point, letyj(0) be as follows:

y(0) = sgn(Y - l - 0T-1/2) - sgn(Yj - f), (3)

then under Assumption 1, for allK, e > 0,

limlimsup P[ sup T-1/2 T|yj(0)-yj(0') - j) + Ey(0') | > e J =0.

0 T-to \0,0'e[-K,K]:|0-0'< j=1 )

Proof For yj(0) as in (23), we have

E \2F (l - ¡1) - 2F (f + 0T-1/2 - ¡1), j < ko, (25)

yj(0) [2F(f - ¡2)-2F(f + 0T-1/2- ¡2), t > ko. ( )

Then for T large enough such that KT-1/2 < n, under the alternative hypothesis H1,

sup T-1I2J2 |Eyj(0)-Eyj(0')|

0,0':|0-0'< j=1

< sup T-1/2^|Eyj(0)-Eyj(0O | + sup T-1/2 |Eyj(0)-Eyj(0') |

0,0':|0-0'|<5 j=1 0,0/:|0-0'|<a j=k0+1

= I11+ I12, (6)

I11 = 2 sup T-1/2£|Eyj(0) - Eyj(0<) |

0,0':|0-0'|<5 j=1

= 2 sup T-mJ2\FI* + <T-1/2- n)- F(IX* + <'T-1/2- n) \

<p0'\\<-0\<5 y=1

< 2 sup T-1/2£ sup f (x)T-1/2\< - 0'\ < 25 sup f(x), (27)

0o':o-0'\<5 y=1 xe[-m,m] xe[-n1,m]

¡12 = 2 sup T\Ey №) - Ey \

= 2 sup T-1/2 ^ \F(|* + <T-1/2 -12) - F(|* + <'T-1/2 -12) \

< 2 sup T-1/2 ^ sup f (x)T-1/2\< - <'\ < 25 sup f (x), (28)

<,<':\<-<'\<5 j=k0+1 xe[-n2,n2] xe[-n2,n2]

where n stands for different constants at different equations. This establishes equiconti-nuity of ¡11 and ¡12. Similar to (26), we have

sup T-1/2£ \yj(<)-y(<')\

0o'e[-K,K]:<-0'\<5 j=1

k0 \ ( )\ < sup T-1/2£ \y(<)-yj «0 \

0o'e[-K,K]:<-0'\<5 j=1

+ sup T-1I2J2 \y;(<)-y;(<')\

<,<'e[-K,K]:\<-<'\<5 ¡=ko+1

= ¡21+ ¡22. (9)

Since y ¡(0 is non-increasing in <,

k0 \ ( )\

¡21= sup T-1/^\yj(0)-yj(<') \

0,0'e[-K,K]:\<-0'\<5 ¡=1

k0 \ ( )\

= sup sup T-1/2V\y,'(0)-y,'(<') \

-[K/5]-1<i<[K/5] <,0'e[-K,K]n[i5,(i+2)5] ¡=1

k0 \ ( )\

< sup T-1/^ \y(i5)-y, ((i + 2)5) \

-[K/5]-1<i<[K/5] ¡=1

k0 \ ( )\ = op(1) + sup T-1/2J2E\yj(i5) -y((i + 2)5) \

-[K/5]-1<i<[K/5] ¡=1

k0 \ ( )\

< op(1)+ sup T-1/2VE\y;(0)-y;(0')\, (30)

0,0'e[-K,K]:\<-0'\<25 ¡=1

and the last term has been proved earlier to be equicontinuous. Similarly y,(<) is non-increasing in <, we have

I22= sup T-112 J2 |yj(0)-yj(0')|

0,0'e[-K,K]:|0-0'|<5 j=ko+1

= sup sup T-1I2J2 |yj(0)-yj(0')|

-[K/5]-1<i<[K/5] 0,0'e[-K,K]n[i5,(i+2)5] j=ko+1

< sup T-1/^ |yj(i5)-yj((i + 2)5)|

-[k/a]-1<i<[k/a] j=ko+1

= op(1) + sup T-1/2VE|yj(i5) -y,((i + 2)5)|

-[K/5]-1<i<[K/5] j=1

< op(1)+ sup T-1/2 v E|yj(0)-yj(0')|, (31)

0,0'e[-K,K]:|0-0'|<25 j=ko+1

and the last term has been proved earlier to be equicontinuous too. By the triangle inequality, for all e > o,

p( sup T-1/2 T|yj(0)-yj(0') -Eyj(0)+Ey;(0')| >e)

\0,0'e[-K,K]:|0-0'|<5 j=1 )

< p( sup T-1/2 |yj(0) -y;(0') -Eyj(0) + Eyj(0') | > e/2)

\0,0'e[-K,K]:|0-0'|<5 j=1 )

+ p( sup T-1/2 T |yj(0) - yj(0') - Eyj(0) + Eyj(0') | > e/2 J

\0,0'e[-K,K]:|0-0'|<5 j=ko+1 j

= I31+132, (2)

131 = p( sup T-1/2 ¿|y,(0) -yj(0') -Eyj(0) + Eyj(0') | > e/2)

\0,0'e[-K,K]:|0-0'|<5 j=1 j

< p( sup T-1/2 ¿|y,(0)-y,(0')|> e/4)

\0,0'e[-K,K]:|0-0'|<5 j=1 j

+ p( sup T-1/2 v|Eyj(0)-Eyj(0')|> e/4)

\0,0'e[-K,K]:|0-0'|<5 j=1 j

< op(1)+ p( sup T-1/2 ¿|Eyj(0)-Eyj(0')|> e/A (33)

\0,0'e[-K,K]:|0-0'|<5 j=1 j

the last term converges to o as 5 — o by the equicontinuity of (27). Similarly, we can show

132 = P( sup T-1/2 T |yj(0)-yj(0') -Eyj(0)+Ey;(0')| > e/2 J

\0,0'e[-K,K]:|0-0'|<5 j=ko+1 )

< P( sup T-1/2 T |yj(0) -yj(0') | > e/4 J

\0,0'e[-K,K]:|0-0'|<5 j=ko+1 j

+ l( sup T-1/2 ¿|Eyj(|)-Ey,(|')|> e/4 j

\|,|'e[-K,K]:\|-|'\<s j=ko+1 )

< Op(1) + 4 sup T-1/2 T |Eyj(|)-Ey,(|') | > e/4 ), (34)

\|,|'e[-K,K]:\|-|'\<s j=ko+1 /

the last term converges to 0 as S — 0 by the equicontinuity of (28) too. Now, we have completed the proof of Lemma 5. □

Lemma 6 If the alternative hypothesis H1 holds, letyj(l) be as in (23), and

[T t ]

Gt (t , |) = T-1/2£ yj(|), (35)

then under Assumption 1, for any K > 0,

sup sup |GT(t,l)-EGT(t,|)| — 0. (36)

t e[0,1] |e[-K,K]

Proof Just as De Jong etal. [14], we can obtain from Kim and Pollard [16, Theorem 2.1] sup sup |Gt(t, I) -EGt(t, |)| 0 (7)

t e[0,1] |e[-K,K]

through the arguments for the finite-dimensional convergence for each | e [-K, K] and the stochastic equicontinuity of supTe[01] \GT(t, |)-EGT(t, |)\. For every | e [-K,K], by Lemma 1, for T large enough such that KT-1/2 < n, we have

| |2 E sup |GT(t, l)-EGT(t, |)|

t e[0,1]

T ( ) ( )

< CT-1J2 |MYj - f * - IT-1/2) - sgn(Yj - f *) ||r

k0 ( ) ( )

= CT1 sgn(Xj + f 1 - f * - IT- sgn(Xj + f 1 - f *) |2

+ CT-1 £ ||sg^( Xj + f2 - f* - IT-m) - sgn(Xj + f2 - f*) 12

j=k0+1

< C'T0 |F(f * - f 1 + KT-1/2) - F(f * - f 1 - KT-1/2) |2/r

+ C'(1 - T0) |F(f * - f 2 + KT-1/2) - F(f * - f 2 - KT-1/2) |2/r

< C"t^ sup f (x)2KT-1/^2/r

+ C"(1- T0)( sup f (x)2KT-1/2^2/r, (8)

where constants C, C', C" > o. Now we have shown the finite-dimensional convergence for each 0 e [-K, K]. By the triangle inequality,

sup 1GT(t,0)-EGT(t,0)1 - sup !Gt(t,0') -EGT(r,0')1

t e[o,1] r e[o,1]

< sup 1GT(r,0)-EGT(r,0)-GT(r,0') + EGT(r,0')1

r e[o,1]

< T-1I2J2|yj(0) -yj(0O -Eyj(0) + Eyj(0') |. (39)

Now stochastic equicontinuity follows from Lemma 5. □

Lemma 7 f the alternative hypothesis H1 holds, then under Assumption 1,

T1/2 (mT - ¡*) = Op (1), (4o)

where ¡* is defined as the median of (7). Proof For T large enough such that T > K2n-2,

sup T-1/2 T sgn(Y - ¡* - 0T-1/2)

0>K j=1 T ( )

< Tsgn(Yj - ¡* - KT-1/2)

= T-1/2 ^ sgn(Xj + ¡1 -¡* - KT-1/2) j=1

T ( ) + T-1/2 ^ sgn(Xj + ¡2 - ¡* -KT-1/2)

j=ko+1

< op(1) + T-1/2 - 2F(KT-1/2 + ¡* - n))

T ( ( )) + op(1) + T(1 - 2F(KT-1/2 + ¡* - ¡2))

j=ko+1

= op (1) + T1/2 (1 - 2F >* + KT-1/2))

= op(1) - 2K inf f *(x), (41)

-n<(x-^*)<n

which implies that limsupT—TOp(T1/2(mT - ¡*) > K) can be made arbitrarily small by choosing K large enough under the alternative hypothesis H1. For p(T1/2(mT - ¡*) < -K), a similar result can be derived, which proves that mT - ¡* = Op(T-1/2) under the alternative hypothesis H1. □

proof of Theorem 2 We just consider the case k = [T r ] > ko = [T ro], the case k < ko can be analyzed similarly. According to Lemma 7, we can find K large enough so that -K <

T l/2(mT - p*) < K will happen with arbitrarily large probability. Then

[T t ]

T-1St,[tt ] = T sgn(Y - mT) j=i

= Tsgn((Xj + pi - p*) - (mT - p*))

[T t ]

+ T-i £ sgn((Xj + - p*) - {mT - p*))

j=[T to]+l

= T-1/2(Gt(t, T1/2(mT - p*^ -£GT(t, Tl/2(mT - ^*)))

[T t ]

+ T-1J2 sgn(Yj - p*) + T-1/2£Gt(t, T1/2 (mT - p*))

= /l+ /2+ /3. (2)

Then /l = Op(l) by Lemma 6, /3 = oP(l) by (25), with Proposition 2.8 of Fan and Yao [17], we have

T-1St,[tt] — ToF(p* - pi) +(t - To)F(p* - p2). Similarly, we can obtain

T-lS2r,[Tt] = TJ2 sgn(Y - mT)

j=[T t ]+l

— (l- t)F(p* - P2). (3)

By the definition of ET(t),

3t(t) — (l - t)To |F(p* - pi) - F(p* - P2) | (44)

as T — to, and t > t0, so

sup |3t(t)| — To(l- To)|F(p* - pi)- F(p* - p^|. (45)

t e(0,l) Q

Proof of Theorem 3 Under the hypothesis Ho, there is no shift in the mean, so the proof of (i) is nearly similar to the proof of the consistency of aT, so we just gave the details of the proof of (ii). Noting that for yj defined in (23),

sgn(Y - mT) = (yj(T1/2(mT - p*)) -Ey(Tm(mT - p*))) + Eyj(Tm(mT - p*)) + sgn(Y - p*)

= aTj + bTj + Cj, (6)

T T ( )

a2 = T-1 k ((i - j)/YT) (an + bK + Ci)(aT, + br, + cj). (47)

i=1 j=1

Under the assumption that yt — to, YT/T — 0, y|/T — 0 as T — to, for yj(|) defined

in (23), by Lemma 7, we have bTi = 0P(T-1/2), then

T T ( ) T T

T-1 E Z)k((i - j)/YT)bTiarj < T\ajj\ £ k(t/yr) x Op(1) = Op(yt/T),

i=1 j=1 j=1 t=-T

T T ( ) T T ( ) ( )

T-1 EEk((i-j)/Yr)bTibTj < C2T-1 EEk((i-»/yt) = O^T-1y^,

i=1 j=1 i=1 j=1

and that

T T ( ) T T ( )

T-1 EEk((i-/)/Y^)bnCj < T-3/2 £„£k((i -,)/Yt) x Op(1)

i=1 j=1 j=1 i=1

e(T-3/2 Cj¿k(j-s)/yr)) < CT-3 £

j=1 s=1 j=1

c^k((/ - s)/yt )

< CT-3 £ (¿k(j - s)/YT))

j=1 s=1

T / T \ 2

< C"T£k(t/Yr)

j=1 t=-T

< C"T-2y^.

Therefore under Assumption 1 and the alternative hypothesis H1, a2 is asymptotically equivalent to

T 1 ^ ^k ((t - s)/yt) (art + Cs)(ars + cs).

t=1 s=1

Furthermore,

T T ( )

T((t - s)/Yr)artCs

t=1 s=1

= f I T-1 ^T^artCsexp(if(t-s)) J f (f)df

t=1 s=1

< T\art \f \T-mJ2 Cs exp(-isf/ytM f (f) df,

J-to \ „i /

t=1 '-to \ s=1

and T 1/2 J2h \aTt I = oP(1) by Lemma 6 under Assumption 1,andthesecondtermis OP(1) because

i( T-1/21

J-TO \ .=1

|f(t)| dt

c, exp(-is| /yt )j f (i) dt

Tcs exp(-is|/yt )

Finally,

T"1 E Ek((t - s)/YT)aTtaTs <i°° | f (I) | dJ T"1/2

t=1 s=1 \ t=1

T2 \aTtI I

by the last term is oP (1) by Lemma 6 and Assumptions 1, 2, so we have shown that a2 asymptotically equals

t_1EEk№ -d/ytw

i=1 j=1

Under Assumptions 1, 2 and the alternative hypothesis H1, {sgn(Yj - )} satisfies the assumptions of Theorem 2.1 in De Jong and Davidson [18], so

T^EEk((i -MctC, - Cj,

i=1 j=1

so the proof of Theorem 2 has been completed.

Competing interests

The authors declare that they have no competing interests. Authors' contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details

1 School of Mathematical Science, Shanxi University, Taiyuan, Shanxi 030006, P.R. China. institute of Management and Decision, Shanxi University, Taiyuan, Shanxi 030006, P.R. China.

Acknowledgements

The authors are grateful to the two anonymous referees for their careful reading of the manuscript and many helpful comments which improve the manuscript greatly. We would like to thank Prof. John Nolan who provided the software for generating the stable innovations in Section 4. By financial support this work is supported by the National Natural Science Foundation of China (No. 11226217) and the Postdoctoral Science Special Foundation (Nos. 2012M510772, 2013T60266), which is gratefully acknowledged.

Received: 12 May 2014 Accepted: 19 January 2015 Published online: 06 February 2015 References

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