More Powerful Test for Homogeneity of Means Under an Order Restriction in Time Series with Stationary Process

: Suppose that an order restriction is imposed among several means in time series. We are interested in testing the homogeneity of these unknown means under this restriction. In the present paper, a test based on the isotonic regression is done for monotonic ordered means in time series with stationary process and short range dependent sequences errors. A test statistic is proposed using the penalized likelihood ratio (PLR) approach. Since the asymptotic null distribution of test statistic is complicated, its critical values are computed by using Monte Carlo simulation method for some values of sample sizes at different significance levels. The power study of our test statistic is provided which is more powerful than that of the test proposed by Brillinger (1989). Finally, to show the application of the proposed test, it is applied to real dataset contains monthly Iran rainfall records.


Introduction
In the application fields of statistical inference it is important to test stationary of the given time series. The interest centre on the stationary of mean, and the process of interest is assumed to be of the following form , k k k Z X    (1) where the k  ,  , 2 , 1 , 0  k , are the means and k Z is a stationary process with mean 0 and finite covariances ) , for any value of i . For this plan, there has been both classical work on testing for the existence of a trend and more recent work on tests for an abrupt change. The present work falls between these two approaches in developing a test for a change, or trend that is monotonic but otherwise arbitrary. It is assumed that the homogeneity of means hypothesis as the null hypothesis of interest is , : against the alternative order hypothesis , : . A statistical change point problem was first studied in the mid-1950s in the context of quality control in industrial processes. A change point is defined as a point in the time order when the probability distribution of a sequence of observations differs before and after that point. The literature of statistical change point has evolved over time and now includes a significant amount of scholarly work on change point analysis, and genetics, to name a few. The classical change point problem is to test for the existence of a change point and estimate its location if it exits. Picard (1985), presented a few techniques which may be useful in the analysis of time series when a failure is suspected. He presented two categories of tests and investigate their asymptotic properties: one, of nonparametric type, is intended to detect a general failure in spectrum; the other investigates the properties of likelihood ratio tests in parametric models which have a non-standard behaviour in this situation. Siegmund (1986) and Bhattacharya (1994), have written review articles, and Shaban (1980) has complied an annotated bibliography. In many literatures, the change-point problem for linear models has been discussed extensively. While most of the contributions are made from a posteriori point of view (we refer to, e.g., Bai, 1997, Csörgo and Horváth, 1997, and Perron, 2006, recently the sequential or on-line change-point detection has received more and more attention. Chu et al. (1995) and Horváth et al. (2004), who suggested cumulative sum (CUSUM) procedures in different stochastic models. CUSUM procedures work best for relatively early changes but show a slower reaction the later the change occurs. Chu et al. (1995) investigated tests for structural change based on moving sums (MOSUMS) of recursive and least-squares residuals. They obtained the asymptotic critical values of the MOSUM test with recursive residuals and 683 showed that the asymptotic critical values of the MOSUM test with least-squares residuals can easily be obtained from already existing tables for the moving-estimates test. Aue et al. (2009), provided the asymptotic normality of the suitably normalized stopping time of the CUSUM procedure in a similar setting. Their drawback is a strong dependence on the choice of the parameters, in particular the right choice of the window size by the statistician. Kwiatkowski et al. (1992), proposed a test of the null hypothesis that an observable series is stationary around a deterministic trend. The series is expressed as the sum of deterministic trend, random walk. The asymptotic distribution of the statistic is derived under the null and under the alternative that the series is difference-stationary. Also, Finite sample size and power are considered in a Monte Carlo experiment. Tang and Macneill (1993), shown that serial correlation can produce striking effects in distributions of change-point statistics. Failure to account for these effects is shown to invalidate change-point tests, either through increases in the type 1 error rates if low frequency spectral mass predominates in the spectrum of the noise process, or through diminution of the power of the tests when high frequency mass predominates. Lombard and Hart (1994), considered abrupt mean-change models for data with dependent, stationary, errors. No specific distributional assumptions, other than the existence and sum ability of cumulates, are made. Also, a consistency property of the least squares estimator of the change-point is derived. See also Brodsky and Darkhovsky (1993, Chapter 3), where strong mixing conditions are imposed.
Extensions of the simple change point model either allow more general change patterns or relax the independence assumption of } { k Z ; dependence is inevitable in the study of time series. This complicates the testing procedures. Woodward and Gray (1993), perform a test of the existence of a linear trend in autoregressive moving average models with applications to global warming. For example like the Iran rainfall data, linearity is not expected, so that a nonparametric test is desirable; Brodsky and Darkhovsky (1993), present a systematic account of nonparametric methods, and Brillinger (1989) develops a test for monotonic trends with dependent errors. The latter two papers contain many further references to test for trend. Davis, et al. (1995), considered the problem of testing whether or not a change has occurred in the parameter values and order of an autoregressive model. It is shown that if the white noise in the AR model is weakly stationary with finite fourth moments, then under the null hypothesis of no change point, the normalized Gaussian likelihood ratio test statistic converges in distribution to the Gumbel extreme value distribution. Bagshaw and Richard (1997), proposed the procedures for monitoring forecast errors in order to detect changes in a time-series model. These procedures are based on likelihood ratio statistics which consist of cumulative sums. Lavielle and Moulines (1997), consider estimation and testing of multiple changes in the mean of strong mixing random processes. The assumed mean function is piecewise constant with an unknown number of pieces. Almasri (2000), used the wavelet filters for testing the trend in the presence of the fractional difference processes. The test method has been constructed using wavelet analyses which have the ability to decompose a time series into low frequencies (trend) and high frequencies (noise) components. Under the Stationary Process normality assumption the test statistic computed and to investigate the properties of the test statistic, empirical critical values for the test have been generated using Monte Carlo simulations. Inference based on the monotonicity assumption is discussed in Robertson et al. (1988) and Silvapulle and Sen (2005). I show that my test statistic based on isotonic regression is asymptotically more powerful than the one proposed by Brillinger (1989). Gombay  In this paper, a test based on the isotonic regression is done for monotonic ordered means in time series with stationary process and short range dependent sequences errors. A test statistic is proposed using PLR approach. The test is compared this with Brillinger's (1989) method. However, there are more methods to be compared with, but it is shown that my test statistic based on isotonic regression is asymptotic more powerful than the one proposed by Brillinger (1989).
To the author's knowledge, the penalized likelihood ratio test for this problem of testing with an order restriction on the unknown means in time series has not been obtained yet.
The rest of this paper is organized as follows. The test statistic is presented using PLR approach in Section 2. In Section 3, the critical values of the proposed PLR test statistic are computed by using Monte Carlo simulation for some values of sample sizes at different significance levels and the test is illustrated by real example associated with Iran rainfall data. Section 4, contains a power study and a comparison with Brillinger's (1989) test. The conclusions are summarized in Section 5. Finally, the proofs are presented in Appendix.

Preamble: Independent identically distributed normal errors
In this subsection, we assume that (1) are independent normal random variables with mean 0 and known variance 2  . Then the log likelihood function is , where C denotes a generic constant that does not depend on parameter  . The unknown parameter  takes values in the space Under the null hypothesis 0 H , the maximum likelihood estimator is obviously Robertson et al. (1988). As shown below, this test statistic is affected by the so-called spiking problem in large samples, in that 1  is too small while n ˆ is too large. Instead of (3), we shall consider the penalized log likelihood function which is just formula (5)

Short range dependent errors
For the remainder of section 2, let the stationary sequence k Z , for any k , exhibits short range dependence in the following way. First suppose that the covariances  If we don't assume strong mixing, sufficient conditions are that Maxwell and Woodroof (2000).
The asymptotic distribution of r n T , is obtained in theorem 1 and 2 below for local alternatives.    Thus it is necessary to show that the right hand side of (15) converges in distribution to the right hand side of (14). This seems plausible in view of (10) and (12). Details may be found in the Appendix.

Remark 1.
If 0  c , then the right hand side (14) in infinity with probability 1; see Groeneboom and Pyke (1983). This is the spiking problem mentioned above. The left hand side of (14) does not have a nondegenerate asymptotic under null hypothesis.

Remark 2.
In the context of classical change point analysis, there is a similar problem. Suppose that we want to test the following hypothesis . : does not have an asymptotic distribution, since it becomes unstable at the ends. Recall that the unpenalized isotonic regression estimator ˆ is given by (5). Hence for any k , the detrended process is    Figure 1 shows the volume of yearly rainfall in Iran from 1898 to 2010 along with the penalized and unpenalized isotonic regression functions. There is a mild spiking problem: the year 1898 has a low value. The spiking problem is clearly suppressed in the penalized isotonic regression. From an autocorrelation plot, the residuals appear to be short range dependent. Only the eighth of the first twenty autocorrelations is significant at the 05 . 0 level. From (8) and (17)

Iran rainfall data
Z is standard normal. Hence, at the 05 . 0 level of significance, Brillinger's (1989) test is unable to detect the trend. Section 4, contains some power comparisons.

Power study of test statistic
In this section, the power of our test statistic r n T , is compared to that of the proposed by Brillinger (1989 . From the plot, we see that the testing procedure based on isotonic regression is more powerful than Brillinger's (1989) test. In Figure 2(b), the mean function is linear:

Conclusions
The present article considers an approach to testing homogeneity of several means under an order restriction in time series with stationary process and short range dependent sequences errors. This approach is based on penalized likelihood ratio and the test statistic obtained and its critical values computed by using Monte Carlo simulation for some values of sample sizes at different significance levels. To show the application of the proposed test the real dataset contains monthly Iran rainfall records collected from 1898 to 2010. The volume of yearly rainfall in these years shown in Figure 1 along with the penalized and unpenalized isotonic regression functions. In the year 1898 there was a low value of mild spiking problem. The value of test statistic computed along with the data which is significant at the 1 . 0 level. Also, from the Figure 2, it was clear that the testing procedure based on isotonic regression is more powerful than Brillinger's (1989) test. Recall the definitions (12) and (13)