## The Annals of Statistics

### Detecting relevant changes in the mean of nonstationary processes—A mass excess approach

#### Abstract

This paper considers the problem of testing if a sequence of means $(\mu_{t})_{t=1,\ldots ,n}$ of a nonstationary time series $(X_{t})_{t=1,\ldots ,n}$ is stable in the sense that the difference of the means $\mu_{1}$ and $\mu_{t}$ between the initial time $t=1$ and any other time is smaller than a given threshold, that is $|\mu_{1}-\mu_{t}|\leq c$ for all $t=1,\ldots ,n$. A test for hypotheses of this type is developed using a bias corrected monotone rearranged local linear estimator and asymptotic normality of the corresponding test statistic is established. As the asymptotic variance depends on the location of the roots of the equation $|\mu_{1}-\mu_{t}|=c$ a new bootstrap procedure is proposed to obtain critical values and its consistency is established. As a consequence we are able to quantitatively describe relevant deviations of a nonstationary sequence from its initial value. The results are illustrated by means of a simulation study and by analyzing data examples.

#### Article information

Source
Ann. Statist., Volume 47, Number 6 (2019), 3578-3608.

Dates
Revised: January 2019
First available in Project Euclid: 31 October 2019

https://projecteuclid.org/euclid.aos/1572487403

Digital Object Identifier
doi:10.1214/19-AOS1811

Mathematical Reviews number (MathSciNet)
MR4025752

#### Citation

Dette, Holger; Wu, Weichi. Detecting relevant changes in the mean of nonstationary processes—A mass excess approach. Ann. Statist. 47 (2019), no. 6, 3578--3608. doi:10.1214/19-AOS1811. https://projecteuclid.org/euclid.aos/1572487403

#### References

• [1] Álvarez-Esteban, P. C., del Barrio, E., Cuesta-Albertos, J. A. and Matrán, C. (2008). Trimmed comparison of distributions. J. Amer. Statist. Assoc. 103 697–704.
• [2] Álvarez-Esteban, P. C., del Barrio, E., Cuesta-Albertos, J. A. and Matrán, C. (2012). Similarity of samples and trimming. Bernoulli 18 606–634.
• [3] Andrews, D. W. K. (1993). Tests for parameter instability and structural change with unknown change point. Econometrica 61 821–856.
• [4] Aue, A., Hörmann, S., Horváth, L. and Reimherr, M. (2009). Break detection in the covariance structure of multivariate time series models. Ann. Statist. 37 4046–4087.
• [5] Aue, A. and Horváth, L. (2013). Structural breaks in time series. J. Time Series Anal. 34 1–16.
• [6] Bai, J. and Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica 66 47–78.
• [7] Baíllo, A. (2003). Total error in a plug-in estimator of level sets. Statist. Probab. Lett. 65 411–417.
• [8] Berman, S. M. (1992). Sojourns and Extremes of Stochastic Processes. The Wadsworth & Brooks/Cole Statistics/Probability Series. Wadsworth & Brooks/Cole, Pacific Grove, CA.
• [9] Brown, R. L., Durbin, J. and Evans, J. M. (1975). Techniques for testing the constancy of regression relationships over time. J. Roy. Statist. Soc. Ser. B 37 149–163.
• [10] Cadre, B. (2006). Kernel estimation of density level sets. J. Multivariate Anal. 97 999–1023.
• [11] Champ, C. W. and Woodall, W. H. (1987). Exact results for Shewhart control charts with supplementary runs rules. Technometrics 29 393–399.
• [12] Chandler, G. and Polonik, W. (2006). Discrimination of locally stationary time series based on the excess mass functional. J. Amer. Statist. Assoc. 101 240–253.
• [13] Cheng, M.-Y. and Hall, P. (1998). Calibrating the excess mass and dip tests of modality. J. R. Stat. Soc. Ser. B. Stat. Methodol. 60 579–589.
• [14] Chernozhukov, V., Fernández-Val, I. and Galichon, A. (2009). Improving point and interval estimators of monotone functions by rearrangement. Biometrika 96 559–575.
• [15] Chernozhukov, V., Fernández-Val, I. and Galichon, A. (2010). Quantile and probability curves without crossing. Econometrica 78 1093–1125.
• [16] Chow, G. C. (1960). Tests of equality between sets of coefficients in two linear regressions. Econometrica 28 591–605.
• [17] Cuevas, A., González-Manteiga, W. and Rodríguez-Casal, A. (2006). Plug-in estimation of general level sets. Aust. N. Z. J. Stat. 48 7–19.
• [18] Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1–37.
• [19] Dette, H., Neumeyer, N. and Pilz, K. F. (2006). A simple nonparametric estimator of a strictly monotone regression function. Bernoulli 12 469–490.
• [20] Dette, H. and Volgushev, S. (2008). Non-crossing non-parametric estimates of quantile curves. J. R. Stat. Soc. Ser. B. Stat. Methodol. 70 609–627.
• [21] Dette, H. and Wied, D. (2016). Detecting relevant changes in time series models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 78 371–394.
• [22] Dette, H. and Wu, W. (2019). Supplement to “Detecting relevant changes in the mean of nonstationary processes—A mass excess approach.” DOI:10.1214/19-AOS1811SUPP.
• [23] Gijbels, I., Lambert, A. and Qiu, P. (2007). Jump-preserving regression and smoothing using local linear fitting: A compromise. Ann. Inst. Statist. Math. 59 235–272.
• [24] Hartigan, J. A. and Hartigan, P. M. (1985). The dip test of unimodality. Ann. Statist. 13 70–84.
• [25] Jandhyala, V., Fotopoulos, S., MacNeill, I. and Liu, P. (2013). Inference for single and multiple change-points in time series. J. Time Series Anal. 34 423–446.
• [26] Krämer, W., Ploberger, W. and Alt, R. (1988). Testing for structural change in dynamic models. Econometrica 56 1355–1369.
• [27] Mason, D. M. and Polonik, W. (2009). Asymptotic normality of plug-in level set estimates. Ann. Appl. Probab. 19 1108–1142.
• [28] Mercurio, D. and Spokoiny, V. (2004). Statistical inference for time-inhomogeneous volatility models. Ann. Statist. 32 577–602.
• [29] Müller, D. W. and Sawitzki, G. (1991). Excess mass estimates and tests for multimodality. J. Amer. Statist. Assoc. 86 738–746.
• [30] Nason, G. P., von Sachs, R. and Kroisandt, G. (2000). Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. J. R. Stat. Soc. Ser. B. Stat. Methodol. 62 271–292.
• [31] Ombao, H., von Sachs, R. and Guo, W. (2005). SLEX analysis of multivariate nonstationary time series. J. Amer. Statist. Assoc. 100 519–531.
• [32] Page, E. S. (1954). Continuous inspection schemes. Biometrika 41 100–115.
• [33] Polonik, W. (1995). Measuring mass concentrations and estimating density contour clusters—An excess mass approach. Ann. Statist. 23 855–881.
• [34] Polonik, W. and Wang, Z. (2005). Estimation of regression contour clusters—An application of the excess mass approach to regression. J. Multivariate Anal. 94 227–249.
• [35] Qiu, P. (2003). A jump-preserving curve fitting procedure based on local piecewise-linear kernel estimation. J. Nonparametr. Stat. 15 437–453.
• [36] Rinaldo, A. and Wasserman, L. (2010). Generalized density clustering. Ann. Statist. 38 2678–2722.
• [37] Samworth, R. J. and Wand, M. P. (2010). Asymptotics and optimal bandwidth selection for highest density region estimation. Ann. Statist. 38 1767–1792.
• [38] Schucany, W. R. and Sommers, J. P. (1977). Improvement of kernel type density estimators. J. Amer. Statist. Assoc. 72 420–423.
• [39] Spokoiny, V. (2009). Multiscale local change point detection with applications to value-at-risk. Ann. Statist. 37 1405–1436.
• [40] Takács, L. (1996). Sojourn times. J. Appl. Math. Stoch. Anal. 9 415–426.
• [41] Tsybakov, A. B. (1997). On nonparametric estimation of density level sets. Ann. Statist. 25 948–969.
• [42] Vogt, M. (2012). Nonparametric regression for locally stationary time series. Ann. Statist. 40 2601–2633.
• [43] Wellek, S. (2010). Testing Statistical Hypotheses of Equivalence and Noninferiority, 2nd ed. CRC Press, Boca Raton, FL.
• [44] Woodall, W. H. and Montgomery, D. C. (1999). Research issues and ideas in statistical process control. J. Qual. Technol. 31 376–386.
• [45] Wu, W. B. and Pourahmadi, M. (2009). Banding sample autocovariance matrices of stationary processes. Statist. Sinica 19 1755–1768.
• [46] Wu, W. B. and Zhao, Z. (2007). Inference of trends in time series. J. R. Stat. Soc. Ser. B. Stat. Methodol. 69 391–410.
• [47] Zhou, Z. and Wu, W. B. (2009). Local linear quantile estimation for nonstationary time series. Ann. Statist. 37 2696–2729.
• [48] Zhou, Z. and Wu, W. B. (2010). Simultaneous inference of linear models with time varying coefficients. J. R. Stat. Soc. Ser. B. Stat. Methodol. 72 513–531.

#### Supplemental materials

• Supplement to “Detecting relevant changes in the mean of nonstationary processes—A mass excess approach”. We provide proof for (B.17) and (B.18) for Theorem 4.1, proof of Theorem 4.4 and technical lemmas in the Supplementary Material.