Bernoulli

  • Bernoulli
  • Volume 24, Number 2 (2018), 1266-1306.

Bump detection in heterogeneous Gaussian regression

Farida Enikeeva, Axel Munk, and Frank Werner

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We analyze the effect of a heterogeneous variance on bump detection in a Gaussian regression model. To this end, we allow for a simultaneous bump in the variance and specify its impact on the difficulty to detect the null signal against a single bump with known signal strength. This is done by calculating lower and upper bounds, both based on the likelihood ratio.

Lower and upper bounds together lead to explicit characterizations of the detection boundary in several subregimes depending on the asymptotic behavior of the bump heights in mean and variance. In particular, we explicitly identify those regimes, where the additional information about a simultaneous bump in variance eases the detection problem for the signal. This effect is made explicit in the constant and/or the rate, appearing in the detection boundary.

We also discuss the case of an unknown bump height and provide an adaptive test and some upper bounds in that case.

Article information

Source
Bernoulli, Volume 24, Number 2 (2018), 1266-1306.

Dates
Received: May 2015
Revised: May 2016
First available in Project Euclid: 21 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.bj/1505980896

Digital Object Identifier
doi:10.3150/16-BEJ899

Mathematical Reviews number (MathSciNet)
MR3706794

Zentralblatt MATH identifier
06778365

Keywords
change point detection heterogeneous Gaussian regression minimax testing theory

Citation

Enikeeva, Farida; Munk, Axel; Werner, Frank. Bump detection in heterogeneous Gaussian regression. Bernoulli 24 (2018), no. 2, 1266--1306. doi:10.3150/16-BEJ899. https://projecteuclid.org/euclid.bj/1505980896


Export citation

References

  • [1] Arias-Castro, E., Candès, E.J., Helgason, H. and Zeitouni, O. (2008). Searching for a trail of evidence in a maze. Ann. Statist. 36 1726–1757.
  • [2] Arias-Castro, E. and Wang, M. (2013). Distribution-free tests for sparse heterogeneous mixtures. Preprint. Available at arXiv:1308.0346.
  • [3] Arlot, S. and Celisse, A. (2011). Segmentation of the mean of heteroscedastic data via cross-validation. Stat. Comput. 21 613–632.
  • [4] Bai, J. and Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica 66 47–78.
  • [5] Bai, J. and Perron, P. (2003). Computation and analysis of multiple structural change models. J. Appl. Econometrics 18 1–22.
  • [6] Baraud, Y. (2002). Non-asymptotic minimax rates of testing in signal detection. Bernoulli 8 577–606.
  • [7] Ben-Tal, A., El Ghaoui, L. and Nemirovski, A. (2009). Robust Optimization. Princeton Series in Applied Mathematics. Princeton, NJ: Princeton Univ. Press.
  • [8] Birgé, L. (2001). An alternative point of view on Lepski’s method. In State of the Art in Probability and Statistics (Leiden, 1999). Institute of Mathematical Statistics Lecture Notes—Monograph Series 36 113–133. Beachwood, OH: IMS.
  • [9] Birgé, L. and Massart, P. (2001). Gaussian model selection. J. Eur. Math. Soc. (JEMS) 3 203–268.
  • [10] Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford: Oxford Univ. Press.
  • [11] Boutahar, M. (2012). Testing for change in mean of independent multivariate observations with time varying covariance. J. Probab. Stat. Art. ID 969753, 17.
  • [12] Boysen, L., Kempe, A., Liebscher, V., Munk, A. and Wittich, O. (2009). Consistencies and rates of convergence of jump-penalized least squares estimators. Ann. Statist. 37 157–183.
  • [13] Braun, J.V., Braun, R.K. and Müller, H.-G. (2000). Multiple changepoint fitting via quasilikelihood, with application to DNA sequence segmentation. Biometrika 87 301–314.
  • [14] Cai, T.T., Jeng, X.J. and Jin, J. (2011). Optimal detection of heterogeneous and heteroscedastic mixtures. J. R. Stat. Soc. Ser. B. Stat. Methodol. 73 629–662.
  • [15] Cai, T.T. and Wu, Y. (2014). Optimal detection of sparse mixtures against a given null distribution. IEEE Trans. Inform. Theory 60 2217–2232.
  • [16] Carlstein, E., Müller, H.-G. and Siegmund, D., eds. (1994). Change-Point Problems. Institute of Mathematical Statistics Lecture Notes—Monograph Series 23. Hayward, CA: IMS.
  • [17] Castro, R.M., Haupt, J., Nowak, R. and Raz, G.M. (2008). Finding needles in noisy haystacks. In IEEE International Conference on Acoustics, Speech and Signal Processing, 2008. ICASSP 2008 5133–5136. New York: IEEE.
  • [18] Chan, H.P. and Walther, G. (2013). Detection with the scan and the average likelihood ratio. Statist. Sinica 23 409–428.
  • [19] Csörgő, M. and Horváth, L. (1997). Limit Theorems in Change-Point Analysis. Chichester: Wiley.
  • [20] Donoho, D. and Jin, J. (2004). Higher criticism for detecting sparse heterogeneous mixtures. Ann. Statist. 32 962–994.
  • [21] Du, C., Kao, C.-L.M. and Kou, S.C. (2016). Stepwise signal extraction via marginal likelihood. J. Amer. Statist. Assoc. 111 314–330.
  • [22] Dümbgen, L. and Spokoiny, V.G. (2001). Multiscale testing of qualitative hypotheses. Ann. Statist. 29 124–152.
  • [23] Dümbgen, L. and Walther, G. (2008). Multiscale inference about a density. Ann. Statist. 36 1758–1785.
  • [24] Frick, K., Munk, A. and Sieling, H. (2014). Multiscale change point inference. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 495–580.
  • [25] Goldenshluger, A., Juditsky, A. and Nemirovski, A. (2015). Hypothesis testing by convex optimization. Electron. J. Stat. 9 1645–1712.
  • [26] Harchaoui, Z. and Lévy-Leduc, C. (2010). Multiple change-point estimation with a total variation penalty. J. Amer. Statist. Assoc. 105 1480–1493.
  • [27] Hsu, D., Kakade, S.M. and Zhang, T. (2012). A tail inequality for quadratic forms of subgaussian random vectors. Electron. Commun. Probab. 17 no. 52, 6.
  • [28] Huang, W.T. and Chang, Y.P. (1993). Nonparametric estimation in change-point models. J. Statist. Plann. Inference 35 335–347.
  • [29] Ingster, Yu.I. (1993). Asymptotically minimax hypothesis testing for nonparametric alternatives. I. Math. Methods Statist. 2 85–114.
  • [30] Ingster, Y.I. and Suslina, I.A. (2002). On detection of a signal of known shape in multi-channel system. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 294 88–112.
  • [31] Jacod, J. and Todorov, V. (2010). Do price and volatility jump together? Ann. Appl. Probab. 20 1425–1469.
  • [32] Jeng, X.J., Cai, T.T. and Li, H. (2010). Optimal sparse segment identification with application in copy number variation analysis. J. Amer. Statist. Assoc. 105 1156–1166.
  • [33] Killick, R., Fearnhead, P. and Eckley, I.A. (2012). Optimal detection of changepoints with a linear computational cost. J. Amer. Statist. Assoc. 107 1590–1598.
  • [34] Korostelev, A. and Korosteleva, O. (2011). Mathematical Statistics: Asymptotic Minimax Theory. Graduate Studies in Mathematics 119. Providence, RI: Amer. Math. Soc.
  • [35] Laurent, B., Loubes, J.-M. and Marteau, C. (2012). Non asymptotic minimax rates of testing in signal detection with heterogeneous variances. Electron. J. Stat. 6 91–122.
  • [36] Lavielle, M. (2005). Using penalized contrasts for the change-point problem. Signal Process. 85 1501–1510.
  • [37] Lehmann, E.L. and Romano, J.P. (2005). Testing Statistical Hypotheses, 3rd ed. Springer Texts in Statistics. New York: Springer.
  • [38] Muggeo, V. and Adelfio, G. (2010). Efficient change point detection for genomic sequences of continuous measurements. Bioinformatics 27 161–166.
  • [39] Munk, A. and Werner, F. (2015). Discussion of “Hypotheses testing by convex optimization” [MR3379005]. Electron. J. Stat. 9 1720–1722.
  • [40] Neher, E. and Sakmann, B. (1995). Single-Channel Recording. New York: Plenum Press.
  • [41] Rivera, C. and Walther, G. (2013). Optimal detection of a jump in the intensity of a Poisson process or in a density with likelihood ratio statistics. Scand. J. Stat. 40 752–769.
  • [42] Rohde, A. and Dümbgen, L. (2013). Statistical inference for the optimal approximating model. Probab. Theory Related Fields 155 839–865.
  • [43] Schirmer, T. (1998). General and specific porins from bacterial outer membranes. J. Struct. Biol. 121 101–109.
  • [44] Shorack, G.R. (2000). Probability for Statisticians. New York: Springer.
  • [45] Siegmund, D. (2013). Change-points: From sequential detection to biology and back. Sequential Anal. 32 2–14.
  • [46] Siegmund, D. and Venkatraman, E.S. (1995). Using the generalized likelihood ratio statistic for sequential detection of a change-point. Ann. Statist. 23 255–271.
  • [47] Siegmund, D., Yakir, B. and Zhang, N.R. (2011). Detecting simultaneous variant intervals in aligned sequences. Ann. Appl. Stat. 5 645–668.
  • [48] Sigworth, F.J. (1985). Open channel noise. I. Noise in acetylcholine receptor currents suggests conformational fluctuations. Biophys. J. 47 709–720.
  • [49] Spokoiny, V. and Zhilova, M. (2013). Sharp deviation bounds for quadratic forms. Math. Methods Statist. 22 100–113.
  • [50] Tsybakov, A.B. (2009). Introduction to Nonparametric Estimation. New York: Springer.
  • [51] Yao, Y.-C. (1988). Estimating the number of change-points via Schwarz’ criterion. Statist. Probab. Lett. 6 181–189.