Adaptive estimation for bifurcating Markov chains

[7] Bertoin, J. (2006). <i>Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics</i> <b>102</b>. Cambridge: Cambridge Univ. Press.

[1] Alquier, P. and Wintenberger, O. (2012). Model selection for weakly dependent time series forecasting. <i>Bernoulli</i> <b>18</b> 883–913.

[2] Athreya, K.B. and Kang, H.-J. (1998). Some limit theorems for positive recurrent branching Markov chains. I. <i>Adv. in Appl. Probab.</i> <b>30</b> 693–710.

[3] Basawa, I.V. and Zhou, J. (2004). Non-Gaussian bifurcating models and quasi-likelihood estimation. <i>J. Appl. Probab.</i> <b>41A</b> 55–64.

[5] Bercu, B. and Blandin, V. (2015). A Rademacher–Menchov approach for random coefficient bifurcating autoregressive processes. <i>Stochastic Process. Appl.</i> <b>125</b> 1218–1243.

[6] Bercu, B., de Saporta, B. and Gégout-Petit, A. (2009). Asymptotic analysis for bifurcating autoregressive processes via a martingale approach. <i>Electron. J. Probab.</i> <b>14</b> 2492–2526.

[8] Bitseki Penda, S.V. and Djellout, H. (2014). Deviation inequalities and moderate deviations for estimators of parameters in bifurcating autoregressive models. <i>Ann. Inst. Henri Poincaré Probab. Stat.</i> <b>50</b> 806–844.

[9] Bitseki Penda, S.V., Djellout, H. and Guillin, A. (2014). Deviation inequalities, moderate deviations and some limit theorems for bifurcating Markov chains with application. <i>Ann. Appl. Probab.</i> <b>24</b> 235–291.

[10] Bitseki Penda, S.V., Escobar-Bach, M. and Guillin, A. Transportation cost-information and concentration inequalities for bifurcating Markov chains. Available at <a href="arXiv:1501.06693">arXiv:1501.06693</a>.

[11] Bitseki Penda, S.V. and Olivier, A. Nonparametric estimation of the autoregressive functions in bifurcating autoregressive models. Available at <a href="arXiv:1506.01842">arXiv:1506.01842</a>.

[12] Blandin, V. (2014). Asymptotic results for random coefficient bifurcating autoregressive processes. <i>Statistics</i> <b>48</b> 1202–1232.

[13] Clémençon, S.J.M. (2000). Adaptive estimation of the transition density of a regular Markov chain. <i>Math. Methods Statist.</i> <b>9</b> 323–357.

[15] Cowan, R. and Staudte, R.G. (1986). The bifurcating autoregressive model in cell lineage studies. <i>Biometrics</i> <b>42</b> 769–783.

[16] de Saporta, B., Gégout-Petit, A. and Marsalle, L. (2011). Parameters estimation for asymmetric bifurcating autoregressive processes with missing data. <i>Electron. J. Stat.</i> <b>5</b> 1313–1353.

[17] de Saporta, B., Gégout-Petit, A. and Marsalle, L. (2012). Asymmetry tests for bifurcating auto-regressive processes with missing data. <i>Statist. Probab. Lett.</i> <b>82</b> 1439–1444.

[18] de Saporta, B., Gégout-Petit, A. and Marsalle, L. (2014). Random coefficients bifurcating autoregressive processes. <i>ESAIM Probab. Stat.</i> <b>18</b> 365–399.

[19] Delmas, J.-F. and Marsalle, L. (2010). Detection of cellular aging in a Galton–Watson process. <i>Stochastic Process. Appl.</i> <b>120</b> 2495–2519.

[20] DeVore, R.A., Konyagin, S.V. and Temlyakov, V.N. (1998). Hyperbolic wavelet approximation. <i>Constr. Approx.</i> <b>14</b> 1–26.

[21] Donoho, D.L. and Johnstone, I.M. (1994). Ideal spatial adaptation by wavelet shrinkage. <i>Biometrika</i> <b>81</b> 425–455.

[22] Donoho, D.L. and Johnstone, I.M. (1995). Adapting to unknown smoothness via wavelet shrinkage. <i>J. Amer. Statist. Assoc.</i> <b>90</b> 1200–1224.

[23] Donoho, D.L., Johnstone, I.M., Kerkyacharian, G. and Picard, D. (1995). Wavelet shrinkage: Asymptopia? <i>J. Roy. Statist. Soc. Ser. B</i> <b>57</b> 301–369.

[24] Donoho, D.L., Johnstone, I.M., Kerkyacharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding. <i>Ann. Statist.</i> <b>24</b> 508–539.

[25] Doumic, M., Hoffmann, M., Krell, N. and Robert, L. (2015). Statistical estimation of a growth-fragmentation model observed on a genealogical tree. <i>Bernoulli</i> <b>21</b> 1760–1799.

[27] Gao, F., Guillin, A. and Wu, L. (2014). Bernstein-type concentration inequalities for symmetric Markov processes. <i>Theory Probab. Appl.</i> <b>58</b> 358–382.

[28] Guyon, J. (2007). Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging. <i>Ann. Appl. Probab.</i> <b>17</b> 1538–1569.

[29] Hairer, M. and Mattingly, J.C. (2011). Yet another look at Harris’ ergodic theorem for Markov chains. In <i>Seminar on Stochastic Analysis</i>, <i>Random Fields and Applications VI. Progress in Probability</i> <b>63</b> 109–117. Basel: Birkhäuser.

[30] Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). <i>Wavelets</i>, <i>Approximation</i>, <i>and Statistical Applications. Lecture Notes in Statistics</i> <b>129</b>. New York: Springer.

[31] Hoffmann, M. (1999). Adaptive estimation in diffusion processes. <i>Stochastic Process. Appl.</i> <b>79</b> 135–163.

[32] Huggins, R.M. and Basawa, I.V. (1999). Extensions of the bifurcating autoregressive model for cell lineage studies. <i>J. Appl. Probab.</i> <b>36</b> 1225–1233.

[33] Huggins, R.M. and Basawa, I.V. (2000). Inference for the extended bifurcating autoregressive model for cell lineage studies. <i>Aust. N. Z. J. Stat.</i> <b>42</b> 423–432.

[34] Kerkyacharian, G. and Picard, D. (2000). Thresholding algorithms, maxisets and well-concentrated bases. <i>Test</i> <b>9</b> 283–344.

[35] Lacour, C. (2007). Adaptive estimation of the transition density of a Markov chain. <i>Ann. Inst. Henri Poincaré Probab. Stat.</i> <b>43</b> 571–597.

[36] Lacour, C. (2008). Nonparametric estimation of the stationary density and the transition density of a Markov chain. <i>Stochastic Process. Appl.</i> <b>118</b> 232–260.

[37] Merlevède, F., Peligrad, M. and Rio, E. (2009). Bernstein inequality and moderate deviations under strong mixing conditions. In <i>High Dimensional Probability V</i>: <i>The Luminy Volume. Inst. Math. Stat. Collect.</i> <b>5</b> 273–292. Beachwood, OH: IMS.

[40] Paulin, D. (2015). Concentration inequalities for Markov chains by Marton couplings and spectral methods. <i>Electron. J. Probab.</i> <b>20</b> no. 79, 32.

[42] Robert, L., Hoffmann, M., Krell, N., Aymerich, S., Robert, J. and Doumic, M. Division control in Escherichia coli is based on a size-sensing rather than a timing mechanism. <i>BMC Biol.</i> <b>02/2014 12(1)</b> 17.

[43] Roussas, G.G. (1991). Estimation of transition distribution function and its quantiles in Markov processes: Strong consistency and asymptotic normality. In <i>Nonparametric Functional Estimation and Related Topics</i> (<i>Spetses</i>, 1990). <i>NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.</i> <b>335</b> 443–462. Dordrecht: Kluwer Academic.

[44] Takacs, C. (2002). Strong law of large numbers for branching Markov chains. <i>Markov Process. Related Fields</i> <b>8</b> 107–116.

[45] Wintenberger, O. (2015). Weak transport inequalities and applications to exponential and oracle inequalities. <i>Electron. J. Probab.</i> <b>20</b> no. 114, 27.

[46] Zhou, J. and Basawa, I.V. (2005). Maximum likelihood estimation for a first-order bifurcating autoregressive process with exponential errors. <i>J. Time Ser. Anal.</i> <b>26</b> 825–842.

[14] Cohen, A. (2000). <i>Handbook of Numerical Analysis. Vol. VII</i> (P.G. Ciarlet and J.L. Lions, eds.). <i>Handbook of Numerical Analysis</i>, <i>VII</i>. Amsterdam: North-Holland.

[26] Francq, C. and Zakoïan, J.-M. (2010). <i>GARCH Models</i>: <i>Structure</i>, <i>Statistical Inference and Financial Applications</i>. Chichester: Wiley.

[38] Meyer, Y. (1990). <i>Ondelettes et Opérateurs. I. Actualités Mathématiques.</i> [<i>Current Mathematical Topics</i>]. Paris: Hermann.

[39] Meyn, S.P. and Tweedie, R.L. (1993). <i>Markov Chains and Stochastic Stability. Communications and Control Engineering Series</i>. London: Springer.

[41] Perthame, B. (2007). <i>Transport Equations in Biology. Frontiers in Mathematics</i>. Basel: Birkhäuser.

[4] Benjamini, I. and Peres, Y. (1994). Markov chains indexed by trees. <i>Ann. Probab.</i> <b>22</b> 219–243.