Robust estimation in controlled branching processes: Bayesian estimators via disparities

This paper is concerned with Bayesian inferential methods for data from controlled branching processes that account for model robustness through the use of disparities. Under regularity conditions, we establish that estimators built on disparity-based posterior, such as expectation and maximum a posteriori estimates, are consistent and efficient under the posited model. Additionally, we show that the estimates are robust to model misspecification and presence of aberrant outliers. To this end, we develop several fundamental ideas relating minimum disparity estimators to Bayesian estimators built on the disparity-based posterior, for dependent tree-structured data. We illustrate the methodology through a simulated example and apply our methods to a real data set from cell kinetics.

[1]  Jayanta K. Ghosh,et al.  Higher Order Asymptotics , 1994 .

[2]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.

[3]  Anand N. Vidyashankar,et al.  Minimum Hellinger distance estimation for randomized play the winner design , 2006 .

[4]  David B. Dunson,et al.  Robust Bayesian Inference via Coarsening , 2015, Journal of the American Statistical Association.

[5]  A. Basu,et al.  General Robust Bayes Pseudo-Posterior: Exponential Convergence results with Applications , 2017, 1708.09692.

[6]  Kohur Gowrisankaran,et al.  Measurability of functions in product spaces , 1972 .

[7]  B. Lindsay.  Efficiency versus robustness : the case for minimum Hellinger distance and related methods , 1994 .

[8]  Kellen Petersen August.  Real Analysis , 2009 .

[9]  Á. Corral,et al.  Exact Derivation of a Finite-Size Scaling Law and Corrections to Scaling in the Geometric Galton-Watson Process , 2016, PloS one.

[10]  G. Debreu.  Integration of correspondences , 1967 .

[11]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[12]  A. Yakovlev,et al.  Branching Processes as Models of Progenitor Cell Populations and Estimation of the Offspring Distributions , 2008 .

[13]  M. Tanner,et al.  Gibbs posterior for variable selection in high-dimensional classification and data mining , 2008, 0810.5655.

[14]  T. N. Sriram,et al.  Minimum Hellinger distance estimation for supercritical Galton-Watson processes , 2000 .

[15]  Ollivier Hyrien,et al.  Stochastic modeling of oligodendrocyte generation in cell culture: model validation with time-lapse data , 2006, Theoretical Biology and Medical Modelling.

[16]  Marek Kimmel,et al.  Branching processes in biology , 2002 .

[17]  A. Basu,et al.  Robust Bayes estimation using the density power divergence , 2016 .

[18]  Jason Lowther.  Statistical Inference for Branching Processes , 1992 .

[19]  Dino Di Carlo,et al.  Increased Asymmetric and Multi-Daughter Cell Division in Mechanically Confined Microenvironments , 2012, PloS one.

[20]  Giles Hooker,et al.  Bayesian model robustness via disparities , 2011, 1112.4213.

[21]  Ayanendranath Basu,et al.  MINIMUM DISPARITY ESTIMATION : ASYMPTOTIC NORMALITY AND BREAKDOWN POINT RESULTS , 2004 .

[22]  M. Habib.  Probabilistic methods for algorithmic discrete mathematics , 1998 .

[23]  Daniel H. Wagner.  Survey of Measurable Selection Theorems , 1977 .

[24]  James O. Berger,et al.  Ordered group reference priors with application to the multinomial problem , 1992 .

[25]  Timothy R. C. Read,et al.  Multinomial goodness-of-fit tests , 1984 .

[26]  L. Brown,et al.  Measurable Selections of Extrema , 1973 .