Robust inference for variance components models for single trees of cell lineage data. (English) Zbl 0865.62016

Summary: R. M. Huggins and R. G. Staudte [J. Am. Stat. Assoc. 89, No. 425, 19-29 (1994)] examined robust estimators for a variance components formulation of the bifurcating autoregressive model for cell lineage data. They gave asymptotic properties of the estimators if a large number of trees were observed. However, for single trees the derivation of these asymptotic properties is more complex.
Here the asymptotic distributions of robust estimators of parameters associated with the stationary bifurcating autoregressive process as a single tree becomes large are obtained. These results follow from the formulation of the estimating functions as the product of a nonrandom matrix and the sum of vectors of functions of an infinite sequence of exchangeable random variables.


62F35 Robustness and adaptive procedures (parametric inference)
62E20 Asymptotic distribution theory in statistics
62P10 Applications of statistics to biology and medical sciences; meta analysis
62J10 Analysis of variance and covariance (ANOVA)
62M99 Inference from stochastic processes
60G09 Exchangeability for stochastic processes
Full Text: DOI


[1] Aitchison, J. and Silvey, S. D. (1958). Maximum likelihood estimation of parameters subject to restraints. Ann. Math. Statist. 29 813-828. · Zbl 0092.36704 · doi:10.1214/aoms/1177706538
[2] Cowan, R. and Staudte, R. G. (1986). The bifurcating autoregression model in cell lineage studies. Biometrics 42 769-783. · Zbl 0622.62105 · doi:10.2307/2530692
[3] Crowder, M. J. (1976). Maximum likelihood estimation for dependent observations. J. Roy. Statist. Soc. Ser. B 38 45-53. JSTOR: · Zbl 0324.62023
[4] Crowder, M. J. (1986). On consistency and inconsistency of estimating equations. Econometric Theory 2 305-330. Huggins, R. M. (1993a). On the robust analysis of variance components models for pedigree data. Austral. J. Statist. 35 43-57. Huggins, R. M. (1993b). A robust approach to the analysis of repeated measures. Biometrics 49 715-720.
[5] Huggins R. M. (1995). A law of large numbers for the bifurcating autoregressive process. Comm. Statist. Stochastic Models 11 273-278. · Zbl 0824.60027 · doi:10.1080/15326349508807345
[6] Huggins, R. M. (1996). On the identifiability of measurement error in the bifurcating autoregressive model. Statist. Probab. Lett. 27 17-23. · Zbl 0885.62085 · doi:10.1016/0167-7152(95)00038-0
[7] Huggins, R. M. and Marschner, I. C. (1991). Robust analysis of the bifurcating autoregressive model in cell lineage studies. Austral. J. Statist. 33 209-220. · Zbl 0781.62176 · doi:10.1111/j.1467-842X.1991.tb00428.x
[8] Huggins, R. M. and Staudte, R. G. (1994). Variance components models for dependent cell populations. J. Amer. Statist. Assoc. 89 19-29. · Zbl 0800.62730 · doi:10.2307/2291197
[9] Kelker, D. (1970). Distribution theory of spherical distributions and a location-scale parameter generalisation. Sankhy\?a Ser. A 32 419-430. · Zbl 0223.60008
[10] Kingman, J. F. C. (1972). On random sequences with spherical sy mmetry. Biometrika 59 492-493. JSTOR: · Zbl 0238.60025 · doi:10.1093/biomet/59.2.492
[11] Klimko, L A. and Nelson, P. I. (1978). On conditional least squares estimation for stochastic processes. Ann. Statist. 6 629-642. · Zbl 0383.62055 · doi:10.1214/aos/1176344207
[12] Marschner, I. C. (1991). Robust estimation for epidemic models. Austral. J. Statist. 33 221-240. · Zbl 0766.62063 · doi:10.1111/j.1467-842X.1991.tb00429.x
[13] Miller, J. J. (1977). Asy mptotic properties of maximum likelihood estimates in the mixed model of the analysis of variance. Ann. Statist. 5 746-762. · Zbl 0406.62017 · doi:10.1214/aos/1176343897
[14] Powell, E. O. (1955). Some features of the generation times of individual bacteria. Biometrika 42 16-44.
[15] Powell, E. O. (1956). An improved culture chamber for the study of living bacteria. Journal of the Roy al Microscopical Society 75 235.
[16] Powell, E. O. (1958). An outline of the pattern of bacterial generation times. Journal of General Microbiology 18 382-417.
[17] Powell, E. O. and Errington, F. P. (1963). Generation times of individual bacteria: some corroborative measurements. Journal of General Microbiology 31 315-327.
[18] Shiry ayev, A. N. (1984). Probability. Springer, New York. · Zbl 0536.60001
[19] Staudte, R. G., Guiget, M. and Colly n d’Hooge, M. (1984). Additive models for dependent cell populations. Journal of Theoretical Biology 109 127-146.
[20] Stewart, G. W. (1973). Introduction to Matrix Computations. Academic Press, New York. · Zbl 0302.65021
[21] Tay lor, R. L., Daffer, P. Z. and Patterson, R. F. (1985). Limit Theorems for Sums of Exchangeable Random Variables. Rowman and Allanheld, Totowa, NJ. · Zbl 0668.60003
[22] Weiss, L. (1973). Asy mptotic properties of maximum likelihood estimators in some non-standard cases. II. J. Amer. Statist. Assoc. 68 428-430. JSTOR: · Zbl 0272.62011 · doi:10.2307/2284091
[23] Weiss, L. (1975). The asy mptotic distribution of the likelihood ratio in some non-standard cases. J. Amer. Statist. Assoc. 70 204-208. JSTOR: · Zbl 0318.62015 · doi:10.2307/2285404
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.