Iterated Logarithm Laws on GLM Randomly Censored with Random Regressors and Incomplete Information

Qiang Zhu, Zhihong Xiao, Guanglian Qin, Fang Ying

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DOI: 10.4236/am.2011.23043                

Abstract

In this paper, we define the generalized linear models (GLM) based on the observed data with incomplete information and random censorship under the case that the regressors are stochastic. Under the given conditions, we obtain a law of iterated logarithm and a Chung type law of iterated logarithm for the maximum likelihood estimator (MLE) in the present model.

Keywords

Generalized Linear Model, Incomplete Information, Stochastic Regressor, Iterated Logarithm Laws

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	J. A. Nelder and R. W. Wedderburn, “Generalized Linear models,” Journal of the Royal Statistical Society: Series A, 135, Part 3, 1972, pp. 370-384. doi:10.2307/2344614
[2]	L. Fahrmeir and H. Kaufmann, “Consistency and Asymtotic Normality of the Maximum Likelihood Estimator in Generalized Linear Models,” Annals of Statistics, Vol. 13, No. 1, 1985, pp. 342-368. doi:10.1214/aos/1176346597
[3]	T. Elperin and I. Gertsbak, “Estimation in a Random Censoring Model with Incomplete Information: Exponential Lifetime Distribution,” IEEE Transactions on Reliability, Vol. 37, No. 2, 1988, pp. 223-229. doi:10.1109/24.3745
[4]	T. L. Lai and C. Z. Wei, “Least Squares Estimates in Stochastic Regression Models with Applications to Identification and Control of Dynamic Systems,” Annals of Statistics, Vol. 10, No. 1, 1982, pp. 154-186. doi:10.1214/aos/1176345697
[5]	S. L. Zeger and M. R. Karim “Generalized Linear Models with Random Effects; A Gibbs' Sampling Approach,” Journal of the American Statistical Association, Vol. 86, No. 413, 1991, pp. 79-86. doi:10.2307/2289717
[6]	Z. H. Xiao and L. Q. Liu, “MLE of Generalized Linear Model Randomly Censored with Incomplete Information,” Acta Mathematica Scientia, Vol. 3(A), 2008, pp. 553-564.
[7]	Z. H. Xiao and L. Q. Liu. “Laws of Iterated Logarithm for Quasi-Maximum Likelihood Estimator in Generalized Linear Model,” Journal of statistical planning and inference, Vol. 138, No. 3, 2008, pp. 611-617. doi:10.1016/j.jspi.2006.12.006
[8]	Z. H. Xiao and L. Q. Liu, “Laws of Iterated Logarithm for MLE of Generalized Linear Model Randomly Censored with Incomplete Information,” Statistics and Probability Letters, Vol. 79, No. 6, 2009, pp. 789-796. doi:10.1016/j.spl.2008.11.016
[9]	J. L. Ding and X. R. Chen, “Asymptotic Properties of the Maximum Likelihood Estimate in Generalized Linear Models with Stochastic Regressors,” Acta Mathematica Sinica, Vol. 22, No. 6, 2006, pp. 1679-1686. doi:10.1007/s10114-005-0693-3