Web & Social Media Analytics Previous Year Question Paper.pdf
Thesis Defense
1. Georgia State University EMPIRICAL LIKELIHOOD INFERENCE FOR THE ACCELERATED FAILURE TIME MODEL USING KENDALL ESTIMATING EQUATUION By Yinghua Lu June 29th 2009
18. Qin and Jing (2001) and Li and Wang (2003): the limiting distribution EL ratio is a weighted chi-square distribution.
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20. Main Procedure – Preliminaries We can rewrite it as a U-statistic with symmetric kernel, Similar to Fygenson and Ritov (1994), where R and J are defined similarly in Fygenson and Ritov (1994).
21. Main Procedure – Preliminaries The asymptotic variance of generalized estimate of β is The numerator can be estimated by The denominator can be estimated by Then we can construct the confidence interval as
22. Main Procedure – Empirical Likelihood Let and Apply the idea of Sen (1960), we define where W’s are independently distributed.
23. Main Procedure – Empirical Likelihood Let be a probability vector. Then the empirical likelihood function at the value β is given by For this function, reaches its maximum when Thus, the empirical likelihood ratio at β is defined by
24. Main Procedure – Empirical Likelihood By Lagrange Multiplier method for logarithm transformation of above equation, we write Setting the partial derivative of G with respect to p to 0, we have then
25. Main Procedure – Empirical Likelihood Plug into the previous equation, we obtain So, for all the p’s We have
26. Main Procedure – Empirical Likelihood Theorem 1 Under the above conditions, converges in distribution to , where is a chi-square random variable with p degrees of freedom. Confidence region for β is given by EL confidence region for the q sub-vector Of Theorem 2 Under the above conditions, converges in distribution to , where is a chi-square random variable with q degrees of freedom. confidence region for is given by
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28. The censoring time C ~ Uniform distribution in [0, c], where c controls the censoring rate.
51. Real Application We consider the following four variables: Disease Group (3 groups) Waiting Time to Transplant in Days (from 24 to 2616 days, mean=275 days) Recipient and Donor Age (from 7 to 52 and from 2 to 56) French-American-British (FAB): classification based on standard morphological criteria.
53. Real Application Results: Two methods show similar results. Two exceptions may due to asymmetric CI of the EL. Average lengths of the EL are a little longer than that of the NA. Same results with the simulation study.
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55. The coverage probabilities of the EL are closer to the nominal levels than NA, especially when the sample size is very small and censoring rate is heavy.