Consistency of the Semi-parametric MLE under the Cox Model with Right-Censored Data

Objective: We studied the consistency of the semi-parametric maximum likelihood estimator (SMLE) under the Cox regression model with right-censored (RC) data. Methods: Consistency proofs of the MLE are often based on the Shannon-Kolmogorov inequality, which requires finite E(lnL), where L is the likelihood function. Results: The results of this study show that one property of the semi-parametric MLE (SMLE) is established. Conclusion: Under the Cox model with RC data, E(lnL) may not exist. We used the Kullback-Leibler information inequality in our proof.

A More Suitable Definition of Quadratic Surds

Background: The properties of the group PGL(2,C) on the Upper Poincar´e Half Plane have been analyzed. Objective: In particular, the classification of points and geodesics has been achieved by considering the solution to the free Hamiltonian associated problem. Methods: The free Hamiltonian associated problem implies to discard the symmetry sl(2,Z) for the definition of reduced geodesics. By means of the new definition and classification of reduced geodesics, new construction for tori, punctured tori, and the tessellation of the Upper Poincar´e Half Plane is found. Results: A definition of quadratic surds is proposed, for which the folding group corresponds to the tiling group, (also) for Hamiltonian systems on the Hyperbolic Plane (also realized as the Upper Poincar´e Half Plane (UPHP)). Conclusion: The initial conditions determine the result of the folding of the trajectories as tiling punctured tori and for tori.

On The Distribution of Partial Sums of Randomly Weighted Powers of Uniform Spacings

Objectives: To study the asymptotic theory of the randomly wieghted partial sum process of powers of k-spacings from the uniform distribution. Methods: Earlier results on the distribution of the uniform incremental randomly weighted sums. Methods: Based on theorems on weak and strong approximations of partial sum processes. Results and conculsions: Our main contribution is to prove the weak convergence of weighted sum of powers of uniform spacings.