Emad-Eldin A. A. Aly

Some tests for comparing cumulative incidence functions and cause-specific hazard rates

Abstract We consider the competing risks problem with the available data in the form of times and causes of failure. In many practical situations (e.g., in reliability testing) it is important to know whether two risks are equal or whether one is “more serious” than the other. We propose some distribution-free tests for comparing cumulative incidence functions and cause-specific hazard rates against ordered alternatives without making any assumptions on the nature of dependence between the risks. Both the censored and the uncensored cases are studied. The performance of the proposed tests is assessed in a simulation study. As an illustration, we compare the risks of two types of cancer mortality (thymic lymphoma and reticulum cell carcinoma) in a strain of laboratory mice.

First order autoregressive time series with negative binomial and geometric marginals

In this paper we present first order autoregressive (AR(1)) time series with negative binomial and geometric marginals. These processes are the discrete analogues of the gamma and exponential processes introduced by Sim (1990). Many properties of the processes discussed here, such as autocorrelation, regression and joint distributions, are studied.

Strong Approximations of the Quantile Process of the Product-Limit Estimator

The quantile process of the product-limit estimator (PL-quantile process) in the random censorship model from the right is studied via strong approximation methods. Some almost sure fluctuation properties of the said process are studied. Sections 3 and 4 contain strong approximations of the PL-quantile process by a generalized Kiefer process. The PL and PL-quantile processes by the same appropriate Kiefer process are approximated and it is demonstrated that this simultaneous approximation cannot be improved in general. Section 5 contains functional LIL for the PL-quantile process and also three methods of constructing confidence bands for theoretical quantiles in the random censorship model from the right.

Explicit stationary distributions for some Galton-Watson processes with immigration

The purpose of this paper is to introduce and study a class of Galton-Watson processes with immigration. This class contains several recent valued AR 1 processes proposed by Al-Osh and Alzaid, Al-Osh and Aly, and McKenzie

Stationary solutions for integer-valued autoregressive processes

The purpose of this paper is to introduce and develop a family of ℤ + -valued autoregressive processes of order p ( INAR ( p ) ) by using the generalized multiplication ⊙ F of van Harn and Steutel (1982). We obtain various distributional and regression properties for these models. A number of stationary INAR ( p ) processes with specific marginals are presented and are shown to generalize several existing models.

A simple test for dispersive ordering

A simple asymptotically distribution free test is proposed for testing for dispersive ordering in the two sample case. A one sample version of the test is also given.

Strong approximations of the Q-Q process

We obtain strong approximation results for the empirical Q-Q process. Moreover, some Glivenko-Cantelli-type results are also obtained. These results are applicable to construct confidence bands for Q-Q plots.

On Geometric Infinite Divisibility and Stability

The purpose of this paper is to study geometric infinite divisibility and geometric stability of distributions with support in Z+ and R+. Several new characterizations are obtained. We prove in particular that compound-geometric (resp. compound-exponential) distributions form the class of geometrically infinitely divisible distributions on Z+ (resp. R+). These distributions are shown to arise as the only solutions to a stability equation. We also establish that the Mittag-Leffler distributions characterize geometric stability. Related stationary autoregressive processes of order one (AR(1)) are constructed. Importantly, we will use Poisson mixtures to deduce results for distributions on R+ from those for their Z+-counterparts.

Quantile-Quantile plots under random censorship

Abstract We obtain strong approximation results for the product-limit Quantile-Quantile (PL-Q-Q) process. In addition, product-limit confidence bands for the theoretical QQ plot are constructed.

Nonparametric Tests for Comparing Two Mean Residual Life Functions

We present nonparametric tests for the comparison of two mean residual life functions. We propose a new graphical approach for the simultaneous comparison of two mean residual life functions and their corresponding failure rate functions. We also consider the problem of testing against crossing mean residual life functions.