Search

Necessary and sufficient conditions for the weak convergence of the generalized range, midrange, extremal quotient, and extremal product are obtained. The classes of possible non degenerate limit distribution functions of these simple statistics are characterized. Comparison study between these statistics with some examples for the most important distribution functions are given. [ABSTRACT FROM PUBLISHER]

In survival analysis, the classical Koziol-Green random censorship model is commonly used to describe informative censoring. Hereby, it is assumed that the distribution of the censoring time is a power of the distribution of the survival time. In this article, we extend this model by assuming a general function between these distributions. We determine this function from a relationship between the observable random variables which is described by a copula family that depends on an unknown parameter θ. For this setting, we develop a semi-parametric estimator for the distribution of the survival time in which we propose a pseudo-likelihood estimator for the copula parameter θ. As results, we show first the consistency and asymptotic normality of the estimator for θ. Afterwards, we prove the weak convergence of the process associated to the semi-parametric distribution estimator. Furthermore, we investigate the finite sample performance of these estimators through a simulation study and finally apply it to a practical data set on survival with malignant melanoma. [ABSTRACT FROM AUTHOR]

The Marshall and Olkin (1997) parametrization operation is given by [image omitted] where F1 is a given distribution function and α > 0, [image omitted] = 1 - α. In this article, we show that both Fα = Pα(F1) and the generating distribution F1 belong to the same domain of maximal (minimal) (upper record value) (lower record value) attraction. Moreover, it is shown that the weak convergence of any non extreme order statistic (order statistic with variable rank) based on F1, under given normalizing constants, to a non degenerate limit type implies that the same non extreme order statistic based on the family Fα = Pα(F1), α≠1, under the same normalizing constants, does not converge to any non degenerate distribution and vice versa. [ABSTRACT FROM AUTHOR]

Let [image omitted] be a triangular array of independent elliptical random vectors in d,d ≥ 2. In a recent article, we show for the case d = 2 that the bivariate sample maxima of this triangular array is attracted by a max-infinitely divisible bivariate distribution function, provided that the components of the triangular array become asymptotically dependent and further the random radius associated to the elliptical random vectors has distribution function in the Weibull max-domain of attraction. [ABSTRACT FROM AUTHOR]

Self-consistency equations are established for the distribution functions of right- and left-censored one- and two-dimensional variables and sojourn times of a Markov renewal process. They have a unique solution that equals the product-limit estimator if a hazard function may be defined. Under right-censoring, the results presented here provide new formulations of known estimators. If the left- and right-censoring times are dependent, no estimators were available and simple algorithms are defined. All the estimators are [image omitted]-consistent and converge weakly to centered Gaussian processes. [ABSTRACT FROM AUTHOR]

We construct a univariate exponential dispersion model comprised of discrete infinitely divisible distributions. This model emerges in the theory of branching processes. We obtain a representation for the Lévy measure of relevant distributions and characterize their laws as Poisson mixtures and/or compound Poisson distributions. The regularity of the unit variance function of this model is employed for the derivation of approximations by the Poisson-exponential model. We emphasize the role of the latter class. We construct local approximations relating them to properties of special functions and branching diffusions. [ABSTRACT FROM AUTHOR]

We derive new properties for the power-variance family of probability distributions, which are also referred to as Tweedie laws. They emerge in the theory of generalized linear models and have a wide range of applications. We present decomposition criteria for Tweedie distributions which generalize Cochran's and Raikov's theorems. These criteria are interpreted in terms of the additivity of a shape parameter. The intersection of the power-variance family with the class of distributions which have no indecomposable components is determined. We reveal continuity of this family and obtain convenient common formulas for the cumulants. [ABSTRACT FROM AUTHOR]

We consider one dimensional probability distributions μ having a continuous and positive probability density function. We find the asymptotic of the size and the mass of the Voronoi cells and we prove that the local distortion associated with stationary or optimal quantizers is asymptotically uniform. Numerical simulations and computations illustrate the theoretical results and lead to the design of some good-fit test for the stationary equilibria. [ABSTRACT FROM AUTHOR]

The smooth nonparametric estimator &Qtilde;[sub n](p) of a quantile function Q(p) is defined as the solution of &Ftilde;[sub n] (&Qtilde;[sub n] (p)) = p, where &Ftilde;[sub n] is the distribution function corresponding to a kernel estimator of a density function. The asymptotic properties of the smooth quantile process, √n(&Qtilde;[sub n] (p) - Q(p)), based on randomly right censored lifetime data are studied. The bootstrap approaches to approximate the distributions of the smooth quantile processes are investigated and are used to construct simultaneous confidence bands for quantile functions. Data-based selection of the bandwidth required for computing &Ftilde;[sub n] is also investigated using bootstrap methods. A Monte Carlo simulation is carried out to assess small sample performance of the proposed confidence bands. An application to construct confidence bands for the quantile function of the time between a manuscript's submission and its first review is provided using a JASA data set. The developed results can be applied to construct simultaneous confidence bands for the difference of two quantile functions and to check whether there is a location shift or scale change for two distributions under study. [ABSTRACT FROM AUTHOR]