Multivariate radial symmetry of copula functions: finite sample comparison in the i.i.d case

Given a d-dimensional random vector X = (X 1, . . ., X d ), if the standard uniform vector U obtained by the component-wise probability integral transform (PIT) of X has the same distribution of its point reflection through the center of the unit hypercube, then X is said to have copula radial symmetry. We generalize to higher dimensions the bivariate test introduced in [11], using three different possibilities for estimating copula derivatives under the null. In a comprehensive simulation study, we assess the finite-sample properties of the resulting tests, comparing them with the finite-sample performance of the multivariate competitors introduced in [17] and [1].

Robust Reliability Estimation for Lindley Distribution—A Probability Integral Transform Statistical Approach

In the modeling and analysis of reliability data via the Lindley distribution, the maximum likelihood estimator is the most commonly used for parameter estimation. However, the maximum likelihood estimator is highly sensitive to the presence of outliers. In this paper, based on the probability integral transform statistic, a robust and efficient estimator of the parameter of the Lindley distribution is proposed. We investigate the relative efficiency of the new estimator compared to that of the maximum likelihood estimator, as well as its robustness based on the breakdown point and influence function. It is found that this new estimator provides reasonable protection against outliers while also being simple to compute. Using a Monte Carlo simulation, we compare the performance of the new estimator and several well-known methods, including the maximum likelihood, ordinary least-squares and weighted least-squares methods in the absence and presence of outliers. The results reveal that the new estimator is highly competitive with the maximum likelihood estimator in the absence of outliers and outperforms the other methods in the presence of outliers. Finally, we conduct a statistical analysis of four reliability data sets, the results of which support the simulation results.

Marginal standardization of upper semicontinuous processes. With application to max-stable processes

Extreme value theory for random vectors and stochastic processes with continuous trajectories is usually formulated for random objects where the univariate marginal distributions are identical. In the spirit of Sklar's theorem from copula theory, such marginal standardization is carried out by the pointwise probability integral transform. Certain situations, however, call for stochastic models whose trajectories are not continuous but merely upper semicontinuous (USC). Unfortunately, the pointwise application of the probability integral transform to a USC process does not, in general, preserve the upper semicontinuity of the trajectories. In this paper we give sufficient conditions to enable marginal standardization of USC processes and we state a partial extension of Sklar's theorem for USC processes. We specialize the results to max-stable processes whose marginal distributions and normalizing sequences are allowed to vary with the coordinate.

New Results on Goodness-of-Fit Tests for the Power-Law Process and Application to Software Reliability

The Power-Law process, also known as the Duane model, is a nonhomogeneous Poisson process which is widely used in reliability growth modeling. Goodness-of-fit tests for this process have been proposed by Crow, Rigdon, Klefsjö–Kumar and Park–Kim. In this paper, we propose several new tests based on chi-square, Laplace and conditional probability integral transform (CPIT) methodologies. We show that the TTT-plot test reduces to the Laplace test with Durbin's modification. We compare the power of 15 goodness-of-fit tests for the Power-Law process, and apply them to real software reliability data.