On smoothness of Tukey depth contours

The smoothness of Tukey depth contours is a regularity condition often encountered in asymptotic theory, among others. This condition ensures that the Tukey depth fully characterizes the underlying multivariate probability distribution. In this paper we demonstrate that this regularity condition is rarely satisfied. It is shown that even well-behaved probability distributions with symmetrical, smooth and (strictly) quasi-concave densities may have non-smooth Tukey depth contours, and that the smoothness behaviour of depth contours is fairly unpredictable.     

Estimation of a Change Point in a Hazard Function Based on Censored Data

The hazard function plays an important role in reliability or survival studies since it describes the instantaneous risk of failure of items at a time point, given that they have not failed before. In some real life applications, abrupt changes in the hazard function are observed due to overhauls, major operations or specific maintenance activities. In such situations it is of interest to detect the location where such a change occurs and estimate the size of the change. In this paper we consider the problem of estimating a single change point in a piecewise constant hazard function when the observed variables are subject to random censoring. We suggest an estimation procedure that is based on certain structural properties and on least squares ideas. A simulation study is carried out to compare the performance of this estimator with two estimators available in the literature: an estimator based on a functional of the Nelson-Aalen estimator and a maximum likelihood estimator. The proposed least squares estimator tums out to be less biased than the other two estimators, but has a larger variance. We illustrate the estimation method on some real data sets.     

Social security reforms and early retirement

Many reform proposals of the social security systems in various OECD economies suggest to scale down the non-actuarial parts of the pension systems. These reforms have a flavor of increased efficiency at the costs of welfare losses for low-income individuals. Assessing the economic effects, we investigate five different reform proposals by means of a numerical overlapping generations model for the Norwegian economy. The model features an endogenous retirement age and heterogeneous individuals within generations. It turns out that the various reforms, which scale down the public non-actuarial pension system, lead to increases in the retirement age and steady-state welfare gains for all income classes. Received: 7 December 2000/Accepted: 29 January 2002 All correspondence to ?ystein Th?gersen. Financial support from the Research Council of Norway (The Economic Research Program on Taxation) is gratefully acknowledged. We are indebted to Lans Bovenberg, John Ermisch, Erling Steigum and two referees for valuable comments and useful discussion. Responsible editor: John F. Ermisch.     

Local linear fitting and improved estimation near peaks

The authors look into the problem of estimating regression functions that exhibit jump irregularities in the first derivative. They investigate the behaviour of the bias in the local linear fit and show the superior performance of appropriate one‐sided versions of the local linear fit near such irregularities. They then propose an improved estimation procedure based on data‐driven selection of a conventional or one‐sided local linear fit according to a residual sum of squares type of criterion. The authors provide theoretical results and illustrate the method both on simulated and real‐life data examples. The Canadian Journal of Statistics 37: 453–475; 2009 © 2009 Statistical Society of Canada     

Positive quadrant dependence tests for copulas

In this paper the interest is in testing the null hypothesis of positive quadrant dependence (PQD) between two random variables. Such a testing problem is important since prior knowledge of PQD is a qualitative restriction that should be taken into account in further statistical analysis, for example, when choosing an appropriate copula function to model the dependence structure. The key methodology of the proposed testing procedures consists of evaluating a “distance” between a nonparametric estimator of a copula and the independence copula, which serves as a reference case in the whole set of copulas having the PQD property. Choices of appropriate distances and nonparametric estimators of copula are discussed, and the proposed methods are compared with testing procedures based on bootstrap and multiplier techniques. The consistency of the testing procedures is established. In a simulation study the authors investigate the finite sample size and power performances of three types of test statistics, Kolmogorov–Smirnov, Cramér–von‐Mises, and Anderson–Darling statistics, together with several nonparametric estimators of a copula, including recently developed kernel type estimators. Finally, they apply the testing procedures on some real data. The Canadian Journal of Statistics 38: 555–581; 2010 © 2010 Statistical Society of Canada     

Copula directed acyclic graphs

A new methodology for selecting a Bayesian network for continuous data outside the widely used class of multivariate normal distributions is developed. The ‘copula DAGs’ combine directed acyclic graphs and their associated probability models with copula C/D-vines. Bivariate copula densities introduce flexibility in the joint distributions of pairs of nodes in the network. An information criterion is studied for graph selection tailored to the joint modeling of data based on graphs and copulas. Examples and simulation studies show the flexibility and properties of the method.     

Nonparametric testing for no covariate effects in conditional copulas

In dependence modelling using conditional copulas, one often imposes the working assumption that the covariate influences the conditional copula solely through the marginal distributions. This so-called (pairwise) simplifying assumption is almost standardly made in vine copula constructions. However, in recent literature evidence was provided that such an assumption might not be justified. Among the first issues is thus to test for its appropriateness. In this paper nonparametric tests for the null hypothesis of the simplifying assumption are proposed, and their asymptotic behaviours, under the null hypothesis and under some local alternatives, are established. The tests are fully nonparametric in nature: not requiring choices of copula families nor knowledge of the marginals. In a simulation study, the finite-sample size and power performances of the tests are investigated, and compared with these of the few available tests. A real data application illustrates the use of the tests.     

Positive quadrant dependence testing and constrained copula estimation

Positive quadrant dependence is a specific dependence structure that is of practical importance in for example modelling dependencies in insurance and actuarial sciences. This dependence structure imposes a constraint on the copula function. The interest in this paper is to test for positive quadrant dependence. One way to assess the distribution of the test statistics under the null hypothesis of positive quadrant dependence is to resample from a constrained copula. This requires constrained estimation of a copula function. We show that this use of resampling under a constrained copula improves considerably the power performance of existing testing procedures. We propose two resampling procedures, one based on a parametric constrained copula estimation and one relying on nonparametric estimation of a positive quadrant dependence copula, and discuss their properties. The finite‐sample performances of the resulting testing procedures are evaluated via a simulation study that also includes comparisons with existing tests. Finally, a data set of Danish fire insurance claims is tested for positive quadrant dependence. The Canadian Journal of Statistics 41: 36–64; 2013 © 2012 Statistical Society of Canada     

On Geometry and Scale of a Stochastic Chemostat

A chemostat is a fixed volume bioreactor in which micro–organisms are grown in a continuously renewed liquid medium. We propose a stochastic model for the evolution of the concentrations in the single species and single substrate case. It is obtained as a diffusion approximation of a pure jump Markov process, whose increments are comparable in mean with the deterministic model. A specific time scale, related to the noise intensity, is considered for each source of variation. The geometric structure of the problem, usable by identification procedures, is preserved both in the drift and diffusion term. We study the properties of this model by numerical experiments.     

About estimation of ARIMA process with strong mixing MA part

A central limit theorem is provided for the least squares estimates of the autoregressive parameters in an ARIMA process with strong mixing moving average part.