Designing sampling plans to capture rare objects

This article focuses on two‐phase sampling designs for a population with unknown number of rare objects. The first phase is used to estimate the number of rare or potentially rare objects in a population, and the second phase to design sampling plans to capture a certain number or a certain proportion of such type of objects. A hypergeometric‐binomial model is applied to infer the number of rare or potentially rare objects and Monte Carlo simulation based approaches are developed to calculate needed sample sizes. Simulations and real data applications are discussed. The Canadian Journal of Statistics 37: 417–434; 2009 © 2009 Statistical Society of Canada     

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     

Parallelism,uniqueness, and large‐sample asymptotics for the Dantzig selector

The Dantzig selector (Candès & Tao, 2007) is a popular $\ell^{1}$ ‐regularization method for variable selection and estimation in linear regression. We present a very weak geometric condition on the observed predictors which is related to parallelism and, when satisfied, ensures the uniqueness of Dantzig selector estimators. The condition holds with probability 1, if the predictors are drawn from a continuous distribution. We discuss the necessity of this condition for uniqueness and also provide a closely related condition which ensures the uniqueness of lasso estimators (Tibshirani, 1996). Large sample asymptotics for the Dantzig selector, that is, almost sure convergence and the asymptotic distribution, follow directly from our uniqueness results and a continuity argument. The limiting distribution of the Dantzig selector is generally non‐normal. Though our asymptotic results require that the number of predictors is fixed (similar to Knight & Fu, 2000), our uniqueness results are valid for an arbitrary number of predictors and observations. The Canadian Journal of Statistics 41: 23–35; 2013 © 2012 Statistical Society of Canada     

Statistical inference for multivariate partially linear regression models

In this paper we study a class of multivariate partially linear regression models. Various estimators for the parametric component and the nonparametric component are constructed and their asymptotic normality established. In particular, we propose an estimator of the contemporaneous correlation among the multiple responses and develop a test for detecting the existence of such contemporaneous correlation without using any nonparametric estimation. The performance of the proposed estimators and test is evaluated through some simulation studies and an analysis of a real data set is used to illustrate the developed methodology. The Canadian Journal of Statistics 41: 1–22; 2013 © 2013 Statistical Society of Canada     

Variable selection and estimation for multivariate panel count data via the seamless‐${\it L}_{{\rm 0}}$ penalty

This paper considers regression analysis of multivariate panel count data with the focus on variable selection and estimation of significant covariate effects. For the problem, we adopt the penalized estimating equation approach with a focus on the use of the seamless‐$L_0$ penalty. The proposed approach selects variables and estimates regression coefficients simultaneously and the asymptotic properties of the resulting estimates are established. The procedure can be easily carried out with the Newton–Raphson algorithm and is evaluated by simulation studies. Also it is applied to a motivating data set arising from a skin cancer study. The Canadian Journal of Statistics 41: 368–385; 2013 © 2013 Statistical Society of Canada     

Weak convergence of residual median process for dependent or randomly censored samples

This article studies the weak convergence of the residual median process (i) when the observations follow a strictly stationary ø-mixing process and (ii) when hte observations are randomly censored. In both these cased the residual median prodeas converges weakly to a gaussian process.     

MULTIVARIATE GENERALIZED POLYA DISTRIBUTION OF ORDER k

ABSTRACT

An order k (or cluster) generalized Polya distribution and a multivariate generalized Polya-Eggenberger one where derived in (Sen, K.; Jain, R. Cluster Generalized Negative Binomial Distribution. In Probability Models and Statistics, A. J. Medhi Festschrift on the Occasion of his 70th Birthday; Borthakur, A.C. et al., Eds.; New Age International Publishers: New Delhi, 1996; 227–241 and Sen, K.; Jain, R. A Multivariate Generalized Polya-Eggenberger Probability Model-First Passage Approach. Communications in Statistics—Theory and Methods 1997, 26, 871–884). Presently, both distributions are generalized to a multivariate generalized Polya distribution of order k by means of an appropriate sampling scheme and a first passage event. This new distribution includes as special cases new multivariate Polya and inverse Polya distributions of order k and the multivariate generalized negative binomial distribution of the same order derived recently in (Tripsiannis, G.A.; Philippou, A.N.; Papathanasiou, A.A. Multivariate Generalized Distributions of Order k. Medical Statistics Technical Report #41: Democritus University of Thrace, Greece, 2001). Limiting cases are considered and applications are indicated.     

The covariance of the number of renewals in a fixed time and the ensuing excess life

The covariance of the number of renewals in a fixed time N ^t and the ensuing excess life time Y ^t is derived using matrix-analytic methods for the stationary PH-renewal process. Specific results for the Erlang and hyperexponential processes are provided to illustrate the ease of computation. Properties concerning the sign and the behavior of the covariance as t→∞ are provided throughout. Parameter estimation for renewal processes which cannot be fully observed serves as the motivation for our derivations. These statistical applications as well as links to estimation for service time distributions in queues shed light on the type of problems for which the covariance is useful.