The Cox Proportional Hazards Model with a Partly Linear Relative Risk Function

The conventional Cox proportional hazards regression model contains a loglinear relative risk function, linking the covariate information to the hazard ratio with a finite number of parameters. A generalization, termed the partly linear Cox model, allows for both finite dimensional parameters and an infinite dimensional parameter in the relative risk function, providing a more robust specification of the relative risk function. In this work, a likelihood based inference procedure is developed for the finite dimensional parameters of the partly linear Cox model. To alleviate the problems associated with a likelihood approach in the presence of an infinite dimensional parameter, the relative risk is reparameterized such that the finite dimensional parameters of interest are orthogonal to the infinite dimensional parameter. Inference on the finite dimensional parameters is accomplished through maximization of the profile partial likelihood, profiling out the infinite dimensional nuisance parameter using a kernel function. The asymptotic distribution theory for the maximum profile partial likelihood estimate is established. It is determined that this estimate is asymptotically efficient; the orthogonal reparameterization enables employment of profile likelihood inference procedures without adjustment for estimation of the nuisance parameter. An example from a retrospective analysis in cancer demonstrates the methodology.     

Optimal Ordering Policies for Inventory Problems With Dynamic Information Delays

Information delays exist when the most recent inventory information available to the Inventory Manager (IM) is dated. In other words, the IM observes only the inventory level that belongs to an earlier period. Such situations are not uncommon, and they arise when it takes a while to process the demand data and pass the results to the IM. We introduce dynamic information delays as a Markov process into the standard multiperiod stochastic inventory problem with backorders. We develop the concept of a reference inventory position. We show that this position along with the magnitude of the latest observed delay and the age of this observation are sufficient statistics for finding the optimal order quantities. Furthermore, we establish that the optimal ordering policy is of state‐dependent base‐stock type with respect to the reference inventory position (or state‐dependent (s, S) type if there is a fixed ordering cost). The optimal base stock and (s, S) levels depend on the magnitude of the latest observed delay and the age of this observation. Finally, we study the sensitivity of the optimal base stock and the optimal cost with respect to the sufficient statistics.     

Spatial Matérn Fields Driven by Non‐Gaussian Noise

The article studies non‐Gaussian extensions of a recently discovered link between certain Gaussian random fields, expressed as solutions to stochastic partial differential equations (SPDEs), and Gaussian Markov random fields. The focus is on non‐Gaussian random fields with Matérn covariance functions, and in particular, we show how the SPDE formulation of a Laplace moving‐average model can be used to obtain an efficient simulation method as well as an accurate parameter estimation technique for the model. This should be seen as a demonstration of how these techniques can be used, and generalizations to more general SPDEs are readily available.     

Testing for Equivalence of Means under Heteroskedasticity by an Approximate Solution of a Partial Differential Equation of Infinite Order

Abstract. A test for two‐sided equivalence of means has been developed under the assumption of normally distributed populations with heterogeneous variances. Its rejection region is limited by functions ± h that depend on the empirical variances. h is stated implicitly by a partial differential equation, an exact solution of which would provide a test that is exactly similar at the boundary of the null hypothesis of non‐equivalence. h is approximated by Taylor series up to third powers in the reciprocal number of degrees of freedom. This suffices to obtain error probabilities of the first kind that are very close to a nominal level of α = 0 . 05 at the boundary of the null hypothesis. For more than 10 data points in each group, they range between 0.04995 and 0.05005, and are thus much more precise than those obtained by other authors.     

Partial sufficient dimension reduction on joint model of recurrent and terminal events

Joint modeling of recurrent and terminal events has attracted considerable interest and extensive investigations by many authors. The assumption of low-dimensional covariates has been usually applied in the existing studies, which is however inapplicable in many practical situations. In this paper, we consider a partial sufficient dimension reduction approach for a joint model with high-dimensional covariates. Some simulations as well as three real data applications are presented to confirm and assess the performance of the proposed model and approach.     

Non‐parametric interval estimation for the partial area under the ROC curve

Accurate diagnosis of disease is a critical part of health care. New diagnostic and screening tests must be evaluated based on their abilities to discriminate diseased conditions from non‐diseased conditions. For a continuous‐scale diagnostic test, a popular summary index of the receiver operating characteristic (ROC) curve is the area under the curve (AUC). However, when our focus is on a certain region of false positive rates, we often use the partial AUC instead. In this paper we have derived the asymptotic normal distribution for the non‐parametric estimator of the partial AUC with an explicit variance formula. The empirical likelihood (EL) ratio for the partial AUC is defined and it is shown that its limiting distribution is a scaled chi‐square distribution. Hybrid bootstrap and EL confidence intervals for the partial AUC are proposed by using the newly developed EL theory. We also conduct extensive simulation studies to compare the relative performance of the proposed intervals and existing intervals for the partial AUC. A real example is used to illustrate the application of the recommended intervals. The Canadian Journal of Statistics 39: 17–33; 2011 © 2011 Statistical Society of Canada     

Identifying Treatment Effects Under Data Combination

We consider the identification of counterfactual distributions and treatment effects when the outcome variables and conditioning covariates are observed in separate data sets. Under the standard selection on observables assumption, the counterfactual distributions and treatment effect parameters are no longer point identified. However, applying the classical monotone rearrangement inequality, we derive sharp bounds on the counterfactual distributions and policy parameters of interest.     

A Practical Two‐Step Method for Testing Moment Inequalities

This paper considers the problem of testing a finite number of moment inequalities. We propose a two‐step approach. In the first step, a confidence region for the moments is constructed. In the second step, this set is used to provide information about which moments are “negative.” A Bonferonni‐type correction is used to account for the fact that, with some probability, the moments may not lie in the confidence region. It is shown that the test controls size uniformly over a large class of distributions for the observed data. An important feature of the proposal is that it remains computationally feasible, even when the number of moments is large. The finite‐sample properties of the procedure are examined via a simulation study, which demonstrates, among other things, that the proposal remains competitive with existing procedures while being computationally more attractive.     

Entropic Latent Variable Integration via Simulation

This paper introduces a general method to convert a model defined by moment conditions that involve both observed and unobserved variables into equivalent moment conditions that involve only observable variables. This task can be accomplished without introducing infinite‐dimensional nuisance parameters using a least favorable entropy‐maximizing distribution. We demonstrate, through examples and simulations, that this approach covers a wide class of latent variables models, including some game‐theoretic models and models with limited dependent variables, interval‐valued data, errors‐in‐variables, or combinations thereof. Both point‐ and set‐identified models are transparently covered. In the latter case, the method also complements the recent literature on generic set‐inference methods by providing the moment conditions needed to construct a generalized method of moments‐type objective function for a wide class of models. Extensions of the method that cover conditional moments, independence restrictions, and some state‐space models are also given.     

Incomplete probability assessments in decision analysis

Summary In analysing a decision problem, in a situation ofpartial knowledge, a decision maker may be reluctant to assign acomplete probability distribution on the relevant states of nature. In order to face this difficulty, several methods, based onindeterminate probabilities or probabilityintervals, have been proposed in the literature. In this paper, arguing that it is meaningless to judge probabilistic assessments ascorrect orwrong, it is maintained that onlycoherence has anobjective andsignificant role. Then to overcome practical difficulties, an approach based on thesubjective methodology and on the use ofnumerical andqualitative probabilities, is outlined.