Recent and classical goodness-of-fit tests for the Poisson distribution

We give a critical synopsis of classical and recent tests for Poissonity, our emphasis being on procedures which are consistent against general alternatives. Two classes of weighted Cramér–von Mises type test statistics, based on the empirical probability generating function process, are studied in more detail. Both of them generalize already known test statistics by introducing a weighting parameter, thus providing more flexibility with regard to power against specific alternatives. In both cases, we prove convergence in distribution of the statistics under the null hypothesis in the setting of a triangular array of rowwise independent and identically distributed random variables as well as consistency of the corresponding test against general alternatives. Therefore, a sound theoretical basis is provided for the parametric bootstrap procedure, which is applied to obtain critical values in a large-scale simulation study. Each of the tests considered in this study, when implemented via the parametric bootstrap method, maintains a nominal level of significance very closely, even for small sample sizes. The procedures are applied to four well-known data sets.     

Bayes linear analysis for graphical models: The geometric approach to local computation and interpretive graphics

This paper concerns the geometric treatment of graphical models using Bayes linear methods. We introduce Bayes linear separation as a second order generalised conditional independence relation, and Bayes linear graphical models are constructed using this property. A system of interpretive and diagnostic shadings are given, which summarise the analysis over the associated moral graph. Principles of local computation are outlined for the graphical models, and an algorithm for implementing such computation over the junction tree is described. The approach is illustrated with two examples. The first concerns sales forecasting using a multivariate dynamic linear model. The second concerns inference for the error variance matrices of the model for sales, and illustrates the generality of our geometric approach by treating the matrices directly as random objects. The examples are implemented using a freely available set of object-oriented programming tools for Bayes linear local computation and graphical diagnostic display.     

A new stratified three-stage unrelated randomized response model for estimating a rare sensitive attribute based on the Poisson distribution

This article suggests an efficient method of estimating a rare sensitive attribute which is assumed following Poisson distribution by using three-stage unrelated randomized response model instead of the Land et al. model (2011 Land, M., S. Singh, and S. A. Sedory. 2011. Estimation of a rare sensitive attribute using poisson distribution. Statistics 46 (3):351–60. doi:10.1080/02331888.2010.524300.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) when the population consists of some different sized clusters and clusters selected by probability proportional to size(:pps) sampling. A rare sensitive parameter is estimated by using pps sampling and equal probability two-stage sampling when the parameter of a rare unrelated attribute is assumed to be known and unknown.

We extend this method to the case of stratified population by applying stratified pps sampling and stratified equal probability two-stage sampling. An empirical study is carried out to show the efficiency of the two proposed methods when the parameter of a rare unrelated attribute is assumed to be known and unknown.     

A modified LGB method for detecting significant effects based on a half-normal probability plot

In analyzing data from unreplicated factorial designs, the half-normal probability plot is commonly used to screen for the ‘vital few’ effects. Recently, many formal methods have been proposed to overcome the subjectivity of this plot. Lawson (1998) (hereafter denoted as LGB) suggested a hybrid method based on the half-normal probability plot, which is a blend of Lenth (1989) and Loh (1992) method. The method consists of fitting a simple least squares line to the inliers, which are determined by the Lenth method. The effects exceeding the prediction limits based on the fitted line are candidates for the vital few effects. To improve the accuracy of partitioning the effects into inliers and outliers, we propose a modified LGB method (hereafter denoted as the Mod_LGB method), in which more outliers can be classified by using both the Carling’s modification of the box plot (Carling, 2000) and Lenth method. If no outlier exists or there is a wide range in the inliers as determined by the Lenth method, more outliers can be found by the Carling method. A simulation study is conducted in unreplicated $2^4$ designs with the number of active effects ranging from 1 to 6 to compare the efficiency of the Lenth method, original LGB methods, and the proposed modified version of the LGB method.     

Linear Increments with Non‐monotone Missing Data and Measurement Error

Linear increments (LI) are used to analyse repeated outcome data with missing values. Previously, two LI methods have been proposed, one allowing non‐monotone missingness but not independent measurement error and one allowing independent measurement error but only monotone missingness. In both, it was suggested that the expected increment could depend on current outcome. We show that LI can allow non‐monotone missingness and either independent measurement error of unknown variance or dependence of expected increment on current outcome but not both. A popular alternative to LI is a multivariate normal model ignoring the missingness pattern. This gives consistent estimation when data are normally distributed and missing at random (MAR). We clarify the relation between MAR and the assumptions of LI and show that for continuous outcomes multivariate normal estimators are also consistent under (non‐MAR and non‐normal) assumptions not much stronger than those of LI. Moreover, when missingness is non‐monotone, they are typically more efficient.     

On conditional risk estimation considering model risk

Usually, parametric procedures used for conditional variance modelling are associated with model risk. Model risk may affect the volatility and conditional value at risk estimation process either due to estimation or misspecification risks. Hence, non-parametric artificial intelligence models can be considered as alternative models given that they do not rely on an explicit form of the volatility. In this paper, we consider the least-squares support vector regression (LS-SVR), weighted LS-SVR and Fixed size LS-SVR models in order to handle the problem of conditional risk estimation taking into account issues of model risk. A simulation study and a real application show the performance of proposed volatility and VaR models.     

A Proportional Hazards Regression Model for the Subdistribution with Covariates‐adjusted Censoring Weight for Competing Risks Data

With competing risks data, one often needs to assess the treatment and covariate effects on the cumulative incidence function. Fine and Gray proposed a proportional hazards regression model for the subdistribution of a competing risk with the assumption that the censoring distribution and the covariates are independent. Covariate‐dependent censoring sometimes occurs in medical studies. In this paper, we study the proportional hazards regression model for the subdistribution of a competing risk with proper adjustments for covariate‐dependent censoring. We consider a covariate‐adjusted weight function by fitting the Cox model for the censoring distribution and using the predictive probability for each individual. Our simulation study shows that the covariate‐adjusted weight estimator is basically unbiased when the censoring time depends on the covariates, and the covariate‐adjusted weight approach works well for the variance estimator as well. We illustrate our methods with bone marrow transplant data from the Center for International Blood and Marrow Transplant Research. Here, cancer relapse and death in complete remission are two competing risks.     

Combining Inverse Probability Weighting and Multiple Imputation to Improve Robustness of Estimation

Inverse probability weighting (IPW) and multiple imputation are two widely adopted approaches dealing with missing data. The former models the selection probability, and the latter models data distribution. Consistent estimation requires correct specification of corresponding models. Although the augmented IPW method provides an extra layer of protection on consistency, it is usually not sufficient in practice as the true data‐generating process is unknown. This paper proposes a method combining the two approaches in the same spirit of calibration in sampling survey literature. Multiple models for both the selection probability and data distribution can be simultaneously accounted for, and the resulting estimator is consistent if any model is correctly specified. The proposed method is within the framework of estimating equations and is general enough to cover regression analysis with missing outcomes and/or missing covariates. Results on both theoretical and numerical investigation are provided.