We discuss two novel approaches to inter-distributional comparisons in the classical two-sample problem. Our starting point is properly standardised and combined, very popular in several areas of statistics and data analysis, ordinal dominance and receiver characteristic curves, denoted by ODC and ROC, respectively. The proposed new curves are termed the comparison curves. Their estimates, being weighted rank processes on (0,1), form the basis of inference. These weighted processes are intuitive, well-suited for visual inspection of data at hand and are also useful for constructing some formal inferential procedures. They can be applied to several variants of two-sample problem. Their use can help improve some existing procedures both in terms of power and the ability to identify the sources of departures from the postulated model. To simplify interpretation of finite sample results, we restrict attention to values of the processes on a finite grid of points. This results in the so-called bar plots (B-plots), which readably summarise the information contained in the data. What is more, we show that B-plots along with adjusted simultaneous acceptance regions provide principled information about where the model departs from the data. This leads to a framework that facilitates identification of regions with locally significant differences.
We show an implementation of the considered techniques to a standard stochastic dominance testing problem. Some min-type statistics are introduced and investigated. A simulation study compares two tests pertinent to the comparison curves to well-established tests in the literature and demonstrates the strong and competitive performance of the former in many typical situations. Some real data applications illustrate simplicity and practical usefulness of the proposed approaches. A range of other applications of considered weighted processes is briefly discussed too.
Estimating the size of a hard-to-count population is a challenging matter. We consider uni-list approaches in which the count of identifications per unit is the basis of analysis. Unseen units have a zero count and do not occur in the sample leading to a zero-truncated setting. Because of various mechanisms, one-inflation is often an occurring phenomena that can lead to seriously biased estimates of population size. The current work reviews some recent advances on one-inflation and zero-truncation modelling, and furthermore focuses here on the impact it has on population size estimation. The zero-truncated one-inflated and the one-inflated zero-truncated model is compared (also with the model ignoring one-inflation) in terms of Horvitz–Thompson estimation of population size. The simulation work shows clearly the biasing effect of ignoring one-inflation. Both models, the zero-truncated one-inflated and the one-inflated zero-truncated one, are suitable to model ongoing one-inflation. It is also important to choose an appropriate base-line distributional model. Finally, all models derived in the paper are illustrated on a number of case studies.
Longitudinal studies are subject to nonresponse when individuals fail to provide data for entire waves or particular questions of the survey. We compare approaches to nonresponse bias analysis (NRBA) in longitudinal studies and illustrate them on the Early Childhood Longitudinal Study, Kindergarten Class of 2010–2011 (ECLS-K:2011). Wave nonresponse with attrition often yields a monotone missingness pattern, and the missingness mechanism can be missing at random (MAR) or missing not at random (MNAR). We discuss weighting, multiple imputation (MI), incomplete data modelling and Bayesian approaches to NRBA for monotone patterns. Weighting adjustments can be effective when the constructed weights are correlated with the survey outcome of interest. MI allows for variables with missing values to be included in the imputation model, yielding potentially less biased and more efficient estimates. We add offsets in the MAR results to provide sensitivity analyses to assess MNAR deviations. We conduct NRBA for descriptive summaries and analytic model estimates in the ECLS-K:2011 application. The strength of evidence about our NRBA depends on the strength of the relationship between the fully observed variables and the key survey outcomes, so the key to a successful NRBA is to include strong predictors.
Since the pioneering work of sliced inverse regression, sufficient dimension reduction has been growing into a mature field in statistics and it has broad applications to regression diagnostics, data visualisation, image processing and machine learning. In this paper, we provide a review of several popular inverse regression methods, including sliced inverse regression (SIR) method and principal hessian directions (PHD) method. In addition, we adopt a conditional characteristic function approach and develop a new class of slicing-free methods, which are parallel to the classical SIR and PHD, and are named weighted inverse regression ensemble (WIRE) and weighted PHD (WPHD), respectively. Relationship with recently developed martingale difference divergence matrix is also revealed. Numerical studies and a real data example show that the proposed slicing-free alternatives have superior performance than SIR and PHD.
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