A model selection criterion based on the BHHJ measure of divergence

The aim of this work is the discussion and investigation of measures of divergence and model selection criteria. A recently introduced measure of divergence, the so-called BHHJ measure (Basu, A., Harris, I.R., Hjort, N.L., Jones, M.C., 1998. Robust and efficient estimation by minimising a density power divergence. Biometrika 85, 549–559) is investigated and a new model selection criterion the divergence information criterion (DIC) based on this measure is proposed. Simulations are performed to check the appropriateness of the proposed criterion.     

A Bayesian nonparametric causal model

Typically, in the practice of causal inference from observational studies, a parametric model is assumed for the joint population density of potential outcomes and treatment assignments, and possibly this is accompanied by the assumption of no hidden bias. However, both assumptions are questionable for real data, the accuracy of causal inference is compromised when the data violates either assumption, and the parametric assumption precludes capturing a more general range of density shapes (e.g., heavier tail behavior and possible multi-modalities). We introduce a flexible, Bayesian nonparametric causal model to provide more accurate causal inferences. The model makes use of a stick-breaking prior, which has the flexibility to capture any multi-modalities, skewness and heavier tail behavior in this joint population density, while accounting for hidden bias. We prove the asymptotic consistency of the posterior distribution of the model, and illustrate our causal model through the analysis of small and large observational data sets.     

A sign based loss approach to model selection in nonparametric regression

In parametric regression models the sign of a coefficient often plays an important role in its interpretation. One possible approach to model selection in these situations is to consider a loss function that formulates prediction of the sign of a coefficient as a decision problem. Taking a Bayesian approach, we extend this idea of a sign based loss for selection to more complex situations. In generalized additive models we consider prediction of the sign of the derivative of an additive term at a set of predictors. Being able to predict the sign of the derivative at some point (that is, whether a term is increasing or decreasing) is one approach to selection of terms in additive modelling when interpretation is the main goal. For models with interactions, prediction of the sign of a higher order derivative can be used similarly. There are many advantages to our sign-based strategy for selection: one can work in a full or encompassing model without the need to specify priors on a model space and without needing to specify priors on parameters in submodels. Also, avoiding a search over a large model space can simplify computation. We consider shrinkage prior specifications on smoothing parameters that allow for good predictive performance in models with large numbers of terms without the need for selection, and a frequentist calibration of the parameter in our sign-based loss function when it is desired to control a false selection rate for interpretation.     

Graphical approach to model adequacy based on exact and near replicates

SUMMARY In this paper, we present an intuitive graphical approach to model validity, which, although to some extent subjective, can be extremely valuable for both presentation and interpretation purposes. In particular, the idea behind such a procedure arises naturally through the generation of a sequence of elements derived from the residuals about a fitted graduating function, based on datum points that are identical or that are relatively close together in a multi-dimensional factor space.     

A Bayesian nonparametric estimator based on left censored data

Summary This paper introduces a Bayesian nonparametric estimator for an unknown distribution function based on left censored observations. Hjort (1990)/Lo (1993) introduced Bayesian nonparametric estimators derived from beta/beta-neutral processes which allow for right censoring. These processes are taken as priors from the class ofneutral to the right processes (Doksum, 1974). The Kaplan-Meier nonparametric product limit estimator can be obtained from these Bayesian nonparametric estimators in the limiting case of a vague prior. The present paper introduces what can be seen as the correspondingleft beta/beta-neutral process prior which allow for left censoring. The Bayesian nonparametyric estimator is obtained as in the corresponding product limit estimator based on left censored data.     

A nonparametric test of symmetry based on the overlapping coefficient

In this paper, we introduce a new nonparametric test of symmetry based on the empirical overlap coefficient using kernel density estimation. Our investigation reveals that the new test is more powerful than the runs test of symmetry proposed by McWilliams [31]. Intensive simulation is conducted to examine the power of the proposed test. Data from a level I Trauma center are used to illustrate the procedures developed in this paper.     

A target distribution model for nonparametric density estimation

In this paper a model is proposed which represents a wide class of continuous distributions. It is shown how the parameters of this model can be estimated leading to a distribution estimator and a corresponding density estimator. An important property of this estimator is that it can be structured to reflect a priori knowledge of the unknown distribution.

Finally, some examples are shown and some comparisons made with kernel and orthogonal series estimators.     

A nonparametric confidence interval for slope based on spearman's rho

An algorithm is presented for computing an exact nonparametric interval estimate of the slope parameter in a simple linear regression model. The confidence interval is obtained by inverting the hypothesis test for slope that uses Spearman's rho. This method is compared to an exact procedure based on Kendall's tau. The Spearman rho procedure will generally give exact levels of confidence closer to desired levels, especially in small samples. Monte carlo results comparing these two methods with the parametric procedure are given     

A nonparametric allocation scheme for classification based on transvariation probabilities

In this paper, a nonparametric discriminant analysis procedure that is less sensitive than traditional procedures to deviations from the usual assumptions is proposed. The procedure uses the projection pursuit methodology where the projection index is the two-group transvariation probability. Montanari [A. Montanari, Linear discriminant analysis and transvariation, J. Classification 21 (2004), pp. 71–88] proposed and used this projection index to measure group separation but allocated the new observation using distances. Our procedure employs a method of allocation based on group–group transvariation probability to classify the new observation. A simulation study shows that the procedure proposed in this paper provides lower misclassification error rates than classical procedures like linear discriminant analysis and quadratic discriminant analysis and recent procedures like maximum depth and Montanari's transvariation-based classifiers, when the underlying distributions are skewed and/or the prior probabilities are unequal.     

Limit behaviors of the estimator of nonparametric regression model based on martingale difference errors

In the paper, we shall establish some limit theorems for the nonparametric estimator of the regression model, which include $L^p$-convergence, complete convergence, and strong convergence of the estimator. These results supplement and improve some known works.     

Complete consistency for the estimator of nonparametric regression model based on martingale difference errors

Abstract

In this paper, we study the complete consistency for the estimator of nonparametric regression model based on martingale difference errors, and obtain the convergence rates of the complete consistency by using the inequalities for martingale difference sequence. Finally, some simulations are illustrated.     

A statistic to check model adequacy in time series

A simple statistic is suggested to examine if the assumptions on variances in a fitted time series model is valid or not. The properties of the statistic are discussed and examples are considered.     

A nonparametric time-varying coefficient model for panel count data

In this research, we describe a nonparametric time-varying coefficient model for the analysis of panel count data. We extend the traditional panel count data models by incorporating B-splines estimates of time-varying coefficients. We show that the proposed model can be implemented using a nonparametric maximum pseudo-likelihood method. We further examine the theoretical properties of the estimators of model parameters. The operational characteristics of the proposed method are evaluated through a simulation study. For illustration, we analyse data from a study of childhood wheezing, and describe the time-varying effect of an inflammatory marker on the risk of wheezing.     

A Phase I nonparametric Shewhart-type control chart based on the median

A nonparametric Shewhart-type control chart is proposed for monitoring the location of a continuous variable in a Phase I process control setting. The chart is based on the pooled median of the available Phase I samples and the charting statistics are the counts (number of observations) in each sample that are less than the pooled median. An exact expression for the false alarm probability (FAP) is given in terms of the multivariate hypergeometric distribution and this is used to provide tables for the control limits for a specified nominal FAP value (of 0.01, 0.05 and 0.10, respectively) and for some values of the sample size (n) and the number of Phase I samples (m). Some approximations are discussed in terms of the univariate hypergeometric and the normal distributions. A simulation study shows that the proposed chart performs as well as, and in some cases better than, an existing Shewhart-type chart based on the normal distribution. Numerical examples are given to demonstrate the implementation of the new chart.     

A Bayesian nonparametric Markovian model for non-stationary time series

Stationary time series models built from parametric distributions are, in general, limited in scope due to the assumptions imposed on the residual distribution and autoregression relationship. We present a modeling approach for univariate time series data, which makes no assumptions of stationarity, and can accommodate complex dynamics and capture non-standard distributions. The model for the transition density arises from the conditional distribution implied by a Bayesian nonparametric mixture of bivariate normals. This results in a flexible autoregressive form for the conditional transition density, defining a time-homogeneous, non-stationary Markovian model for real-valued data indexed in discrete time. To obtain a computationally tractable algorithm for posterior inference, we utilize a square-root-free Cholesky decomposition of the mixture kernel covariance matrix. Results from simulated data suggest that the model is able to recover challenging transition densities and non-linear dynamic relationships. We also illustrate the model on time intervals between eruptions of the Old Faithful geyser. Extensions to accommodate higher order structure and to develop a state-space model are also discussed.     

Two sample nonparametric tests based on subsamples

The paper introduces a general class of nonparametric tests for the two-sample location problem based on subsamples. Includ- ed in this class is the Mann-Whitney (or the Wilcoxon rank sum) test. General formulas for the Pitman efficacy for different methods of subsampling are derived. A small sample power simu- lation compares the performance of members of this class     

Fast nonparametric classification based on data depth

A new procedure, called D D α-procedure, is developed to solve the problem of classifying d-dimensional objects into q ≥ 2 classes. The procedure is nonparametric; it uses q-dimensional depth plots and a very efficient algorithm for discrimination analysis in the depth space [0,1] ^q . Specifically, the depth is the zonoid depth, and the algorithm is the α-procedure. In case of more than two classes several binary classifications are performed and a majority rule is applied. Special treatments are discussed for ‘outsiders’, that is, data having zero depth vector. The D Dα-classifier is applied to simulated as well as real data, and the results are compared with those of similar procedures that have been recently proposed. In most cases the new procedure has comparable error rates, but is much faster than other classification approaches, including the support vector machine.