A distance based regression model for prediction with mixed data

A multiple regression method based on distance analysis and metric scaling is proposed and studied. This method allow us to predict a continuous response variable from several explanatory variables, is compatible with the general linear model and is found to be useful when the predictor variables are both continuous and categorical. Real data examples are given to illustrate the results obtained.     

SB-robustness of directional mean for circular distributions

In this paper we study the robustness of the directional mean (a.k.a. circular mean) for different families of circular distributions. We show that the directional mean is robust in the sense of finite standardized gross error sensitivity (SB-robust) for the following families: (1) mixture of two circular normal distributions, (2) mixture of wrapped normal and circular normal distributions and (3) mixture of two wrapped normal distributions. We also show that the directional mean is not SB-robust for the family of all circular normal distributions with varying concentration parameter. We define the circular trimmed mean and prove that it is SB-robust for this family. In general the property of SB-robustness of an estimator at a family of probability distributions is dependent on the choice of the dispersion measure. We introduce the concept of equivalent dispersion measures and prove that if an estimator is SB-robust for one dispersion measure then it is SB-robust for all equivalent dispersion measures. Three different dispersion measures for circular distributions are considered and their equivalence studied.     

Multivariate-multiple circular regression

We introduce a fully model-based approach of studying functional relationships between a multivariate circular-dependent variable and several circular covariates, enabling inference regarding all model parameters and related prediction. Two multiple circular regression models are presented for this approach. First, for an univariate circular-dependent variable, we propose the least circular mean-square error (LCMSE) estimation method, and asymptotic properties of the LCMSE estimators and inferential methods are developed and illustrated. Second, using a simulation study, we provide some practical suggestions for model selection between the two models. An illustrative example is given using a real data set from protein structure prediction problem. Finally, a straightforward extension to the case with a multivariate-dependent circular variable is provided.     

Residuals for directional data *

Various types of residual are proposed for use with directional data. Asymptotic distributions are given for large sample sizes and for concentrated error distributions. A form of residual plot is suggested and is illustrated with some practical examples.     

Nonparametric Bayes methods for directional data

A model for directional data in q dimensions is studied. The data are assumed to arise from a distribution with a density on a sphere of q — 1 dimensions. The density is unimodal and rotationally symmetric, but otherwise of unknown form. The posterior distribution of the unknown mode (mean direction) is derived, and small-sample posterior inference is discussed. The posterior mean of the density is also given. A numerical method for evaluating posterior quantities based on sampling a Markov chain is introduced. This method is generally applicable to problems involving unknown monotone functions.     

Small nonparametric tolerance regions for directional data

We present a natural approach, based on minimum volume sets, for constructing nonparametric tolerance regions for directional data. The tolerance regions have desirable features like invariance and are asymptotically minimal under certain conditions. We establish the asymptotic correctness of our tolerance regions by using the theory of empirical processes and generalized quantiles. The results are obtained under minimal conditions. In case of circular data, the finite sample properties of the tolerance arcs are studied through simulations. The method is also applied to a real data example.     

Nonparametric regression for threshold data

Consider a detector which records the times at which the endogenous variable of a nonparametric regression model exceeds a certain threshold. If the error distribution is known, the regression function can still be identified from these threshold data. The author constructs estimators for the regression function that are transformations of kernel estimators. She determines the bandwidth that minimizes the asymptotic mean average squared error. Her investigation was motivated by recent work on stochastic resonance in neuroscience and signal detection theory, where it was observed that detection of a subthreshold signal is enhanced by the addition of noise. The author compares her model with several others that have been proposed in the recent past.     

Panel data regression for counts

A new panel data model for count data is introduced. We suggest alternative estimators, such as pseudo maximum likelihood and generalized method of moments, of structural and nuisance parameters. In addition, different test statistics of independence and overdispersion are obtained. The small sample performance of the estimators and tests are evaluated in Monte Carlo experiments. The model is applied to the number of days absent in Sweden 1981–1991 for a panel of Swedish male workers.     

Bayesian consistency for regression models under a supremum distance

This paper studies the consistency of Bayesian nonparametric regression models. We concentrate on the use of the sup metric and dealing with non-stochastic, i.e. designed, covariate values. We illustrate our results on a normal mean regression function and demonstrate the usefulness of a model based on piecewise constant functions.     

Analysis of variance for circular data

Over the last 25 years, increasing attention has been given to the problem of analysing data arising from circular distributions. The most important circular distribution was introduced by Von Mises (1918) which takes the form:

[Formulas]

where Io(k) is a modified Bessel function, u0 is the mean direction and k is the concentration parameter of the distribution. Watson & Williams (1956) laid the foundation of analysis of variance type techniques for the two-dimensional case of circular data using the Von Mises distribution. Stephens (1962a,b; 1969, 1972). Upton (1974) and Stephens (1982) made further improvements to Watson & Williams’ work. In this paper the authors will discuss the pitfalls of the methods adopted by Stephens (1982) and present a unified analysis of variance type approach for circular data.     

Testing for outliers in circular data

Bagchi & Guttman (1990) have recently suggested a new procedure for the detection of outliers in circular data. It is shown that there are problems with the implementation of both their procedure and the earlier procedures suggested by Collett (1980). It is argued that at present the most appropriate procedure is that of Mardia (1975).     

On projection-based tests for directional and compositional data

A new class of nonparametric tests, based on random projections, is proposed. They can be used for several null hypotheses of practical interest, including uniformity for spherical (directional) and compositional data, sphericity of the underlying distribution and homogeneity in two-sample problems on the sphere or the simplex. The proposed procedures have a number of advantages, mostly associated with their flexibility (for example, they also work to test “partial uniformity” in a subset of the sphere), computational simplicity and ease of application even in high-dimensional cases.     

Sliced inverse regression for survival data

We apply the univariate sliced inverse regression to survival data. Our approach is different from the other papers on this subject. The right-censored observations are taken into account during the slicing of the survival times by assigning each of them with equal weight to all of the slices with longer survival. We test this method with different distributions for the two main survival data models, the accelerated lifetime model and Cox’s proportional hazards model. In both cases and under different conditions of sparsity, sample size and dimension of parameters, this non-parametric approach finds the data structure and can be viewed as a variable selector.     

Inverse regression estimation for censored data

An inverse regression methodology for assessing predictor performance in the censored data setup is developed along with inference procedures and a computational algorithm. The technique developed here allows for conditioning on the unobserved failure time along with a weighting mechanism that accounts for the censoring. The implementation is nonparametric and computationally fast. This provides an efficient methodological tool that can be used especially in cases where the usual modeling assumptions are not applicable to the data under consideration. It can also be a good diagnostic tool that can be used in the model selection process. We have provided theoretical justification of consistency and asymptotic normality of the methodology. Simulation studies and two data analyses are provided to illustrate the practical utility of the procedure.     

Exponentiated Weibull regression for time-to-event data

The Weibull, log-logistic and log-normal distributions are extensively used to model time-to-event data. The Weibull family accommodates only monotone hazard rates, whereas the log-logistic and log-normal are widely used to model unimodal hazard functions. The increasing availability of lifetime data with a wide range of characteristics motivate us to develop more flexible models that accommodate both monotone and nonmonotone hazard functions. One such model is the exponentiated Weibull distribution which not only accommodates monotone hazard functions but also allows for unimodal and bathtub shape hazard rates. This distribution has demonstrated considerable potential in univariate analysis of time-to-event data. However, the primary focus of many studies is rather on understanding the relationship between the time to the occurrence of an event and one or more covariates. This leads to a consideration of regression models that can be formulated in different ways in survival analysis. One such strategy involves formulating models for the accelerated failure time family of distributions. The most commonly used distributions serving this purpose are the Weibull, log-logistic and log-normal distributions. In this study, we show that the exponentiated Weibull distribution is closed under the accelerated failure time family. We then formulate a regression model based on the exponentiated Weibull distribution, and develop large sample theory for statistical inference. We also describe a Bayesian approach for inference. Two comparative studies based on real and simulated data sets reveal that the exponentiated Weibull regression can be valuable in adequately describing different types of time-to-event data.     

Logistic regression analyses for indirect data

The article’s topic is logistic regression for direct data on the covariates, but indirect data on the endogenous variable. The indirect data may result from a privacy-protecting survey procedure for sensitive characteristics or from statistical disclosure control. Various procedures to generate the indirect data exist. However, we show that it is possible to develop a general approach for logistic regression analyses with indirect data that covers many procedures. We first derive a general algorithm for the maximum likelihood estimation and a general procedure for variance estimation. Subsequently, lots of examples demonstrate the broad applicability of our general framework.     

Quantile regression for interval censored data

As direct generalization of the quantile regression for complete observed data, an estimation method for quantile regression models with interval censored data is proposed, and the property of consistency is obtained. The property of asymptotic normality is also established with a bias converging to zero, and to reduce the bias, two bias correction methods are proposed. Methods proposed in this paper do not require the censoring vectors to be identically distributed, and can be applied to models with various covariates. Simulation results show that the proposed methods work well.