A simple,automatic and adaptive bivariate density estimator based on conditional densities

The standard approach to non-parametric bivariate density estimation is to use a kernel density estimator. Practical performance of this estimator is hindered by the fact that the estimator is not adaptive (in the sense that the level of smoothing is not sensitive to local properties of the density). In this paper a simple, automatic and adaptive bivariate density estimator is proposed based on the estimation of marginal and conditional densities. Asymptotic properties of the estimator are examined, and guidance to practical application of the method is given. Application to two examples illustrates the usefulness of the estimator as an exploratory tool, particularly in situations where the local behaviour of the density varies widely. The proposed estimator is also appropriate for use as a pilot estimate for an adaptive kernel estimate, since it is relatively inexpensive to calculate.     

Generalized Leverage and its Applications

The generalized leverage of an estimator is defined in regression models as a measure of the importance of individual observations. We derive a simple but powerful result, developing an explicit expression for leverage in a general M -estimation problem, of which the maximum likelihood problems are special cases. A variety of applications are considered, most notably to the exponential family non-linear models. The relationship between leverage and local influence is also discussed. Numerical examples are given to illustrate our results     

Spatial variation in risk of disease: a nonparametric binary regression approach

A common problem in environmental epidemiology is the estimation and mapping of spatial variation in disease risk. In this paper we analyse data from the Walsall District Health Authority, UK, concerning the spatial distributions of cancer cases compared with controls sampled from the population register. We formulate the risk estimation problem as a nonparametric binary regression problem and consider two different methods of estimation. The first uses a standard kernel method with a cross-validation criterion for choosing the associated bandwidth parameter. The second uses the framework of the generalized additive model (GAM) which has the advantage that it can allow for additional explanatory variables, but is computationally more demanding. For the Walsall data, we obtain similar results using either the kernel method with controls stratified by age and sex to match the age–sex distribution of the cases or the GAM method with random controls but incorporating age and sex as additional explanatory variables. For cancers of the lung or stomach, the analysis shows highly statistically significant spatial variation in risk. For the less common cancers of the pancreas, the spatial variation in risk is not statistically significant.     

On the shrinkage of local linear curve estimators

Local linear curve estimators are typically constructed using a compactly supported kernel, which minimizes edge effects and (in the case of the Epanechnikov kernel) optimizes asymptotic performance in a mean square sense. The use of compactly supported kernels can produce numerical problems, however. A common remedy is ridging, which may be viewed as shrinkage of the local linear estimator towards the origin. In this paper we propose a general form of shrinkage, and suggest that, in practice, shrinkage be towards a proper curve estimator. For the latter we propose a local linear estimator based on an infinitely supported kernel. This approach is resistant against selection of too large a shrinkage parameter, which can impair performance when shrinkage is towards the origin. It also removes problems of numerical instability resulting from using a compactly supported kernel, and enjoys very good mean squared error properties.     

A new lifetime distribution

Many if not most lifetime distributions are motivated only by mathematical interest. Here, a new three-parameter distribution motivated mainly by lifetime issues is introduced. Some properties of the new distribution including estimation procedures, univariate generalizations and bivariate generalizations are derived. A real data application is described to show its superior performance versus at least that of 15 of the known lifetime models.     

Assessment of modeling longitudinal binary data based on graphical methods

Longitudinal categorical data are commonly applied in a variety of fields and are frequently analyzed by generalized estimating equation (GEE) method. Prior to making further inference based on the GEE model, the assessment of model fit is crucial. Graphical techniques have long been in widespread use for assessing the model adequacy. We develop alternative graphical approaches utilizing plots of marginal model-checking condition and local mean deviance to assess the GEE model with logit link for longitudinal binary responses. The applications of the proposed procedures are illustrated through two longitudinal binary datasets.     

A method for choosing the smoothing parameter in a semi-parametric model for detecting change-points in blood flow

In a smoothing spline model with unknown change-points, the choice of the smoothing parameter strongly influences the estimation of the change-point locations and the function at the change-points. In a tumor biology example, where change-points in blood flow in response to treatment were of interest, choosing the smoothing parameter based on minimizing generalized cross-validation (GCV) gave unsatisfactory estimates of the change-points. We propose a new method, aGCV, that re-weights the residual sum of squares and generalized degrees of freedom terms from GCV. The weight is chosen to maximize the decrease in the generalized degrees of freedom as a function of the weight value, while simultaneously minimizing aGCV as a function of the smoothing parameter and the change-points. Compared with GCV, simulation studies suggest that the aGCV method yields improved estimates of the change-point and the value of the function at the change-point.     

Likelihood estimation for a general class of inverse exponentiated distributions based on complete and progressively censored data

In this paper, we establish the existence and uniqueness of the maximum-likelihood estimates of the parameters of a general class of inverse exponentiated distributions based on complete as well as progressively Type-I and Type-II censored data.     

The exponential–Weibull lifetime distribution

In this paper, we propose a new three-parameter model called the exponential–Weibull distribution, which includes as special models some widely known lifetime distributions. Some mathematical properties of the proposed distribution are investigated. We derive four explicit expressions for the generalized ordinary moments and a general formula for the incomplete moments based on infinite sums of Meijer's G functions. We also obtain explicit expressions for the generating function and mean deviations. We estimate the model parameters by maximum likelihood and determine the observed information matrix. Some simulations are run to assess the performance of the maximum likelihood estimators. The flexibility of the new distribution is illustrated by means of an application to real data.     

Testing for a change point in a sequence of exponential random variables with repeated values

Two test statistics are proposed for the change-point problem with repeated values when the data follow an exponential distribution. The properties of these two statistics have been studied and their asymptotic distributions under the alternative have been derived. The powers of the two test statistics are compared. Real-data examples are presented to illustrate the application of these tests.