Semiparametric estimation for count data through weighted distributions

This paper is concerned with semiparametric discrete kernel estimators when the unknown count distribution can be considered to have a general weighted Poisson form. The estimator is constructed by multiplying the Poisson estimate with a nonparametric discrete kernel-type estimate of the Poisson weight function. Comparisons are then carried out with the ordinary discrete kernel probability mass function estimators. The Poisson weight function is thus a local multiplicative correction factor, and is considered as the uniform measure to detect departures from the equidispersed Poisson distribution. In this way, the effects of dispersion and zero-proportion with respect to the standard Poisson distribution are also minimized. This method of estimation is also applied to the weighted binomial form for the count distribution having a finite support. The proposed estimators, in addition to being simple, easy-to-implement and effective, also outperform the competing nonparametric and parametric estimators in finite-sample situations. Two examples illustrate this new semiparametric estimation.     

Deletion residuals in the detection of heterogeneity of variances in linear regression

The heterogeneity of error variance often causes a huge interpretive problem in linear regression analysis. Before taking any remedial measures we first need to detect this problem. A large number of diagnostic plots are now available in the literature for detecting heteroscedasticity of error variances. Among them the ‘residuals’ and ‘fits’ (R–F) plot is very popular and commonly used. In the R–F plot residuals are plotted against the fitted responses, where both these components are obtained using the ordinary least squares (OLS) method. It is now evident that the OLS fits and residuals suffer a huge setback in the presence of unusual observations and hence the R–F plot may not exhibit the real scenario. The deletion residuals based on a data set free from all unusual cases should estimate the true errors in a better way than the OLS residuals. In this paper we propose ‘deletion residuals’ and the ‘deletion fits’ (DR–DF) plot for the detection of the heterogeneity of error variances in a linear regression model to get a more convincing and reliable graphical display. Examples show that this plot locates unusual observations more clearly than the R–F plot. The advantage of using deletion residuals in the detection of heteroscedasticity of error variance is investigated through Monte Carlo simulations under a variety of situations.     

Robust location estimation with missing data

In a missing data setting, we have a sample in which a vector of explanatory variables ${\bf x}_i$ is observed for every subject i, while scalar responses $y_i$ are missing by happenstance on some individuals. In this work we propose robust estimators of the distribution of the responses assuming missing at random (MAR) data, under a semiparametric regression model. Our approach allows the consistent estimation of any weakly continuous functional of the response's distribution. In particular, strongly consistent estimators of any continuous location functional, such as the median, L‐functionals and M‐functionals, are proposed. A robust fit for the regression model combined with the robust properties of the location functional gives rise to a robust recipe for estimating the location parameter. Robustness is quantified through the breakdown point of the proposed procedure. The asymptotic distribution of the location estimators is also derived. The proofs of the theorems are presented in Supplementary Material available online. The Canadian Journal of Statistics 41: 111–132; 2013 © 2012 Statistical Society of Canada     

A likelihood ratio test for goodness‐of‐fit of recessive and dominant models for case–control studies

Testing goodness‐of‐fit of commonly used genetic models is of critical importance in many applications including association studies and testing for departure from Hardy–Weinberg equilibrium. Case–control design has become widely used in population genetics and genetic epidemiology, thus it is of interest to develop powerful goodness‐of‐fit tests for genetic models using case–control data. This paper develops a likelihood ratio test (LRT) for testing recessive and dominant models for case–control studies. The LRT statistic has a closed‐form formula with a simple $\chi^{2}(1)$ null asymptotic distribution, thus its implementation is easy even for genome‐wide association studies. Moreover, it has the same power and optimality as when the disease prevalence is known in the population. The Canadian Journal of Statistics 41: 341–352; 2013 © 2013 Statistical Society of Canada     

Nenparametric Two-Group Classification: Concepts and a SAS-Based Software Package

This article introduces BestClass, a set of SAS macros, available in the mainframe and workstation environment, designed for solving two-group classification problems using a class of recently developed nonparametric classification methods. The criteria used to estimate the classification function are based on either minimizing a function of the absolute deviations from the surface which separates the groups, or directly minimizing a function of the number of misclassified entities in the training sample. The solution techniques used by BestClass to estimate the classification rule use the mathematical programming routines of the SAS/OR software. Recently, a number of research studies have reported that under certain data conditions this class of classification methods can provide more accurate classification results than existing methods, such as Fisher's linear discriminant function and logistic regression. However, these robust classification methods have not yet been implemented in the major statistical packages, and hence are beyond the reach of those statistical analysts who are unfamiliar with mathematical programming techniques. We use a limited simulation experiment and an example to compare and contrast properties of the methods included in Best-Class with existing parametric and nonparametric methods. We believe that BestClass contributes significantly to the field of nonparametric classification analysis, in that it provides the statistical community with convenient access to this recently developed class of methods. BestClass is available from the authors.     

Number of observations near the maximum in an F α-scheme

Let K _n (a) be the number of observations in the interval (M _n ,?a, M _n ), where M _n is the maximum value in a sequence of size n. We study the asymptotic properties of K _n (a) under the F ^α-scheme and discuss the influence of the associated sequence α _n on the limit behaviour of this random variable.     

A bibliography of statistical tolerance regions,II

This bibiliography (completing the previous one, JILEK 1981a) contains about 130 references dealing with statistical tolerance regions. References are classified according to the type of probability distribution concerned and according to some other criteria as well     

Percentile Related Bounds and Estimator for the Mean of the Negative Binomial Distribution

Upper and lower bounds are obtained for the mean of the negative binomial distribution. These bounds are simple functions of a percentile determined by the shape parameter. The result is then used to obtain a robust estimate of the mean when the shape parameter is known.