Robust model-based sampling designs

We investigate methods for the design of sample surveys, and address the traditional resistance of survey samplers to the use of model-based methods by incorporating model robustness at the design stage. The designs are intended to be sufficiently flexible and robust that resulting estimates, based on the designer’s best guess at an appropriate model, remain reasonably accurate in a neighbourhood of this central model. Thus, consider a finite population of N units in which a survey variable Y is related to a q dimensional auxiliary variable x. We assume that the values of x are known for all N population units, and that we will select a sample of n≤N population units and then observe the n corresponding values of Y. The objective is to predict the population total $T=\sum_{i=1}^{N}Y_{i}$ . The design problem which we consider is to specify a selection rule, using only the values of the auxiliary variable, to select the n units for the sample so that the predictor has optimal robustness properties. We suppose that T will be predicted by methods based on a linear relationship between Y—possibly transformed—and given functions of x. We maximise the mean squared error of the prediction of T over realistic neighbourhoods of the fitted linear relationship, and of the assumed variance and correlation structures. This maximised mean squared error is then minimised over the class of possible samples, yielding an optimally robust (‘minimax’) design. To carry out the minimisation step we introduce a genetic algorithm and discuss its tuning for maximal efficiency.     

Iterative bias reduction: a comparative study

Multivariate nonparametric smoothers, such as kernel based smoothers and thin plate splines smoothers, are adversely impacted by the sparseness of data in high dimension, also known as the curse of dimensionality. Adaptive smoothers, that can exploit the underlying smoothness of the regression function, may partially mitigate this effect. This paper presents a comparative simulation study of a novel adaptive smoother (IBR) with competing multivariate smoothers available as package or function within the R language and environment for statistical computing. Comparison between the methods are made on simulated datasets of moderate size, from 50 to 200 observations, with two, five or 10 potential explanatory variables, and on a real dataset. The results show that the good asymptotic properties of IBR are complemented by a very good behavior on moderate sized datasets, results which are similar to those obtained with Duchon low rank splines.     

Robust methods of estimation of regression coefficients 1

Recently, cumulative residual entropy (CRE) has been found to be a new measure of information that parallels Shannon's entropy (see Rao et al. [Cumulative residual entropy: A new measure of information, IEEE Trans. Inform. Theory. 50(6) (2004), pp. 1220–1228] and Asadi and Zohrevand [On the dynamic cumulative residual entropy, J. Stat. Plann. Inference 137 (2007), pp. 1931–1941]). Motivated by this finding, in this paper, we introduce a generalized measure of it, namely cumulative residual Renyi's entropy, and study its properties. We also examine it in relation to some applied problems such as weighted and equilibrium models. Finally, we extend this measure into the bivariate set-up and prove certain characterizing relationships to identify different bivariate lifetime models.     

Book review

W.G.Cochran: Sampling Techniques, 3rd. Ed. John Wiley & Sons, New York-Santa Barbara-London-Sydney-Toronto 1977. 428 S., £ 12.50; $21.50.

H.Toutenburg: Vorhersage in linearen Modellen. Akademie-Verlag, Berlin 1975, VIII, 176b S., 3 Tab., 28,– M.

O.Kallenberg: Random Measure. Akademie-Verlag, Berlin 1975; Academic Press, London 1976. 104 pp., 28,– M.     

Characterizations of Student's t-distribution via regressions of order statistics

Utilizing regression properties of order statistics, we characterize a family of distributions introduced by Akhundov et al. [New characterizations by properties of midrange and related statistics, Commun. Stat. Theory Methods 33(12) (2004), pp. 3133–3143], which includes the t-distribution with two degrees of freedom as one of its members. Then we extend this characterization result to t-distribution with more than two degrees of freedom.     

Invariance In the linear model – an argument for χ2 and F in nonnormal situations

For a suitable norm, conservation of the distance between expectation and hypothesis may furnish a basis for data reduction by invariance in the linear, not neces-sarily normal, model. If the norm is Euclidean (i.e. based on some inner product), the maximal invariant is a pair of sums of squares. This provides support for traditional χ² (or F) - methods also in nonnormal cases. If the norm is l^p p≠2, or the supnorm, the maximal invariant is, at the best a air of order statistics.     

Higher-order and non-stationary properties of lampard's stochastic reversible counter system

Summary. In this paper we discuss higher-order and non-stationary properties of LAM-PAED'S (1968) stochastic reversible counter model whose output is a stochastic point process with MABKOV dependent time intervals. Statistical properties of the primed counter system, are also considered. A characterization of the probability density function of the interval lengths and the probability distribution of the counts by means of a mixed POXSSOH process is then derived. The distribution of counts of the primed counter system is shown to be mathematically equivalent to the birth-and-death process with immigration. A generalization of the problem when initial counts and incremental counts have independent distributions is also discussed     

Inference on Overlap for Two Inverse Gaussian Populations: Equal Means Case

The inverse Gaussian distribution is often suited for modeling positive and/or positively skewed data (see Chhikara and Folks, 1989 Chhikara , R. S. , Folks , J. L. ( 1989 ). The Inverse Gaussian Distribution . New York : Marcel Dekker . [Google Scholar]) and presents an interesting alternative to the Gaussian model in such cases. We note here that overlap coefficients and their variants are widely studied in the literature for Gaussian populations (see Mulekar and Mishra, 1994 Mulekar , M. , Mishra , S. N. ( 1994 ). Overlap coefficients of two normal densities: equal means case . J. Japan. Statist. Soc. 24 : 169 – 180 . [Google Scholar], 2000 Mulekar , M. , Mishra , S. N. ( 2000 ). Confidence interval estimation of overlap: equal means case . Computat. Statist. Data Anal. 34 : 121 – 137 .[Crossref], [Web of Science ®] , [Google Scholar], and references therein for further details). This article studies the properties and addresses point estimation for large samples of commonly used measures of overlap when the populations are described by inverse Gaussian distributions. The bias and mean square error properties of the estimators are studied through a simulation study.     

A simple approximation for the doubly noncentral t-distribution

A simple approximation for the doubly noncentral t-distribution, based upon the Fieller-Geary Theorem (1930) and approximate normality of the square root of the noncentral chi-square variable observed by Patnaik (1949), is developed, This approximation and an Edgeworth series expansion associated with it are evaluated. The simple approximation is seen to be reasonably accurate for most practical purposes.     

An Algorithm for Unconstrained Quadratically Penalized Convex Optimization

Estimators are often defined as the solutions to data dependent optimization problems. A common form of objective function (function to be optimized) that arises in statistical estimation is the sum of a convex function V and a quadratic complexity penalty. A standard paradigm for creating kernel-based estimators leads to such an optimization problem. This article describes an optimization algorithm designed for unconstrained optimization problems in which the objective function is the sum of a non negative convex function and a known quadratic penalty. The algorithm is described and compared with BFGS on some penalized logistic regression and penalized L ^3/2 regression problems.