Optimum stratification in sampling with varying probabilities

Summary The paper considers the problem of optimum stratification on an auxiliary variablex when the units from the different strate are selected with probability proportional to the value of the auxiliary variable. Under a super-population set-up assuming the form, of the regression of the estimation variabley on the auxiliary variablex as also the form of the variance functionV(y/x), minimal equations giving optimum strata boundaries have been obtained for the Neyman allocation method. As the minimal equations cannot be solved easily, methods to find approximate solutions have been given. A numerical illustration has also been given to study the effect of optimum stratification.     

An improved exponential estimator of finite population mean in simple random sampling using an auxiliary attribute

In this paper, we propose an exponential ratio type estimator of the finite population mean when auxiliary information is qualitative in nature. Under simple random sampling without replacement scheme, the expressions for the bias and the mean square error of the proposed estimator have been obtained, up to first order of approximation. To show that our proposed estimator is more efficient as compared to the existing estimators, we have made a comparative study with respect to their mean square errors. Theoretically and numerically, we have found that our proposed estimator is always more efficient as compared to its competitor estimators including all the estimators of Abd-Elfattah et al. [1] [A.M. Abd-Elfattah, E.A. El-Sherpieny, S.M. Mohamed, and O.F. Abdou. Improvement in estimating the population mean in simple random sampling using information on auxiliary attribute. Applied Mathematics and Computation, 215 (2010), 4198-4202].     

On consistency of the likelihood moment estimators for a linear process with regularly varying innovations

In 1975 James Pickands III showed that the excesses over a high threshold are approximatly Generalized Pareto distributed. Since then, a variety of estimators for the parameters of this cdf have been studied, but always assuming the underlying data to be independent. In this paper we consider the special case where the underlying data arises from a linear process with regularly varying (i.e. heavy-tailed) innovations. Using this setup, we then show that the likelihood moment estimators introduced by Zhang Aust. N.Z. J. Stat. 49, 69–77 (2007) are consistent estimators for the parameters of the Generalized Pareto distribution.     

Asymptotic normality for two-stage sampling from a finite population

Summary In this paper we consider two-stage sampling from a finite population, and associated estimators of the population total, in a general setting which includes most two-stage procedures in the literature. The main result gives general conditions for asymptotic normality of the estimators. The proof is based on a martingale central limit theorem. It is indicated how the result can be extended to multi-stage procedures.     

On the asymptotic exactness of error estimators for linear triangular finite elements

Summary This paper deals with a-posteriori error estimates for piecewise linear finite element approximations of elliptic problems. We analyze two estimators based on recovery operators for the gradient of the approximate solution. By using superconvergence results we prove their asymptotic exactness under regularity assumptions on the mesh and the solution.One of the estimators can be easily computed in terms of the jumps of the gradient of the finite element approximation. This estimator is equivalent to the error in the energy norm under rather general conditions. However, we show that for the asymptotic exactness, the regularity assumption on the mesh is not merely technical. While doing this, we analyze the relation between superconvergence and asymptotic exactness for some particular examples.     

Unequal probability sampling from a finite population: A multicriteria approach

In this note we address the problem of determining selection probabilities for multipurpose surveys, when the aim is the simultaneous minimization of variances for each variable under study. A characterization of the set of Pareto-optimal designs is given for designs with replacement and also for a class of designs without replacement, namely, Poisson designs.     

An improved class of estimators of a finite population quantile in sample surveys

This work proposes a general class of estimators for a finite population quantile using auxiliary information. This information is provided by the population means of auxiliary variables. The optimum estimator in this class is derived. This result is supported with a numerical example.     

On a class of finite linear spaces with few lines

A famous result of de Bruijn and Erdős (Indag. Math. 10 (1948) 421–423) states that a finite linear space has at least as many lines as points, with equality only if it is a projective plane or a near-pencil. This result led to the problem of characterizing finite linear spaces for which the difference between the number b of lines and the number v of points is assigned.

In this paper finite linear spaces with b−vm, m being the minimum number of lines on a point, are characterized.     

On the calculation of cumulants of estimators arising from a linear time series regression model

Bounds for higher-order cumulants of statistics arising from a linear time series regression model are investigated. A result given in Brillinger is proved and extended. The bounds permit derivation of asymptotic moments and asymptotic normality for estimators of parameters in the model. Two examples are given as illustrations.     

On estimation and inference in a partially linear hazard model with varying coefficients

We study estimation and inference in a marginal proportional hazards model that can handle (1) linear effects, (2) non-linear effects and (3) interactions between covariates. The model under consideration is an amalgamation of three existing marginal proportional hazards models studied in the literature. Developing an estimation and inference procedure with desirable properties for the amalgamated model is rather challenging due to the co-existence of all three effects listed above. Much of the existing literature has avoided the problem by considering narrow versions of the model. The object of this paper is to show that an estimation and inference procedure that accommodates all three effects is within reach. We present a profile partial-likelihood approach for estimating the unknowns in the amalgamated model with the resultant estimators of the unknown parameters being root- \(n\) consistent and the estimated functions achieving optimal convergence rates. Asymptotic normality is also established for the estimators.     

On a nonparametric estimator for the finite time survival probability with zero initial surplus

In this paper, we consider the estimation of the finite time survival probability in the classical risk model when the initial surplus is zero. We construct a nonparametric estimator by Fourier inversion and kernel density estimation method. Under some mild assumptions imposed on the kernel, bandwidth and claim size density, we derive the order of the bias and variance, and show that the estimator has asymptotic normality property. Some simulation studies show that the estimator performs quite well in the finite sample setting.     

Admissibilities of linear estimator in a class of linear models with a multivariate t error variable

This paper discusses admissibilities of estimators in a class of linear models,which include the following common models:the univariate and multivariate linear models,the growth curve model,the extended growth curve model,the seemingly unrelated regression equations,the variance components model,and so on.It is proved that admissible estimators of functions of the regression coefficient β in the class of linear models with multivariate t error terms,called as Model II,are also ones in the case that error terms have multivariate normal distribution under a strictly convex loss function or a matrix loss function.It is also proved under Model II that the usual estimators of β are admissible for p 2 with a quadratic loss function,and are admissible for any p with a matrix loss function,where p is the dimension of β.     

On consistency of a generalized orthoregressive parameter estimator for a linear dynamical system

We obtain some conditions for consistency of generalized orthoregressive estimator for the parameters of a linear dynamical system from the observation of a large number of the independent trajectories of finite length. This leads to the consistency of the Structured Total Least Squares Estimator over the trajectory ensemble.