A consistent nonparametric density estimator for the deconvolution problem

The problem of nonparametric estimation of a probability density function when the sample observations are contaminated with random noise is studied. A particular estimator f?_n(x) is proposed which uses kernel-density and deconvolution techniques. The estimator f?_n(x) is shown to be uniformly consistent, and its appearance and properties are affected by constants M_n and h_n which the user may choose. The optimal choices of M_n and h_n depend on the sample size n, the noise distribution, and the true distribution which is being estimated. Particular selections for M_n and h_n which minimize upper-bound functions of the mean squared error for f?_n(x) are recommended.     

An alternative stochastic restricted Liu estimator in linear regression

In this paper, we introduce an alternative stochastic restricted Liu estimator for the vector of parameters in a linear regression model when additional stochastic linear restrictions on the parameter vector are assumed to hold. The new estimator is a generalization of the ordinary mixed estimator (OME) (Durbin in J Am Stat Assoc 48:799–808, 1953; Theil and Goldberger in Int Econ Rev 2:65–78, 1961; Theil in J Am Stat Assoc 58:401–414, 1963) and Liu estimator proposed by Liu (Commun Stat Theory Methods 22:393–402, 1993). Necessary and sufficient conditions for the superiority of the new stochastic restricted Liu estimator over the OME, the Liu estimator and the estimator proposed by Hubert and Wijekoon (Stat Pap 47:471–479, 2006) in the mean squared error matrix (MSEM) sense are derived. Furthermore, a numerical example based on the widely analysed dataset on Portland cement (Woods et al. in Ind Eng Chem 24:1207–1241, 1932) and a Monte Carlo evaluation of the estimators are also given to illustrate some of the theoretical results.     

On the polynomial structural relationship

In estimating a linear measurement error model, extra information is generally needed to identify the model. Here the authors show that the polynomial structural model with errors in the endogenous and exogenous variables can be identified without any extra information if the degree is greater than one. They also show that a weighted least squares approach for the estimation of the parameters in the model leads to the same estimates as the solutions of a system of estimating equations.     

A small sample estimator for a polynomial regression with errors in the variables

An adjusted least squares estimator, introduced by Cheng and Schneeweiss for consistently estimating a polynomial regression of any degree with errors in the variables, is modified such that it shows good results in small samples without losing its asymptotic properties for large samples. Simulation studies corroborate the theoretical findings.     

An alternative estimator for multiple characteristics in PPS sampling

The problem considered relates to large-scale sample surveys. A new estimator of population total for the characteristics that are poorly correlated with the selection probabilities has been developed for the PPSWR sampling scheme. The relative efficiency of the proposed estimator has been studied under a super-population model. A numerical investigation into the performance of the estimator has also been made.     

An alternative to ratio estimator in post-stratification

This article proposes an alternative to usual ratio estimator of population mean in post-stratified sampling procedure and its properties are analyzed. Both theoretical and empirical findings are encouraging and support the soundness of the proposed procedure for mean estimation over an alternative to ratio estimator in simple random sampling without replacement suggested by Srivenkataramana and Tracy (1980), usual combined ratio estimators suggested by Ige and Tripathi (1989), and usual unbiased estimator in post-stratified sampling scheme. Both theoretical and empirical findings are encouraging and support the soundness of the present study. At the end, a simulation study has been carried out to verify the superiority of the proposed estimator.     

An alternative estimator for multiple characteristics in rhg sampling scheme

The problem considered relates to large scale sample surveys, A new estimator of population total for the characteristics that are poorly correlated with. the selection probabilities,has been developed for the RHC sampling scheme, She relative efficiency of the proposed estimator has been studied under a super-population model, A numerical. Investigation into the performance of the estimator, has also been made.     

Corrected local polynomial estimation in varying‐coefficient models with measurement errors

The authors study a varying‐coefficient regression model in which some of the covariates are measured with additive errors. They find that the usual local linear estimator (LLE) of the coefficient functions is biased and that the usual correction for attenuation fails to work. They propose a corrected LLE and show that it is consistent and asymptotically normal, and they also construct a consistent estimator for the model error variance. They then extend the generalized likelihood technique to develop a goodness of fit test for the model. They evaluate these various procedures through simulation studies and use them to analyze data from the Framingham Heart Study.     

Intensity Estimation for Spatial Point Processes Observed with Noise

Abstract.  This article introduces a kernel estimator of the intensity function of spatial point processes taking into account location errors. The asymptotic properties of the estimator are derived and a bandwidth selection procedure is described. A simulation study compares our results with that of the classical kernel estimator and shows that the edge-corrected deconvoluting kernel estimator is more appropriate.     

An error-bounded polynomial approximation for bivariate normal probabilities

Although the bivariate normal distribution is frequently employed in the development of screening models, the formulae for computing bivariate normal probabilities are quite complicated. A simple and accurate error-bounded, noniterative approximation for bivariate normal probabilities based on a simple univariate normal quadratic or cubic approximation is developed for use in screening applications. The approximation, which is most accurate for large absolute correlation coefficients, is especially suitable for screening applications (e.g., in quality control), where large absolute correlations between performance and screening variables are desired. A special approximation for conditional bivariate normal probabilities is also provided which in quality control screening applications improves the accuracy of estimating the average outgoing product quality. Some anomalies in computing conditional bivariate normal probabilities using BNRDF and NORDF in IMSL are also discussed.     

Penalized contrast estimator for adaptive density deconvolution

The authors consider the problem of estimating the density g of independent and identically distributed variables X_I, from a sample Z₁,… Z_n such that Z_I = X_I + σ? for i = 1,…, n, and E is noise independent of X, with σ? having a known distribution. They present a model selection procedure allowing one to construct an adaptive estimator of g and to find nonasymptotic risk bounds. The estimator achieves the minimax rate of convergence, in most cases where lower bounds are available. A simulation study gives an illustration of the good practical performance of the method.     

Bivariate deconvolution with SIMEX: an application to mapping Alaska earthquake density

Constructing spatial density maps of seismic events, such as earthquake hypocentres, is complicated by the fact that events are not located precisely. In this paper, we present a method for estimating density maps from event locations that are measured with error. The estimator is based on the simulation–extrapolation method of estimation and is appropriate for location errors that are either homoscedastic or heteroscedastic. A simulation study shows that the estimator outperforms the standard estimator of density that ignores location errors in the data, even when location errors are spatially dependent. We apply our method to construct an estimated density map of earthquake hypocenters using data from the Alaska earthquake catalogue.     

An alternative approach to variable selection for prediction

A procedure for selecting a subset of predictor variables in regression analysis is suggested. The procedure is so designed that it leads to the selection of a subset of variables having an adequate degree of informativeness with a directly specified confidence coefficient. Some examples are considered to illustrate the application of the procedure.     

An alternative estimator for randomized response sampling with continuous distributions from a dichotomous population

In this paper, we introduce an alternative estimator of a population proportion from a dichotomous population when using randomized response sampling with continuous randomizing distributions. We also propose the alternative use of exponential randomizing densities. The estimator is obtained by method of moments and is compared with Franklin's (1989) estimator using normal and exponential distributions. The proposed estimator is more efficient than Franklin's (1989) estimator under suitable conditions for the two distributions.     

An estimator for the extreme-value index

Asymptotic normally is derived for a new estimator of the extreme value index under certain conditions.     

An alternative phase II/III design for continuous endpoints

The success rate of drug development has been declined dramatically in recent years and the current paradigm of drug development is no longer functioning. It requires a major undertaking on breakthrough strategies and methodology for designs to minimize sample sizes and to shorten duration of the development. We propose an alternative phase II/III design based on continuous efficacy endpoints, which consists of two stages: a selection stage and a confirmation stage. For the selection stage, a randomized parallel design with several doses with a placebo group is employed for selection of doses. After the best dose is chosen, the patients of the selected dose group and placebo group continue to enter the confirmation stage. New patients will also be recruited and randomized to receive the selected dose or placebo group. The final analysis is performed with the cumulative data of patients from both stages. With the pre‐specified probabilities of rejecting the drug at each stage, sample sizes and critical values for both stages can be determined. As it is a single trial with controlling overall type I and II error rates, the proposed phase II/III adaptive design may not only reduce the sample size but also improve the success rate. An example illustrates the applications of the proposed phase II/III adaptive design. Copyright © 2010 John Wiley & Sons, Ltd.