Affiliation discrete weighted networks with an increasing degree sequence

Affiliation network is one kind of two-mode social network with two different sets of nodes (namely, a set of actors and a set of social events) and edges representing the affiliation of the actors with the social events. The connections in many affiliation networks are only binary weighted between actors and social events that can not reveal the affiliation strength relationship. Although a number of statistical models are proposed to analyze affiliation binary weighted networks, the asymptotic behaviors of the maximum likelihood estimator (MLE) are still unknown or have not been properly explored in affiliation weighted networks. In this paper, we study an affiliation model with the degree sequence as the exclusively natural sufficient statistic in the exponential family distributions. We derive the consistency and asymptotic normality of the maximum likelihood estimator in affiliation finite discrete weighted networks when the numbers of actors and events both go to infinity. Simulation studies and a real data example demonstrate our theoretical results.     

Asymptotic properties of the kernel mode estimator under twice censorship model

In this article, we study the asymptotic properties of the kernel estimator of the mode and density function when the data are twice censored. More specifically, we first establish a strong uniform consistency over a compact set with a rate of the kernel density estimator and then we give the consistency with rate and asymptotic normality for the kernel mode estimator. An application to confidence bands is given.     

Smooth estimation of a reliability function in ranked set sampling

The authors develop a kernel-based estimator of a dynamic reliability measure for use with independent ranked set samples. The estimator is in the form of a ratio, whose numerator and denominator are shown to outperform their rivals based on simple random samples. Some asymptotic properties about the proposed estimator are also established. Simulation studies reveal finite-sample properties of the estimator. The technique is finally applied on an agricultural data set.     

Asymptotic properties of the ols and grls estimator for the replicated functional relationship model

For the simple linear functional relationship model with replication, the asymptotic properties of the ordinary (OLS) and grouping least squares (GRLS) estimator of the slope are investi- gated under the assumption of normally distributed errors with unknown covariance matrix. The relative performance of the OLS and GRLS estimator is compared in terms of the asymptotic mean square error, and a set of critical parameters are identified for determining the dominance of one estimator over the other. It is also shown that the GRLS estimator is asymptoticallyequivalent to the maximum likelihood (ML) estimator under the given assumptions.     

A risk set calibration method for failure time regression by using a covariate reliability sample

Regression parameter estimation in the Cox failure time model is considered when regression variables are subject to measurement error. Assuming that repeat regression vector measurements adhere to a classical measurement model, we can consider an ordinary regression calibration approach in which the unobserved covariates are replaced by an estimate of their conditional expectation given available covariate measurements. However, since the rate of withdrawal from the risk set across the time axis, due to failure or censoring, will typically depend on covariates, we may improve the regression parameter estimator by recalibrating within each risk set. The asymptotic and small sample properties of such a risk set regression calibration estimator are studied. A simple estimator based on a least squares calibration in each risk set appears able to eliminate much of the bias that attends the ordinary regression calibration estimator under extreme measurement error circumstances. Corresponding asymptotic distribution theory is developed, small sample properties are studied using computer simulations and an illustration is provided.     

Inference based on adaptive grid selection of probability transforms

Probability transform-based inference, for example, characteristic function-based inference, is a good alternative to likelihood methods when the probability density function is unavailable or intractable. However, a set of grids needs to be determined to provide an effective estimator based on probability transforms. This paper is concerned with parametric inference based on adaptive selection of grids. By employing a closeness measure to evaluate the asymptotic variance of the transform-based estimator, we propose a statistical inference procedure, accompanied with adaptive grid selection. The selection algorithm aims for a small set of grids, and yet the resulting estimator can be highly efficient. Generally, the asymptotic variance is very close to that of the maximum likelihood estimator.     

Stein-type estimation using ranked set sampling

In this study, we consider the application of the James–Stein estimator for population means from a class of arbitrary populations based on ranked set sample (RSS). We consider a basis for optimally combining sample information from several data sources. We succinctly develop the asymptotic theory of simultaneous estimation of several means for differing replications based on the well-defined shrinkage principle. We showcase that a shrinkage-type estimator will have, under quadratic loss, a substantial risk reduction relative to the classical estimator based on simple random sample and RSS. Asymptotic distributional quadratic biases and risks of the shrinkage estimators are derived and compared with those of the classical estimator. A simulation study is used to support the asymptotic result. An over-riding theme of this study is that the shrinkage estimation method provides a powerful extension of its traditional counterpart for non-normal populations. Finally, we will use a real data set to illustrate the computation of the proposed estimators.     

An Efficient Estimation for Switching Regression Models: A Monte Carlo Study

This article investigates an efficient estimation method for a class of switching regressions based on the characteristic function (CF). We show that with the exponential weighting function, the CF-based estimator can be achieved from minimizing a closed form distance measure. Due to the availability of the analytical structure of the asymptotic covariance, an iterative estimation procedure is developed involving the minimization of a precision measure of the asymptotic covariance matrix. Numerical examples are illustrated via a set of Monte Carlo experiments examining the implementation, finite sample property and the efficiency of the proposed estimator.     

AN ADAPTIVE TRIMMED LIKELIHOOD ALGORITHM FOR IDENTIFICATION OF MULTIVARIATE OUTLIERS

This article describes an algorithm for the identification of outliers in multivariate data based on the asymptotic theory for location estimation as described typically for the trimmed likelihood estimator and in particular for the minimum covariance determinant estimator. The strategy is to choose a subset of the data which minimizes an appropriate measure of the asymptotic variance of the multivariate location estimator. Observations not belonging to this subset are considered potential outliers which should be trimmed. For α less than about 0.5, the correct trimming proportion is taken to be that α > 0 for which the minimum of any minima of this measure of the asymptotic variance occurs. If no minima occur for an α > 0 then the data set will be considered outlier free.     

Tail index and second-order parameters’ semi-parametric estimation based on the log-excesses

In this paper we are interested in the derivation of the asymptotic and finite-sample distributional properties of a ‘quasi-maximum likelihood’ estimator of a ‘scale’ second-order parameter β, directly based on the log-excesses of an available sample. Such estimation is of primordial importance for the adaptive selection of the optimal sample fraction to be used in the classical semi-parametric tail index estimation as well as for the reduced-bias estimation of the tail index, high quantiles and other parameters of extreme or even rare events. An application in the area of survival analysis is provided, on the basis of a data set on males diagnosed with cancer of the tongue.     

Non parametric regression analysis for longitudinal data with time-depending autoregressive error process

This paper considers a non parametric longitudinal model, where the within-subject correlation structure is represented by a time-depending autoregressive error process. An initial estimator without taking into account the within-subject correlation is obtained to fit the time-depending autoregressive error process. With the initial estimator, we construct a two-stage local linear estimator of the mean function. According to the asymptotic normality of the initial and two-stage estimators, it is discovered that the two-stage estimator has a smaller asymptotic variance. The simulation results show us that the two-stage estimation has some good properties. The analysis of a data set demonstrates its application.     

Variance estimation of the Buckley–James estimator under discrete assumptions

The Buckley–James estimator (BJE) is a widely recognized approach in dealing with right-censored linear regression models. There have been a lot of discussions in the literature on the estimation of the BJE as well as its asymptotic distribution. So far, no simulation has been done to directly estimate the asymptotic variance of the BJE. Kong and Yu [Asymptotic distributions of the Buckley–James estimator under nonstandard conditions, Statist. Sinica 17 (2007), pp. 341–360] studied the asymptotic distribution under discontinuous assumptions. Based on their methodology, we recalculate and correct some missing terms in the expression of the asymptotic variance in Theorem 2 of their work. We propose an estimator of the standard deviation of the BJE by using plug-in estimators. The estimator is shown to be consistent. The performance of the estimator is accessed through simulation studies under discrete underline distributions. We further extend our studies to several continuous underline distributions through simulation. The estimator is also applied to a real medical data set. The simulation results suggest that our estimation is a good approximation to the true standard deviation with reference to the empirical standard deviation.     

Asymptotically efficient estimation in response adaptive trials

We examine the issue of asymptotic efficiency of estimation for response adaptive designs of clinical trials, from which the collected data set contains a dependency structure. We establish the asymptotic lower bound of exponential rates for consistent estimators. Under certain regularity conditions, we show that the maximum likelihood estimator achieves the asymptotic lower bound for response adaptive trials with dichotomous responses. Furthermore, it is shown that the maximum likelihood estimator of the treatment effect is asymptotically efficient in the Bahadur sense for response adaptive clinical trials.     

On the estimation of reliabilty function in a Weibull lifetime distribution

The estimation of the reliability function of the Weibull lifetime model is considered in the presence of uncertain prior information (not in the form of prior distribution) on the parameter of interest. This information is assumed to be available in some sort of a realistic conjecture. In this article, we focus on how to combine sample and non-sample information together in order to achieve improved estimation performance. Three classes of point estimatiors, namely, the unrestricted estimator, the shrinkage estimator and shrinkage preliminary test estimator (SPTE) are proposed. Their asymptotic biases and mean-squared errors are derived and compared. The relative dominance picture of the estimators is presented. Interestingly, the proposed SPTE dominates the unrestricted estimator in a range that is wider than that of the usual preliminary test estimator. A small-scale simulation experiment is used to examine the small sample properties of the proposed estimators. Our simulation investigations have provided strong evidence that corroborates with asymptotic theory. The suggested estimation methods are applied to a published data set to illustrate the performance of the estimators in a real-life situation.     

Asymptotics for L2-norm of ARCH innovation density estimator

In this paper we consider the asymptotic properties of the ARCH innovation density estimator. We obtain the asymptotic normality of the Bickel-Rosenblatt test statistic (based on our density estimator) under the null hypothesis, which is the same as in the case of the one sample set up (given in Bickel and Rosenblatt, 1973). We also show the strong consistency of the estimator for the true density in L2-norm.     

Super efficient frequency estimation

In this paper we propose a modified Newton-Raphson method to obtain super efficient estimators of the frequencies of a sinusoidal signal in presence of stationary noise. It is observed that if we start from an initial estimator with convergence rate O_p(n⁻¹) and use Newton-Raphson algorithm with proper step factor modification, then it produces super efficient frequency estimator in the sense that its asymptotic variance is lower than the asymptotic variance of the corresponding least squares estimator. The proposed frequency estimator is consistent and it has the same rate of convergence, namely O_p(n^−3/2), as the least squares estimator. Monte Carlo simulations are performed to observe the performance of the proposed estimator for different sample sizes and for different models. The results are quite satisfactory. One real data set has been analyzed for illustrative purpose.     

Quantile Estimation Using Ranked Set Samples from a Population with Known Mean

Ranked set sampling (RSS) is a cost-efficient technique for data collection when the units in a population can be easily judgment ranked by any cheap method other than actual measurements. Using auxiliary information in developing statistical procedures for inference about different population characteristics is a well-known approach. In this work, we deal with quantile estimation from a population with known mean when data are obtained according to RSS scheme. Through the simple device of mean-correction (subtract off the sample mean and add on the known population mean), a modified estimator is constructed from the standard quantile estimator. Asymptotic normality of the new estimator and its asymptotic efficiency relative to the original estimator are derived. Simulation results for several underlying distributions show that the proposed estimator is more efficient than the traditional one.     

Asymptotic properties of nonparametric regression for long memory random fields

The kernel estimator of spatial regression function is investigated for stationary long memory (long range dependent) random fields observed over a finite set of spatial points. A general result on the strong consistency of the kernel density estimator is first obtained for the long memory random fields, and then, under some mild regularity assumptions, the asymptotic behaviors of the regression estimator are established. For the linear long memory random fields, a weak convergence theorem is also obtained for kernel density estimator. Finally, some related issues on the inference of long memory random fields are discussed through a simulation example.     

On kernel density estimation based on different stratified sampling with optimal allocation

Kernel density estimation is probably the most widely used non parametric statistical method for estimating probability densities. In this paper, we investigate the performance of kernel density estimator based on stratified simple and ranked set sampling. Some asymptotic properties of kernel estimator are established under both sampling schemes. Simulation studies are designed to examine the performance of the proposed estimators under varying distributional assumptions. These findings are also illustrated with the help of a dataset on bilirubin levels in babies in a neonatal intensive care unit.     

Inference of R=P[X<Y] for generalized logistic distribution

This paper deals with the estimation of R=P[X<Y] when X and Y come from two independent generalized logistic distributions with different parameters. The maximum-likelihood estimator (MLE) and its asymptotic distribution are proposed. The asymptotic distribution is used to construct an asymptotic confidence interval of R. Assuming that the common scale parameter is known, the MLE, uniformly minimum variance unbiased estimator, Bayes estimation and confidence interval of R are obtained. The MLE of R, asymptotic distribution of R in the general case, is also discussed. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a real data set has also been presented for illustrative purposes.