Sensitivity of normal-based triple sampling sequential point estimation to the normality assumption

This paper discusses the sensitivity of the sequential normal-based triple sampling procedure for estimating the population mean to departures from normality. We assume that the underlying population has finite absolute sixth moment and find that asymptotically the behavior of the estimator and of the sample size depend on the skewness and kurtosis of the underlying distribution when using a squared error loss function with linear sampling cost. These results enable the effects of non-normality easily to be assessed both qualitatively and quantitatively. We supplement our asymptotic results with a simulation experiment to study the performance of the estimator and the sample size in a range of conditions.     

Inferences from Cross-Sectional,Stochastic Frontier Models

Conventional approaches for inference about efficiency in parametric stochastic frontier (PSF) models are based on percentiles of the estimated distribution of the one-sided error term, conditional on the composite error. When used as prediction intervals, coverage is poor when the signal-to-noise ratio is low, but improves slowly as sample size increases. We show that prediction intervals estimated by bagging yield much better coverages than the conventional approach, even with low signal-to-noise ratios. We also present a bootstrap method that gives confidence interval estimates for (conditional) expectations of efficiency, and which have good coverage properties that improve with sample size. In addition, researchers who estimate PSF models typically reject models, samples, or both when residuals have skewness in the “wrong” direction, i.e., in a direction that would seem to indicate absence of inefficiency. We show that correctly specified models can generate samples with “wrongly” skewed residuals, even when the variance of the inefficiency process is nonzero. Both our bagging and bootstrap methods provide useful information about inefficiency and model parameters irrespective of whether residuals have skewness in the desired direction.     

Monte Carlo Approximations of the Quantiles of a Sample Statistic

We consider in this article the problem of numerically approximating the quantiles of a sample statistic for a given population, a problem of interest in many applications, such as bootstrap confidence intervals. The proposed Monte Carlo method can be routinely applied to handle complex problems that lack analytical results. Furthermore, the method yields estimates of the quantiles of a sample statistic of any sample size though Monte Carlo simulations for only two optimally selected sample sizes are needed. An analysis of the Monte Carlo design is performed to obtain the optimal choices of these two sample sizes and the number of simulated samples required for each sample size. Theoretical results are presented for the bias and variance of the numerical method proposed. The results developed are illustrated via simulation studies for the classical problem of estimating a bivariate linear structural relationship. It is seen that the size of the simulated samples used in the Monte Carlo method does not have to be very large and the method provides a better approximation to quantiles than those based on an asymptotic normal theory for skewed sampling distributions.     

Confidence intervals for the closed population size under inverse sampling without replacement

Inverse sampling is an appropriate design for the second phase of capture-recapture experiments which provides an exactly unbiased estimator of the population size. However, the sampling distribution of the resulting estimator tends to be highly right skewed for small recapture samples, so, the traditional Wald-type confidence intervals appear to be inappropriate. The objective of this paper is to study the performance of interval estimators for the population size under inverse recapture sampling without replacement. To this aim, we consider the Wald-type, the logarithmic transformation-based, the Wilson score, the likelihood ratio and the exact methods. Also, we propose some bootstrap confidence intervals for the population size, including the with-replacement bootstrap (BWR), the without replacement bootstrap (BWO), and the Rao–Wu’s rescaling method. A Monte Carlo simulation is employed to evaluate the performance of suggested methods in terms of the coverage probability, error rates and standardized average length. Our results show that the likelihood ratio and exact confidence intervals are preferred to other competitors, having the coverage probabilities close to the desired nominal level for any sample size, with more balanced error rate for exact method and shorter length for likelihood ratio method. It is notable that the BWO and Rao–Wu’s rescaling methods also may provide good intervals for some situations, however, those coverage probabilities are not invariant with respect to the population arguments, so one must be careful to use them.     

New generalized regression estimator in the presence of non response under unequal probability sampling

In this paper, we propose a new generalized regression estimator for the problem of estimating the population total using unequal probability sampling without replacement. A modified automated linearization approach is applied in order to transform the proposed estimator to estimate variance of population total. The variance and estimated value of the variance of the proposed estimator is investigated under a reverse framework assuming that the sampling fraction is negligible and there are equal response probabilities for all units. We prove that the proposed estimator is an asymptotically unbiased estimator and that it does not require a known or estimated response probability to function.     

Gamma mixture of generalized error distribution

Abstract

A new symmetric heavy-tailed distribution, namely gamma mixture of generalized error distribution is defined by scaling generalized error distribution with gamma distribution, its probability density function, k-moment, skewness and kurtosis are derived. After tedious calculation, we also give the Fisher information matrix, moment estimators and maximum likelihood estimators for the parameters of gamma mixture of generalized error distribution. In order to evaluate the effectiveness of the point estimators and the stability of Fisher information matrix, extensive simulation experiments are carried out in three groups of parameters. Additionally, the new distribution is applied to Apple Inc. stock (AAPL) data and compared with normal distribution, F-S skewed standardized t distribution and generalized error distribution. It is found that the new distribution has better fitting effect on the data under the Akaike information criterion (AIC). To a certain extent, our results enrich the probability distribution theory and develop the scale mixture distribution, which will provide help and reference for financial data analysis.     

网络访问固定样本调查的统计推断研究

如何解决网络访问固定样本调查的统计推断问题,是大数据背景下网络调查面临的严重挑战。针对此问题,提出将网络访问固定样本的调查样本与概率样本结合,利用倾向得分逆加权和加权组调整构造伪权数来估计目标总体,进一步采用基于有放回概率抽样的Vwr方法、基于广义回归估计的Vgreg方法与Jackknife方法来估计方差,并比较不同方法估计的效果。研究表明:无论概率样本的样本量较大还是较小,本研究所提出的总体均值估计方法效果较好,并且在方差估计中Jackknife方法的估计效果最好。     

Confidence intervals for estimating the population signal-to-noise ratio: a simulation study

This paper considered several confidence intervals for estimating the population signal-to-noise ratio based on parametric, non-parametric and modified methods. A simulation study has been conducted to compare the performance of the interval estimators under both symmetric and skewed distributions. We reported coverage probability and average width of the interval estimators. Based on the simulation study, we observed that some of our proposed interval estimators are performing better in the sense of smaller width and coverage probability and have been recommended for the researchers.     

A Sample Selection Model with Skew‐normal Distribution

Non‐random sampling is a source of bias in empirical research. It is common for the outcomes of interest (e.g. wage distribution) to be skewed in the source population. Sometimes, the outcomes are further subjected to sample selection, which is a type of missing data, resulting in partial observability. Thus, methods based on complete cases for skew data are inadequate for the analysis of such data and a general sample selection model is required. Heckman proposed a full maximum likelihood estimation method under the normality assumption for sample selection problems, and parametric and non‐parametric extensions have been proposed. We generalize Heckman selection model to allow for underlying skew‐normal distributions. Finite‐sample performance of the maximum likelihood estimator of the model is studied via simulation. Applications illustrate the strength of the model in capturing spurious skewness in bounded scores, and in modelling data where logarithm transformation could not mitigate the effect of inherent skewness in the outcome variable.     

Heterogeneous skewness in binary choice models: predicting outcomes in the men's NCAA basketball tournament

Several authors have recently explored the estimation of binary choice models based on asymmetric error structures. One such family of skewed models is based on the exponential generalized beta type 2 (EGB2). One model in this family is the skewed logit. Recently, McDonald (1996, 2000) extended the work on the EGB2 family of skewed models to permit heterogeneity in the scale parameter. The aim of this paper is to extend the skewed logit model to allow for heterogeneity in the skewness parameter. By this we mean that, in the model developed, here the skewness parameter is permitted to vary from observation to observation by making it a function of exogenous variables. To demonstrate the usefulness of our model, we examine the issue of the predictive ability of sports seedings. We find that we are able to obtain better probability predictions using the skewed logit model with heterogeneous skewness than can be obtained with logit, probit, or skewed logit.     

The “wrong skewness” problem in stochastic frontier models: A new approach

Stochastic frontier models are widely used to measure, e.g., technical efficiencies of firms. The classical stochastic frontier model often suffers from the empirical artefact that the residuals of the production function may have a positive skewness, whereas a negative one is expected under the model, which leads to estimated full efficiencies of all firms. We propose a new approach to the problem by generalizing the distribution used for the inefficiency variable. This generalized stochastic frontier model allows the sample data to have the wrong skewness while estimating well-defined and nondegenerate efficiency measures. We discuss the statistical properties of the model, and we discuss a test for the symmetry of the error term (no inefficiency). We provide a simulation study to show that our model delivers estimators of efficiency with smaller bias than those of the classical model even if the population skewness has the correct sign. Finally, we apply the model to data of the U.S. textile industry for 1958–2005 and show that for a number of years our model suggests technical efficiencies well below the frontier while the classical one estimates no inefficiency in those years.     

Variance Estimation under Two‐Phase Sampling

We consider the variance estimation of the weighted likelihood estimator (WLE) under two‐phase stratified sampling without replacement. Asymptotic variance of the WLE in many semiparametric models contains unknown functions or does not have a closed form. The standard method of the inverse probability weighted (IPW) sample variances of an estimated influence function is then not available in these models. To address this issue, we develop the variance estimation procedure for the WLE in a general semiparametric model. The phase I variance is estimated by taking a numerical derivative of the IPW log likelihood. The phase II variance is estimated based on the bootstrap for a stratified sample in a finite population. Despite a theoretical difficulty of dependent observations due to sampling without replacement, we establish the (bootstrap) consistency of our estimators. Finite sample properties of our method are illustrated in a simulation study.     

Designing clinical trials with uncertain estimates of variability

An estimated sample size is a function of three components: the required power, the predetermined Type I error rate, and the specified effect size. For Normal data the standardized effect size is taken as the difference between two means divided by an estimate of the population standard deviation. However, in early phase trials one may not have a good estimate of the population variance as it is often based on the results of a few relatively small trials. The imprecision of this estimate should be taken into account in sample size calculations. When estimating a trial sample size this paper recommends that one should investigate the sensitivity of the trial to the assumptions made about the variance and consider being adaptive in one's trial design. Copyright © 2004 John Wiley & Sons Ltd.     

Maximum likelihood estimation for mixtures of two gompertz distributions when censoring occurs

In estimating the proportion ‘cured’ after adjuvant treatment, a population of cancer patients can be assumed to be a mixture of two Gompertz subpopulations, those who will die of other causes with no evidence of disease relapse and those who will die of their primary cancer. Estimates of the parameters of the component dying of other causes can be obtained from census data, whereas maximum likelihood estimates for the proportion cured and for the parameters of the component of patients dying of cancer can be obtained from follow-up data.

This paper examines, through simulation of follow-up data, the feasibility of maximum likelihood estimation of a mixture of two Gompertz distributions when censoring occurs. Means, variances and mean square error of the maximum likelihood estimates and the estimated asymptotic variance-covariance matrix is obtained from the simulated samples. The relationship of these variances with sample size, proportion censored, mixing proportion and population parameters are considered.

Moderate sample size typical of cooperative trials yield clinically acceptable estimates. Both increasing sample size and decreasing proportion of censored data decreases variance and covariance of the unknown parameters. Useful results can be obtained with data which are as much as 50% censored. Moreover, if the sample size is sufficiently large, survival data which are as much as 70% censored can yield satisfactory results.     

L Ranked Set Sampling: A Generalization Procedure for Robust Visual Sampling

In this article, a robust ranked set sampling (LRSS) scheme for estimating population mean is introduced. The proposed method is a generalization for many types of ranked set sampling that introduced in the literature for estimating the population mean. It is shown that the LRSS method gives unbiased estimator for the population mean with minimum variance providing that the underlying distribution is symmetric. However, for skewed distributions a weighted mean is given, where the optimal weights is computed by using Shannon's entropy. The performance of the population mean estimator is discussed along with its properties. Monte Carlo comparisons for detecting outliers are made with the traditional simple random sample and the ranked set sampling for some distributions. The results indicate that the LRSS estimator is superior alternative to the existing methods.     

On describing multivariate skewed distributions: A directional approach

Most multivariate measures of skewness in the literature measure the overall skewness of a distribution. These measures were designed for testing the hypothesis of distributional symmetry; their relevance for describing skewed distributions is less obvious. In this article, the authors consider the problem of characterizing the skewness of multivariate distributions. They define directional skewness as the skewness along a direction and analyze two parametric classes of skewed distributions using measures based on directional skewness. The analysis brings further insight into the classes, allowing for a more informed selection of classes of distributions for particular applications. The authors use the concept of directional skewness twice in the context of Bayesian linear regression under skewed error: first in the elicitation of a prior on the parameters of the error distribution, and then in the analysis of the skewness of the posterior distribution of the regression residuals.     

Estimation Under Purposive Sampling

Purposive sampling is described as a random selection of sampling units within the segment of the population with the most information on the characteristic of interest. Nonparametric bootstrap is proposed in estimating location parameters and the corresponding variances. An estimate of bias and a measure of variance of the point estimate are computed using the Monte Carlo method. The bootstrap estimator of the population mean is efficient and consistent in the homogeneous, heterogeneous, and two-segment populations simulated. The design-unbiased approximation of the standard error estimate differs substantially from the bootstrap estimate in severely heterogeneous and positively skewed populations.     

Remainder Markov systematic sampling

Systematic sampling is the simplest and easiest of the most common sampling methods. However, when the population size N cannot be evenly divided by the sampling size n, systematic sampling cannot be performed. Not only is it difficult to determine the sampling interval k equivalent to the sampling probability of the sampling unit, but also the sample size will be inconstant and the sample mean will be a biased estimator of the population mean. To solve this problem, this paper introduces an improved method for systematic sampling: the remainder Markov systematic sampling method. This new method involves separately finding the first-order and second-order inclusion probabilities. This approach uses the Horvitz-Thompson estimator as an unbiased estimator of the population mean to find the variance of the estimator. This study examines the effectiveness of the proposed method for different super-populations.     

Partially adaptive quantile estimators

This paper contrasts two approaches to estimating quantile regression models: traditional semi-parametric methods and partially adaptive estimators using flexible probability density functions (pdfs). While more general pdfs could have been used, the skewed Laplace was selected for pedagogical purposes. Monte Carlo simulations are used to compare the behavior of the semi-parametric and partially adaptive quantile estimators in the presence of possibly skewed and heteroskedastic data. Both approaches accommodate skewness and heteroskedasticity which are consistent with linear quantiles; however, the partially adaptive estimator considered allows for non linear quantiles and also provides simple tests for symmetry and heteroskedasticity. The methods are applied to the problem of estimating conditional quantile functions for wages corresponding to different levels of education.