On Statistics Independent of a Sufficient Statistic: Basu's Lemma

The determination of optimal sample sizes for estimating the difference between population means to a desired degree of confidence and precision is a question of economic significance. This question, however, is generally not discussed in statistics texts. Sample sizes to minimize linear sampling costs are proportional to the population standard deviations and inversely proportional to the square roots of the unit sampling costs. Sensitivity analysis shows that the impact of the use of equal rather than optimal sample sizes on the amount of sampling and its cost is not great as long as the unit costs and population variances are comparable.     

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.     

Determining Optimal Number of Samples for Constructing Multivariate Control Charts

Normally, an average run length (ARL) is used as a measure for evaluating the detecting performance of a multivariate control chart. This has a direct impact on the false alarm cost in Phase II. In this article, we first conduct a simulation study to calculate both in-control and out-of-control ARLs under various combinations of process shifts and number of samples. Then, a trade-off analysis between sampling inspection and false alarm costs is performed. Both the simulation results and trade-off analysis suggest that the optimal number of samples for constructing a multivariate control chart in Phase I can be determined.     

Optimized Sampling Strategies for Identifying Modes in Length-frequency Distributions

An approach to evaluate sampling strategies to detect modes in length-distributions is presented. Distributions based on various numbers of samples (S) and sample sizes (n) were simulated from the original cod and capelin data in Icelandic waters, incorporating within sample correlations. A peak was discerned if the difference between any simulated and original distribution did not exceed a pre-specified Δ, given a probability. This was achieved with numerous combinations of S and n, and the optimal choice will depend on the sampling costs. Variance-equivalence curves also illustrate the difference between demanding precision of mean lengths versus precision of length distributions.     

Normal Theory and Bootstrap Confidence Interval Estimation in Assessing Diagnostic Performance Gain When Combining Two Diagnostic Tests

Many diagnostic tests may be available to identify a particular disease. Diagnostic performance can be potentially improved by combining. “Either” and “both” positive strategies for combining tests have been discussed in the literature, where a gain in diagnostic performance is measured by a ratio of positive (negative) likelihood ratio of the combined test to that of an individual test. Normal theory and bootstrap confidence intervals are constructed for gains in likelihood ratios. The performance (coverage probability, width) of the two methods are compared via simulation. All confidence intervals perform satisfactorily for large samples, while bootstrap performs better in smaller samples in terms of coverage and width.     

Empirical Likelihood Inference for the Rao‐Hartley‐Cochran Sampling Design

The Hartley‐Rao‐Cochran sampling design is an unequal probability sampling design which can be used to select samples from finite populations. We propose to adjust the empirical likelihood approach for the Hartley‐Rao‐Cochran sampling design. The approach proposed intrinsically incorporates sampling weights, auxiliary information and allows for large sampling fractions. It can be used to construct confidence intervals. In a simulation study, we show that the coverage may be better for the empirical likelihood confidence interval than for standard confidence intervals based on variance estimates. The approach proposed is simple to implement and less computer intensive than bootstrap. The confidence interval proposed does not rely on re‐sampling, linearization, variance estimation, design‐effects or joint inclusion probabilities.     

相关差别指标的区间估计方法比较

在公共疾病控制领域,重大稀有疾病的发病率非常低,符合逆抽样特征,量化分析重大稀有疾病的发病率并对其特点进行分析。为了研究在带有群内相关条件下的整群抽样问题,通过二项分布抽样对比流行病学中相关差别指标的六种渐近置信区间的构造方法研究,综合考虑实际覆盖率与区间长度对各种方法的优劣及适用情况做出对比分析。研究表明,Wald型置信区间与对数变换的置信区间对发病率的估计表现因参数而定,而Bootstrap类方法不稳定。本研究找出了不同区间估计方法的适用场合,应合理看待置信区间这种评估方法在流行病学中的实际应用。     

相关差别指标的区间估计方法比较

在公共疾病控制领域,重大稀有疾病的发病率非常低,符合逆抽样特征,量化分析重大稀有疾病的发病率并对其特点进行分析,为了研究在带有群内相关条件下的整群抽样问题,通过β-二项分布抽样对比流行病学中相关差别指标的六种渐近置信区间的构造方法,综合考虑实际覆盖率与区间长度对各种方法的优劣及适用情况并对比分析。研究表明,Wald型置信区间与对数变换的置信区间对发病率的估计表现因参数而定,而Bootstrap类方法不稳定。本研究找出了不同区间估计方法的适用场合,认为应合理看待置信区间这种评估方法在流行病学中的实际应用。     

Performance of Interval Estimators for the Inverse Hypergeometric Distribution

The inverse hypergeometric distribution is of interest in applications of inverse sampling without replacement from a finite population where a binary observation is made on each sampling unit. Thus, sampling is performed by randomly choosing units sequentially one at a time until a specified number of one of the two types is selected for the sample. Assuming the total number of units in the population is known but the number of each type is not, we consider the problem of estimating this parameter. We use the Delta method to develop approximations for the variance of three parameter estimators. We then propose three large sample confidence intervals for the parameter. Based on these results, we selected a sampling of parameter values for the inverse hypergeometric distribution to empirically investigate performance of these estimators. We evaluate their performance in terms of expected probability of parameter coverage and confidence interval length calculated as means of possible outcomes weighted by the appropriate outcome probabilities for each parameter value considered. The unbiased estimator of the parameter is the preferred estimator relative to the maximum likelihood estimator and an estimator based on a negative binomial approximation, as evidenced by empirical estimates of closeness to the true parameter value. Confidence intervals based on the unbiased estimator tend to be shorter than the two competitors because of its relatively small variance but at a slight cost in terms of coverage probability.     

Using the bootstrap in correlation analysis,with application to a longitudinal data set

The problem of constructing simultaneous confidence intervals for various measures of association is considered. Alternative bootstrap algorithms are given for approximating the sampling distributions of the quantities generating the confidence sets. The small sample performance of the procedures is illustrated using simulated data from 3- and 6-variate normal populations. The results are applied to a large multidimensional longitudinal data set from a study of the relationship between drug use and several behavioral attributes.     

关于捕获再捕获抽样的置信区间

捕获再捕获抽样是一种应用广泛的抽样方法。运用随机模拟,研究捕获再捕获抽样的三种估计量的均值、方差、偏度、峰度、利用近似正态分布构造的置信区间的统计性质。改进了估计量的样本方差计算公式,使得利用近似正态分布构造的置信区间更优。     

Confidence interval estimation for the population coefficient of variation using ranked set sampling: a simulation study

In this paper, an evaluation of the performance of several confidence interval estimators of the population coefficient of variation (τ) using ranked set sampling compared to simple random sampling is performed. Two performance measures are used to assess the confidence intervals for τ, namely: width and coverage probabilities. Simulated data were generated from normal, log-normal, skew normal, Gamma, and Weibull distributions with specified population parameters so that the same values of τ are obtained for each distribution, with sample sizes n=15, 20, 25, 50, 100. A real data example representing birth weight of 189 newborns is used for illustration and performance comparison.     

Weighted empirical likelihood inference for multiple samples

We propose a weighted empirical likelihood approach to inference with multiple samples, including stratified sampling, the estimation of a common mean using several independent and non-homogeneous samples and inference on a particular population using other related samples. The weighting scheme and the basic result are motivated and established under stratified sampling. We show that the proposed method can ideally be applied to the common mean problem and problems with related samples. The proposed weighted approach not only provides a unified framework for inference with multiple samples, including two-sample problems, but also facilitates asymptotic derivations and computational methods. A bootstrap procedure is also proposed in conjunction with the weighted approach to provide better coverage probabilities for the weighted empirical likelihood ratio confidence intervals. Simulation studies show that the weighted empirical likelihood confidence intervals perform better than existing ones.     

Statistical Inferences for the Lifetime Performance Index of the Products with the Gompertz Distribution under Censored Samples

Manufacturers often apply process capability indices in the quality control. This study constructs statistical analysis methods of assessing the lifetime performance index of Gompertz products under progressively type II right censored samples. The maximum likelihood estimator of the index is inferred by data transformation and then utilized to develop a hypothesis testing procedure and a confidence interval to assess product performance. We also give one example and some Monte Carlo simulations to assess the behavior of the testing procedure and confidence interval. The results show that our proposed method can effectively evaluate whether the lifetime of products meet the requirement.     

Bootstrap confidence bands for the CDF using ranked-set sampling

In ranked-set sampling (RSS), a stratification by ranks is used to obtain a sample that tends to be more informative than a simple random sample of the same size. Previous work has shown that if the rankings are perfect, then one can use RSS to obtain Kolmogorov–Smirnov type confidence bands for the CDF that are narrower than those obtained under simple random sampling. Here we develop Kolmogorov–Smirnov type confidence bands that work well whether the rankings are perfect or not. These confidence bands are obtained by using a smoothed bootstrap procedure that takes advantage of special features of RSS. We show through a simulation study that the coverage probabilities are close to nominal even for samples with just two or three observations. A new algorithm allows us to avoid the bootstrap simulation step when sample sizes are relatively small.     

Control charts for location based on different sampling schemes

Control charts are the most important statistical process control tool for monitoring variations in a process. A number of articles are available in the literature for the X? control chart based on simple random sampling, ranked set sampling, median-ranked set sampling (MRSS), extreme-ranked set sampling, double-ranked set sampling, double median-ranked set sampling and median double-ranked set sampling. In this study, we highlight some limitations of the existing ranked set charting structures. Besides, we propose different runs rules-based control charting structures under a variety of sampling strategies. We evaluate the performance of the control charting structures using power curves as a performance criterion. We observe that the proposed merger of varying runs rules schemes with different sampling strategies improve significantly the detection ability of location control charting structures. More specifically, the MRSS performs the best under both single- and double-ranked set strategies with varying runs rules schemes. We also include a real-life example to explain the proposal and highlight its significance for practical data sets.     

Nonparametric confidence regions via the analytic wild bootstrap

The wild bootstrap is a nonparametric tool that can be used to estimate a sampling distribution in the presence of heteroscedastic errors. In particular, the wild bootstrap enables us to compute confidence regions for regression parameters under non-i.i.d. models. While the wild bootstrap may perform well in these settings, its obvious drawback is a lack of computational efficiency. The wild bootstrap requires a large number of bootstrap replications, making the use of this tool impractical when dealing with big data. We introduce the analytic wild bootstrap (ANWB), which provides a nonparametric alternative way of constructing confidence regions for regression parameters. The ANWB is superior to the wild bootstrap from a computational standpoint while exhibiting similar finite-sample performance. We report simulation results for both least squares and ridge regression. Additionally, we test the ANWB on a real dataset and compare its performance with that of other standard approaches.     

Asymptotic confidence interval of power spectrum of a continuous time process through progressively faster sampling

If the power spectral density of a continuous time stationary stochastic process is not limited to a finite bandwidth, data sampled from that process at any uniform sampling rate leads to biased and inconsistent spectrum estimators, which are unsuitable for constructing confidence intervals. In this paper, we use the smoothed periodogram estimator to construct asymptotic confidence intervals shrinking to the true spectra, by allowing the sampling rate to go to infinity suitably fast as the sample size goes to infinity. The proposed method requires minimal computation, as it does not involve bootstrap or other resampling. The method is illustrated through a Monte-Carlo simulation study, and its performance is compared with that of the corresponding method based on uniform sampling at a fixed rate.     

New Confidence Intervals for the Proportion of Interest in One-Sample Correlated Binary Data

Correlated binary data is obtained in many fields of biomedical research. When constructing a confidence interval for the proportion of interest, asymptotic confidence intervals have already been developed. However, such asymptotic confidence intervals are unreliable in small samples. To improve the performance of asymptotic confidence intervals in small samples, we obtain the Edgeworth expansion of the distribution of the studentized mean of beta-binomial random variables. Then, we propose new asymptotic confidence intervals by correcting the skewness in the Edgeworth expansion in one direct and two indirect ways. New confidence intervals are compared with the existing confidence intervals in simulation studies.     

Estimation of rare and clustered population variance in adaptive cluster sampling

Abstract

Many researchers used auxiliary information together with survey variable to improve the efficiency of population parameters like mean, variance, total and proportion. Ratio and regression estimation are the most commonly used methods that utilized auxiliary information in different ways to get the maximum benefits in the form of high precision of the estimators. Thompson first introduced the concept of Adaptive cluster sampling, which is an appropriate technique for collecting the samples from rare and clustered populations. In this article, a generalized exponential type estimator is proposed and its properties have been studied for the estimation of rare and highly clustered population variance using single auxiliary information. A numerical study is carried out on a real and artificial population to judge the performance of the proposed estimator over the competing estimators. It is shown that the proposed generalized exponential type estimator is more efficient than the adaptive and non adaptive estimators under conventional sampling design.