A program to calculate bootstrap confidence intervals for the process capability index Cpk

Franklin and Wasserman (1991) introduced the use of Bootstrap sampling procedures for deriving nonparametric confidence intervals for the process capability index, C_pk, which are applicable for instances when at least twenty data points are available. This represents a significant reduction in the usually recommended sample requirement of 100 observations (see Gunther 1989). To facilitate and encourage the use of these procedures. a FORTRAN program is provided for computation of confidence intervals for C_pk. Three methods are provided for this calculation including the standard method, the percentile confidence interval, and the biased - corrected percentile confidence interval.     

基于贝叶斯的Bootstrap置信区间

在许多领域中,Bootstrap成为一种数据处理的有效方法。很多情况下,模型中感兴趣的参数的置信区间难以构建,为了解决这一问题,文章提出了一个新的贝叶斯Bootstrap置信区间的估计量,并做了蒙特卡洛模拟比较,结果比经典区间估计方法和经典Bootstrap方法更优,并进行了实例分析。     

Lower confidence bounds as precision measure for truncated processes

Process capability index C_p has been the most popular one used in the manufacturing industry to provide numerical measures on process precision. For normally distributed processes with automatic fully inspections, the inspected processes follow truncated normal distributions. In this article, we provide the formulae of moments used for the Edgeworth approximation on the precision measurement C_p for truncated normally distributed processes. Based on the developed moments, lower confidence bounds with various sample sizes and confidence levels are provided and tabulated. Consequently, practitioners can use lower confidence bounds to determine whether their manufacturing processes are capable of preset precision requirements.     

On using percentiles to fit data by a johnson distribution

The well-known Johnson system of distributions was developed by N. L. Johnson (1949). Slifker and Shapiro (1980) presented a criterion for choosing a member from the three distributional classes (S_B,S_L, and S_v) in the Johnson system to fit a set of data. The criterion is based on the value of a quantile ratio which depends on a specified positive z value and the parameters of the distribution. In this paper, we present some properties of the quantile ratio for various distributions and for some selected z values. Some comments are made on using the criterion for selecting a Johnson distribution to fit empirical data.     

A robust process capability index

The existing process capability indices (PCI's) assume that the distribution of the process being investigated is normal. For non-normal distributions, PCI's become unreliable in that PCI's may indicate the process is capable when in fact it is not. In this paper, we propose a new index which can be applied to any distribution. The proposed indexC_f:, is directly related to the probability of non-conformance of the process. For a given random sample, the estimation of C_f boils down to estimating non-parametrically the tail probabilities of an unknown distribution. The approach discussed in this paper is based on the works by Pickands (1975) and Smith (1987). We also discuss the construction of bootstrap confidence intervals of C_f: based on the so-called accelerated bias correction method (BC _a:). Several simulations are carried out to demonstrate the flexibility and applicability of C_f:. Two real life data sets are analyzed using the proposed index.     

On the increasing failure rate criteria

We introduce the t₀ -increasing failure rate (IFR-t₀) class, where the failure rate at age x+t₀is greater then or equal to the failure rate at age x for x≥0. The dual class of t₀- decreasing failure rate (DER-t₀) is defined analogoualy.The relation between the IFR -t₀class and other classes of life distributions is studied. Preservation and nonpreservation properties of the IFR-t₀ and the DFR-t₀ classes under various reliablity operation are presented. The concet of stochastic comparison is utilized to cheracterized the IFR-t ₀ class and to suggest other classes of life distributions for aging.     

Interval estimation for a binomial proportion: a bootstrap approach

This paper discusses the classic but still current problem of interval estimation of a binomial proportion. Bootstrap methods are presented for constructing such confidence intervals in a routine, automatic way. Three confidence intervals for a binomial proportion are compared and studied by means of a simulation study, namely: the Wald confidence interval, the Agresti–Coull interval and the bootstrap-t interval. A new confidence interval, the Agresti–Coull interval with bootstrap critical values, is also introduced and its good behaviour related to the average coverage probability is established by means of simulations.     

Consistency of the Subsample Bootstrap empirical process

In the classical Bootstrap approach the number of distinct observation in the resample is random. To overcome this hitch Rao et al. [Bootstrap by sequential resampling, J. Statist. Plan. Inference 64 (1997), pp. 257–281] have proposed a modified resampling procedure – the so-called Sequential Bootstrap or 0.632-Bootstrap – in which each resample has exactly the same number m ～eq ?0.632 n? of distinct observations. Motivated by this idea we introduce an akin procedure, the Subsample Bootstrap, where additionally even the size of each resample is equal. It will turn out that the Subsample Bootstrap empirical process is consistent for a wide class of Donsker classes.     

Bootstrap confidence intervals for the coefficient of quartile variation

ABSTRACT

In non-normal populations, it is more convenient to use the coefficient of quartile variation rather than the coefficient of variation. This study compares the percentile and t-bootstrap confidence intervals with Bonett's confidence interval for the quartile variation. We show that empirical coverage of the bootstrap confidence intervals is closer to the nominal coverage (0.95) for small sample sizes (n = 5, 6, 7, 8, 9, 10 and 15) for most distributions studied. Bootstrap confidence intervals also have smaller average width. Thus, we propose using bootstrap confidence intervals for the coefficient of quartile variation when the sample size is small.     

New bootstrap confidence intervals for means of positively skewed distributions

ABSTRACT

In this paper, we consider the problem of constructing non parametric confidence intervals for the mean of a positively skewed distribution. We suggest calibrated, smoothed bootstrap upper and lower percentile confidence intervals. For the theoretical properties, we show that the proposed one-sided confidence intervals have coverage probability α + O(n^{? 3/2}). This is an improvement upon the traditional bootstrap confidence intervals in terms of coverage probability. A version smoothed approach is also considered for constructing a two-sided confidence interval and its theoretical properties are also studied. A simulation study is performed to illustrate the performance of our confidence interval methods. We then apply the methods to a real data set.     

Robust confidence regions for multinomial probabilities

Generally, confidence regions for the probabilities of a multinomial population are constructed based on the Pearson χ² statistic. Morales et al. (Bootstrap confidence regions in multinomial sampling. Appl Math Comput. 2004;155:295–315) considered the bootstrap and asymptotic confidence regions based on a broader family of test statistics known as power-divergence test statistics. In this study, we extend their work and propose penalized power-divergence test statistics-based confidence regions. We only consider small sample sizes where asymptotic properties fail and alternative methods are needed. Both bootstrap and asymptotic confidence regions are constructed. We consider the percentile and the bias corrected and accelerated bootstrap confidence regions. The latter confidence region has not been studied previously for the power-divergence statistics much less for the penalized ones. Designed simulation studies are carried out to calculate average coverage probabilities. Mean absolute deviation between actual and nominal coverage probabilities is used to compare the proposed confidence regions.     

Block bootstrap for periodic characteristics of periodically correlated time series

This research is dedicated to the study of periodic characteristics of periodically correlated time series such as seasonal means, seasonal variances and autocovariance functions. Two bootstrap methods are used: the extension of the usual Moving Block Bootstrap (EMBB) and the Generalised Seasonal Block Bootstrap (GSBB). The first approach is proposed, because the usual Moving Block Bootstrap does not preserve the periodic structure contained in the data and cannot be applied for the considered problems. For the aforementioned periodic characteristics the bootstrap estimators are introduced and consistency of the EMBB in all cases is obtained. Moreover, the GSBB consistency results for seasonal variances and autocovariance function are presented. Additionally, the bootstrap consistency of both considered techniques for smooth functions of the parameters of interest is obtained. Finally, the simultaneous bootstrap confidence intervals are constructed. A simulation study to compare their actual coverage probabilities is provided. A real data example is presented.     

Non parametric space-time modeling of SO2 in presence of many missing data

Given pollution measurement from a network of monitoring sites in the area of a city and over an extended period of time, an important problem is to identify the spatial and temporal structure of the data. In this paper we focus on the identification and estimate of a statistical non parametric model to analyse the SO₂ in the city of Padua, where data are collected by some fixed stations and some mobile stations moving without any specific rule in different new locations. The impact of the use of mobile stations is that for each location there are times when data was not collected. Assuming temporal stationarity and spatial isotropy for the residuals of an additive model for the logarithm of SO₂ concentration, we estimate the semivariogram using a kernel-type estimator. Attempts are made to avoid the assumption of spatial isotropy. Bootstrap confidence bands are obtained for the spatial component of the additive model that is a deterministic function which defines the spatial structure. Finally, an example is proposed to design an optimal network for the mobiles monitoring stations in a fixed future time, given all the information available.     

Confidence Intervals for the Current Status Model

We discuss a new way of constructing pointwise confidence intervals for the distribution function in the current status model. The confidence intervals are based on the smoothed maximum likelihood estimator, using local smooth functional theory and normal limit distributions. Bootstrap methods for constructing these intervals are considered. Other methods to construct confidence intervals, using the non‐standard limit distribution of the (restricted) maximum likelihood estimator, are compared with our approach via simulations and real data applications.     

Better nonparametric bootstrap confidence intervals for the correlation coefficient

We respond to criticism leveled at bootstrap confidence intervals for the correlation coefficient by recent authors by arguing that in the correlation coefficient case, non–standard methods should be employed. We propose two such methods. The first is a bootstrap coverage coorection algorithm using iterated bootstrap techniques (Hall, 1986; Beran, 1987a; Hall and Martin, 1988) applied to ordinary percentile–method intervals (Efron, 1979), giving intervals with high coverage accuracy and stable lengths and endpoints. The simulation study carried out for this method gives results for sample sizes 8, 10, and 12 in three parent populations. The second technique involves the construction of percentile–t bootstrap confidence intervals for a transformed correlation coefficient, followed by an inversion of the transformation, to obtain “transformed percentile–t” intervals for the correlation coefficient. In particular, Fisher's z–transformation is used, and nonparametric delta method and jackknife variance estimates are used to Studentize the transformed correlation coefficient, with the jackknife–Studentized transformed percentile–t interval yielding the better coverage accuracy, in general. Percentile–t intervals constructed without first using the transformation perform very poorly, having large expected lengths and erratically fluctuating endpoints. The simulation study illustrating this technique gives results for sample sizes 10, 15 and 20 in four parent populations. Our techniques provide confidence intervals for the correlation coefficient which have good coverage accuracy (unlike ordinary percentile intervals), and stable lengths and endpoints (unlike ordinary percentile–t intervals).     

On Parameter Inference for Step-Stress Accelerated Life Test with Geometric Distribution

In this article, we present the parameter inference in step-stress accelerated life tests under the tampered failure rate model with geometric distribution. We deal with Type-II censoring scheme involved in experimental data, and provide the maximum likelihood estimate and confidence interval of the parameters of interest. With the help of the Monte-Carlo simulation technique, a comparison of precision of the confidence limits is demonstrated for our method, the Bootstrap method, and the large-sample based procedure. The application of two industrial real datasets shows the proposed method efficiency and feasibility.     

Random Weighting Estimation of Confidence Intervals for Quantiles

This paper presents a new random weighting method for confidence interval estimation for the sample ‐quantile. A theory is established to extend ordinary random weighting estimation from a non‐smoothed function to a smoothed function, such as a kernel function. Based on this theory, a confidence interval is derived using the concept of backward critical points. The resultant confidence interval has the same length as that derived by ordinary random weighting estimation, but is distribution‐free, and thus it is much more suitable for practical applications. Simulation results demonstrate that the proposed random weighting method has higher accuracy than the Bootstrap method for confidence interval estimation.     

Bootstrapping a hedonic price index: experience from used cars data

Every hedonic price index is an estimate of an unknown economic parameter. It depends, in practice, on one or more random samples of prices and characteristics of a certain good. Bootstrap resampling methods provide a tool for quantifying sampling errors. Following some general reflections on hedonic elementary price indices, this paper proposes a case-based, a model-based, and a wild bootstrap approach for estimating confidence intervals for hedonic price indices. Empirical results are obtained for a data set on used cars in Switzerland. A simple and an enhanced adaptive semi-logarithmic model are fit to monthly samples, and bootstrap confidence intervals are estimated for Jevons-type hedonic elementary price indices.     

Lower Confidence Limits on the Generalized Exponential Distribution Percentiles Under Progressive Type-I Interval Censoring

In industrial life test and survival analysis, the percentile estimation is always a practical issue with lower confidence bound required for maintenance purpose. Sampling distributions for the maximum likelihood estimators of percentiles are usually unknown. Bootstrap procedures are common ways to estimate the unknown sampling distributions. Five parametric bootstrap procedures are proposed to estimate the confidence lower bounds on maximum likelihood estimators for the generalized exponential (GE) distribution percentiles under progressive type-I interval censoring. An intensive simulation is conducted to evaluate the performances of proposed procedures. Finally, an example of 112 patients with plasma cell myeloma is given for illustration.     

Polya后验方法在有限总体抽样估计中的模拟研究

polya后验方法作为一种无信息贝叶斯估计方法,在有限总体抽样中,通过观测的样本,构造一系列的模拟总体,然后进行统计推断。通过统计模拟研究了polya后验方法估计的一些特点,并和Bootstrap方法进行比较。模拟结果显示:polya后验方法能够很好地估计总体的均值,随着样本量的增大,估计值与真值的差距越来越小。采用polya后验方法构造的置信区间区间长度较小,能够很好地覆盖真值。