Constant width simultaneous confidence bands in multiple linear regression with predictor variables constrained in intervals

In the last fifty years, a great deal of research effort has been made on the construction of simultaneous confidence bands for a linear regression function. Two most frequently quoted confidence bands in the statistics literature are the Scheffé type and constant width bands over a given rectangular region of the predictor variables. For the constant width bands, a method is given by Gafarian [Gafarian, A.V., 1964, Confidence bands in straight line regression. Journal of the American Statistical Association, 59, 182–213.] for the calculation of critical constants only for the special case of one predictor variable. In this article, a method is proposed to construct constant width bands when there are any number of predictor variables. A new criterion for assessing a confidence band is also proposed; it is the probability that a confidence band excludes a false regression function and can be viewed as the power function of a test associated, naturally, with a confidence band. Under this criterion, a numerical comparison between the Scheffé type and constant width bands is then carried out. It emerges from this comparison that the constant width bands can be better than the Scheffé type bands for certain designs.     

Confidence interval estimation for negative binomial group distribution

Negative binomial group distribution was proposed in the literature which was motivated by inverse sampling when considering group inspection: products are inspected group by group, and the number of non-conforming items of a group is recorded only until the inspection of the whole group is finished. The non-conforming probability p of the population is thus the parameter of interest. In this paper, the confidence interval construction for this parameter is investigated. The common normal approximation and exact method are applied. To overcome the drawbacks of these commonly used methods, a composite method that is based on the confidence intervals of the negative binomial distribution is proposed, which benefits from the relationship between negative binomial distribution and negative binomial group distribution. Simulation studies are carried out to examine the performances of our methods. A real data example is also presented to illustrate the application of our method.     

Simultaneous Confidence Tubes in Multivariate Linear Regression

Simultaneous confidence bands have been shown in the statistical literature as powerful inferential tools in univariate linear regression. While the methodology of simultaneous confidence bands for univariate linear regression has been extensively researched and well developed, no published work seems available for multivariate linear regression. This paper fills this gap by studying one particular simultaneous confidence band for multivariate linear regression. Because of the shape of the band, the word ‘tube’ is more pertinent and so will be used to replace the word ‘band’. It is shown that the construction of the tube is related to the distribution of the largest eigenvalue. A simulation‐based method is proposed to compute the 1 ? α quantile of this eigenvalue. With the computation power of modern computers, the simultaneous confidence tube can be computed fast and accurately. A real‐data example is used to illustrate the method, and many potential research problems have been pointed out.     

A comparison of methods forsing loss-based capability index

The process capability index C _pm, sometimes called the loss-based index, has been proposed to the manufacturing industry for measuring process reproduction capability. This index incorporates the variation of production items with respect to the target value and the specification limits preset in the factory. To estimate the loss-based index properly and accurately, certain frequentist and Bayesian perspectives have been proposed to obtain lower confidence bounds (LCBs) for providing minimum process capability. The LCBs not only provide critical information regarding process performance but are also used to determine whether an improvement was made in a capability index and by extension in reducing the fraction of non-conforming items. In this paper, under the assumption of normality, based on frequentist and Bayesian senses, several existing approaches for constructing LCBs of C _pm are presented. Depending on the statistical methods used, we then classify these existing approaches into three categories and compared them in terms of the coverage rates and the mean values of the LCBs via simulations. The relative advantages and disadvantages of these approaches are summarized with some highlights of the relevant findings.     

Corrected profile likelihood confidence interval for binomial paired incomplete data

Clinical trials often use paired binomial data as their clinical endpoint. The confidence interval is frequently used to estimate the treatment performance. Tang et al. (2009) have proposed exact and approximate unconditional methods for constructing a confidence interval in the presence of incomplete paired binary data. The approach proposed by Tang et al. can be overly conservative with large expected confidence interval width (ECIW) in some situations. We propose a profile likelihood‐based method with a Jeffreys' prior correction to construct the confidence interval. This approach generates confidence interval with a much better coverage probability and shorter ECIWs. The performances of the method along with the corrections are demonstrated through extensive simulation. Finally, three real world data sets are analyzed by all the methods. Statistical Analysis System (SAS) codes to execute the profile likelihood‐based methods are also presented. Copyright © 2013 John Wiley & Sons, Ltd.     

Exact Simultaneous Confidence Bands for Quadratic and Cubic Polynomial Regression with Applications in Dose Response Study

Exact simultaneous confidence bands (SCBs) for a polynomial regression model are available only in some special situations. In this paper, simultaneous confidence levels for both hyperbolic and constant width bands for a polynomial function over a given interval are expressed as multidimensional integrals. The dimension of these integrals is equal to the degree of the polynomial. Hence the values can be calculated quickly and accurately via numerical quadrature provided that the degree of the polynomial is small (e.g. 2 or 3). This allows the construction of exact SCBs for quadratic and cubic regression functions over any given interval and for any given design matrix. Quadratic and cubic regressions are frequently used to characterise dose response relationships in addition to many other applications. Comparison between the hyperbolic and constant width bands under both the average width and minimum volume confidence set criteria shows that the constant width band can be much less efficient than the hyperbolic band. For hyperbolic bands, comparison between the exact critical constant and conservative or approximate critical constants indicates that the exact critical constant can be substantially smaller than the conservative or approximate critical constants. Numerical examples from a dose response study are used to illustrate the methods.     

Multiple comparisons with a control for exponential location parameters under heteroscedasticity

In this paper, a new design-oriented two-stage two-sided simultaneous confidence intervals, for comparing several exponential populations with control population in terms of location parameters under heteroscedasticity, are proposed. If there is a prior information that the location parameter of k exponential populations are not less than the location parameter of control population, one-sided simultaneous confidence intervals provide more inferential sensitivity than two-sided simultaneous confidence intervals. But the two-sided simultaneous confidence intervals have advantages over the one-sided simultaneous confidence intervals as they provide both lower and upper bounds for the parameters of interest. The proposed design-oriented two-stage two-sided simultaneous confidence intervals provide the benefits of both the two-stage one-sided and two-sided simultaneous confidence intervals. When the additional sample at the second stage may not be available due to the experimental budget shortage or other factors in an experiment, one-stage two-sided confidence intervals are proposed, which combine the advantages of one-stage one-sided and two-sided simultaneous confidence intervals. The critical constants are obtained using the techniques given in Lam [9,10]. These critical constant are compared with the critical constants obtained by Bonferroni inequality techniques and found that critical constant obtained by Lam [9,10] are less conservative than critical constants computed from the Bonferroni inequality technique. Implementation of the proposed simultaneous confidence intervals is demonstrated by a numerical example.     

Probability density estimation via an infinite Gaussian mixture model: application to statistical process monitoring

Summary.  The primary goal of multivariate statistical process performance monitoring is to identify deviations from normal operation within a manufacturing process. The basis of the monitoring schemes is historical data that have been collected when the process is running under normal operating conditions. These data are then used to establish confidence bounds to detect the onset of process deviations. In contrast with the traditional approaches that are based on the Gaussian assumption, this paper proposes the application of the infinite Gaussian mixture model (GMM) for the calculation of the confidence bounds, thereby relaxing the previous restrictive assumption. The infinite GMM is a special case of Dirichlet process mixtures and is introduced as the limit of the finite GMM, i.e. when the number of mixtures tends to ∞. On the basis of the estimation of the probability density function, via the infinite GMM, the confidence bounds are calculated by using the bootstrap algorithm. The methodology proposed is demonstrated through its application to a simulated continuous chemical process, and a batch semiconductor manufacturing process.     

SHOULD WE ESTIMATE THE PROBABILITY OF CORRECT SELECTION?

Various authors, given k location parameters, have considered lower confidence bounds on (standardized) dserences between the largest and each of the other k - 1 parameters. They have then used these bounds to put lower confidence bounds on the probability of correct selection (PCS) in the same experiment (as was used for finding the lower bounds on differences). It is pointed out that this is an inappropriate inference procedure. Moreover, if the PCS refers to some later experiment it is shown that if a non-trivial confidence bound is possible then it is already possible to conclude, with greater confidence, that correct selection has occurred in the first experiment. The short answer to the question in the title is therefore ‘No’, but this should be qualified in the case of a Bayesian analysis.     

Construction of fixed width confidence intervals for a Bernoulli success probability using sequential sampling: a simulation study

This article considers the construction of level 1?α fixed width 2d confidence intervals for a Bernoulli success probability p, assuming no prior knowledge about p and so p can be anywhere in the interval [0, 1]. It is shown that some fixed width 2d confidence intervals that combine sequential sampling of Hall [Asymptotic theory of triple sampling for sequential estimation of a mean, Ann. Stat. 9 (1981), pp. 1229–1238] and fixed-sample-size confidence intervals of Agresti and Coull [Approximate is better than ‘exact’ for interval estimation of binomial proportions, Am. Stat. 52 (1998), pp. 119–126], Wilson [Probable inference, the law of succession, and statistical inference, J. Am. Stat. Assoc. 22 (1927), pp. 209–212] and Brown et al. [Interval estimation for binomial proportion (with discussion), Stat. Sci. 16 (2001), pp. 101–133] have close to 1?α confidence level. These sequential confidence intervals require a much smaller sample size than a fixed-sample-size confidence interval. For the coin jamming example considered, a fixed-sample-size confidence interval requires a sample size of 9457, while a sequential confidence interval requires a sample size that rarely exceeds 2042.     

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.     

Hypothesis Testing and Confidence Regions for the Mean Sojourn Time of an M/M/1 Queueing System

This article discusses testing hypotheses and confidence regions with correct levels for the mean sojourn time of an M/M/1 queueing system. The uniformly most powerful unbiased tests for three usual hypothesis testing problems are obtained and the corresponding p values are provided. Based on the duality between hypothesis tests and confidence sets, the uniformly most accurate confidence bounds are derived. A confidence interval with correct level is proposed.     

Post hoc false positive control for structured hypotheses

In a high-dimensional multiple testing framework, we present new confidence bounds on the false positives contained in subsets S of selected null hypotheses. These bounds are post hoc in the sense that the coverage probability holds simultaneously over all S, possibly chosen depending on the data. This article focuses on the common case of structured null hypotheses, for example, along a tree, a hierarchy, or geometrically (spatially or temporally). Following recent advances in post hoc inference, we build confidence bounds for some prespecified forest-structured subsets and deduce a bound for any subset S by interpolation. The proposed bounds are shown to improve substantially previous ones when the signal is locally structured. Our findings are supported both by theoretical results and numerical experiments. Moreover, our bounds can be obtained by an algorithm (with complexity bilinear in the sizes of the reference hierarchy and of the selected subset) that is implemented in the open-source R package sansSouci available from https://github.com/pneuvial/sanssouci , making our approach operational.     

Simultaneous Confidence Bands for Linear Regression with Covariates Constrained in Intervals

Abstract. The focus of this article is on simultaneous confidence bands over a rectangular covariate region for a linear regression model with k>1 covariates, for which only conservative or approximate confidence bands are available in the statistical literature stretching back to Working & Hotelling (J. Amer. Statist. Assoc. 24 , 1929; 73–85). Formulas of simultaneous confidence levels of the hyperbolic and constant width bands are provided. These involve only a k‐dimensional integral; it is unlikely that the simultaneous confidence levels can be expressed as an integral of less than k‐dimension. These formulas allow the construction for the first time of exact hyperbolic and constant width confidence bands for at least a small k(>1) by using numerical quadrature. Comparison between the hyperbolic and constant width bands is then addressed under both the average width and minimum volume confidence set criteria. It is observed that the constant width band can be drastically less efficient than the hyperbolic band when k>1. Finally it is pointed out how the methods given in this article can be applied to more general regression models such as fixed‐effect or random‐effect generalized linear regression models.     

Bias in linear model power and sample size calculation due to estimating noncentrality

Data analysts frequently calculate power and sample size for a planned study using mean and variance estimates from an initial trial. Hence power,or the sample size needed to achieve a fixed power, varies randomly. Such claculations can be very inaccurate in the General Linear Univeriate Model (GLUM). Biased noncentrality estimators and censored power calculations create inaccuracy. Censoring occurs if only certain outcomes of an initial trial lead to a power calculation. For example, a confirmatory study may be planned (and a sample size estimated) only following a significant resulte in the initial trial.

Computing accurate point estimates or confidence bounds of GLUM noncentrality, power, or sample size in the presence of censoring involves truncated noncentral F distributions. We recommed confidence bounds, whether or not censoring occurs. A power analysis of data from humans exposed to carbon monoxide demonstrates the substantial impact on samle size that may occur. The results highlight potential; biases and should aid study planning and interpretation.     

Parameter estimation of the flexible Weibull distribution for type I censored samples

Often in practice one is interested in the situation where the lifetime data are censored. Censorship is a common phenomenon frequently encountered when analyzing lifetime data due to time constraints. In this paper, the flexible Weibull distribution proposed in Bebbington et al. [A flexible Weibull extension, Reliab. Eng. Syst. Safety 92 (2007), pp. 719–726] is studied using maximum likelihood technics based on three different algorithms: Newton Raphson, Levenberg Marquardt and Trust Region reflective. The proposed parameter estimation method is introduced and proved to work from theoretical and practical point of view. On one hand, we apply a maximum likelihood estimation method using complete simulated and real data. On the other hand, we study for the first time the model using simulated and real data for type I censored samples. The estimation results are approved by a statistical test.     

A note on the asymptotic distribution of the process capability index C pmk

Process capability indices have been widely used to evaluate the process performance to the continuous improvement of quality and productivity. The distribution of the estimator of the process capability index C _pmk is very complicated and the asymptotic distribution is proposed by Chen and Hsu [The asymptotic distribution of the processes capability index C _pmk , Comm. Statist. Theory Methods 24(5) (1995), pp. 1279–1291]. However, we found a critical error for the asymptotic distribution when the population mean is not equal to the midpoint of the specification limits. In this paper, a correct version of the asymptotic distribution is given. An asymptotic confidence interval of C _pmk by using the correct version of asymptotic distribution is proposed and the lower bound can be used to test if the process is capable. A simulation study of the coverage probability of the proposed confidence interval is shown to be satisfactory. The relation of six sigma technique and the index C _pmk is also discussed in this paper. An asymptotic testing procedure to determine if a process is capable based on the index of C _pmk is also given in this paper.     

The simultaneous confidence intervals for all distances from the extreme populations for two-parameter exponential populations based on the multiply type II censored samples

Among k independent two-parameter exponential distributions which have the common scale parameter, the lower extreme population (LEP) is the one with the smallest location parameter and the upper extreme population (UEP) is the one with the largest location parameter. Given a multiply type II censored sample from each of these k independent two-parameter exponential distributions, 14 estimators for the unknown location parameters and the common unknown scale parameter are considered. Fourteen simultaneous confidence intervals (SCIs) for all distances from the extreme populations (UEP and LEP) and from the UEP from these k independent exponential distributions under the multiply type II censoring are proposed. The critical values are obtained by the Monte Carlo method. The optimal SCIs among 14 methods are identified based on the criteria of minimum confidence length for various censoring schemes. The subset selection procedures of extreme populations are also proposed and two numerical examples are given for illustration.     

Testing process capability based on Cpm in the presence of random measurement errors

Process capability indices have been widely used in the manufacturing industry providing numerical measures on process performance. The index C_p provides measures on process precision (or product consistency). The index C_pm, sometimes called the Taguchi index, meditates on process centring ability and process loss. Most research work related to C_p and C_pm assumes no gauge measurement errors. This assumption insufficiently reflects real situations even with highly advanced measuring instruments. Conclusions drawn from process capability analysis are therefore unreliable and misleading. In this paper, we conduct sensitivity investigation on process capability C_p and C_pm in the presence of gauge measurement errors. Due to the randomness of variations in the data, we consider capability testing for C_p and C_pm to obtain lower confidence bounds and critical values for true process capability when gauge measurement errors are unavoidable. The results show that the estimator with sample data contaminated by the measurement errors severely underestimates the true capability, resulting in imperceptible smaller test power. To obtain the true process capability, adjusted confidence bounds and critical values are presented to practitioners for their factory applications.     

Sequential testing with interval censored data ∗

Let X ₁,…,X _n be the lifetimes of n items put on testing at the same time. It is not possible to observe the actual lifetimes. However, it is possible to inspect the items at a finite number of time intervals. At each time of inspection, the number of failures can be recorded. Only these numbers of failures at times of inspections will be available to make decision on the distribution of the lifetimes. Decision can be made at the time of inspection. A “sequential” statistical test is developed to test the mean levels of the lifetimes when the probability distribution is assumed to be exponential. Some numerical results will be presented. The power and the expected time for the decision are compared with those for the idealized situation when each and every actual lifetime is recorded. They are also compared with those for the case when one and only one inspection is allowed to make the decision.