Confidence prediction of the mean values of multiple observations

The confidence prediction of the mean value ofmultiple responses in a linear multivariate normal regression model is considered. In order to solve it, confidence intervals of the mean value of multiple responses and its predicted value are obtained. They are numerically modeled and analyzed in comparison with known analogues for regression and individual response.     

Multivariate estimation in regenerative simulation

A definition is given of regenerative simulation that applies to simulation that applies to simulations with multiple responses. Then, a multivariate regenerative estimator of the stationary mean response is presented and shown to be asymptotically normal. Methods for computing confidence regions and simultaneous confidence intervals are also presented.     

A Simultaneous Tube of the Mean Value of Multiple Observations in Multivariate Regression

Moscow University Computational Mathematics and Cybernetics - The problem of constructing a simultaneous confidence tube of the mean value of multiple responses in a multivariate linear normal...     

Simulation-Based Simultaneous Confidence Bands in Multiple Linear Regression With Predictor Variables Constrained in Intervals

This article presents a method for the construction of a simultaneous confidence band for the normal-error multiple linear regression model. The confidence bands considered have their width proportional to the standard error of the estimated regression function, and the predictor variables are allowed to be constrained in intervals. Past articles in this area gave exact bands only for the simple regression model. When there is more than one predictor variable, only conservative bands are proposed in the statistics literature. This article advances this methodology by providing simulation-based confidence bands for regression models with any number of predictor variables. Additionally, a criterion is proposed to assess the sensitivity of a simultaneous confidence band. This criterion is defined to be the probability that a false linear regression model is excluded from the band at least at one point and hence this false linear regression model is correctly declared as a false model by the band. Finally, the article considers and compares several computational algorithms for obtaining the confidence band.     

单个多元正态总体均值向量联合置信区间的比较

单个多元正态总体均值向量的Bonferroni和Scheff\'{e}联合置信区间在实际中经常用到,本文主要采用解析的办法比较这两个置信区间的长短, 证明了当均值向量的维数$2\le p\le 12$时, Bonferroni联合置信区间比Scheff\'{e}联合置信区间短.     

Mean–Mean Multiple Comparison Displays for Families of Linear Contrasts

Traditional tabular and graphical displays of results of simultaneous confidence intervals or hypothesis tests are deficient in several respects. Expanding on earlier work, we present new mean–mean multiple comparison graphs that succinctly and compactly display the results of traditional procedures for multiple comparisons of population means or linear contrasts involving means. The MMC plot can be used with unbalanced, multifactor designs with covariates. After reviewing the construction of these displays in the S language (S-Plus and R), we demonstrate their application to four multiple comparison scenarios.     

Multiple comparisons of several homoscedastic multivariate populations

The limiting joint distribution of correlated Hotelling’s T ² statistics associated with multiple comparisons with a control in multivariate one-way layout model is a multivariate central nonsingular chi-square distribution with one-factorial correlation matrix, which has the distribution function expressed in a closed form as an integral of a product of noncentral chi-square distribution functions with respect to a central chi-square density function. For pairwise comparisons, it is a multivariate central singular chi-square distribution whose distribution function is generally intricate. To overcome the complexity of the (exact or asymptotic) distribution theory of -type statistics appeared in simultaneous confidence intervals of mean vectors, improved Bonferroni-type inequalities are applied to construct asymptotically conservative simultaneous confidence intervals for pairwise comparisons as well as comparisons with a control.     

Confidence regions for the intensity function of a cyclic Poisson process

A classical approach to constructing simultaneous confidence intervals (i.e., confidence bands or regions) for a function is via establishing a limiting process of the appropriately normalized difference between the function and its empirical estimator. In the present paper we depart from this approach and construct confidence bands for the intensity function of a cyclic Poisson process via extreme value type asymptotic results for the appropriately normalized supremum of the difference between the intensity function and its empirical estimator.      

Exact confidence coefficients of simultaneous confidence intervals for multinomial proportions

Simultaneous confidence intervals for multinomial proportions are useful in many areas of science. Since 1964, approximate simultaneous 1-α confidence intervals have been proposed for multinomial proportions. Although at each point in the parameter space, these confidence sets have asymptotic 1-α coverage probability, the exact confidence coefficients of these simultaneous confidence intervals for a fixed sample size are unknown before.In this paper, we propose a procedure for calculating exact confidence coefficients for simultaneous confidence intervals of multinomial proportions for any fixed sample size. With this methodology, exact confidence coefficients can be clearly derived, and the point at which the infimum of the coverage probability occurs can be clearly identified.     

核实数据下响应变量缺失的线性EV模型经验似然推断

考虑响应变量随机缺失而协变量带有误差的线性模型,借助于核实数据和借补方法,构造了回归系数的两种经验似然比,证明了所提出的估计的经验对数似然比渐近于一个自由度为1的独立χ2变量的加权和;而经调整后所得的调整经验对数似然比渐近于自由度为p的χ2分布,该结果可以用来构造未知参数的置信域.此外,我们也构造了响应均值的调整经验对数似然比统计量,并证明了所提出的统计量渐近于x2分布,可用此结果构造响应均值的置信域.通过模拟研究比较了置信域的精度及其平均区间长度.     

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在φ混合的随机误差下,本文研究了固定设计及响应变量有缺失的非参数回归模型中回归函数的经验似然置信区间的构造.首先采用非参数回归填补法对缺失的数据进行填补,其次利用补足后得到的"完全样本"构造了非参数回归函数的经验似然比统计量,并证明了经验似然比统计量的极限分布为卡方分布,利用此结果可以构造非参数回归函数的经验似然置信区间.     

On simultaneous confidence intervals for all contrasts in the means of the intraclass correlation model with missing data

In this paper, we consider simultaneous confidence intervals for all contrasts in the means when the observations are missing at random in the intraclass correlation model. An exact test statistic for the equality of the means and Scheffé, Bonferroni and Tukey types of simultaneous confidence intervals are given by an extension of Bhargava and Srivastava [On Tukey's confidence intervals for the contrasts in the means of the intraclass correlation model, J. Royal Statist. Soc. B35 (1973) 147-152] when the missing observations are of the monotone type. Finally, numerical results of simultaneous confidence intervals are presented.     

Remarks on between estimator in the intraclass correlation model with missing data

In this paper, we consider the between estimator under the intraclass correlation model with missing data. We give a necessary and sufficient condition for existing exact simultaneous confidence intervals for all contrasts in the means under the between transformed model, which indicates the F-test statistic and simultaneous confidence intervals, constructed by Seo et al. [T. Seo, J. Kikuchi, K. Koizumi, On simultaneous confidence intervals for all contracts in the means of the intraclass correlation model with missing data, J. Multivariate Anal. 97 (2006) 1976–1983] based on the between estimator, is invalid. Furthermore, using the distribution of the between estimator, we present the exact test statistics and confidence intervals for partial contrasts.     

Comparison of hyperbolic and constant width simultaneous confidence bands in multiple linear regression under MVCS criterion

A simultaneous confidence band provides useful information on the plausible range of the unknown regression model, and different confidence bands can often be constructed for the same regression model. For a simple regression line, Liu and Hayter [W. Liu, A.J. Hayter, Minimum area confidence set optimality for confidence bands in simple linear regression, J. Amer. Statist. Assoc. 102 (477) (2007) pp. 181–190] proposed the use of the area of the confidence set corresponding to a confidence band as an optimality criterion in comparison of confidence bands; the smaller the area of the confidence set, the better the corresponding confidence band. This minimum area confidence set (MACS) criterion can be generalized to a minimum volume confidence set (MVCS) criterion in the study of confidence bands for a multiple linear regression model. In this paper hyperbolic and constant width confidence bands for a multiple linear regression model over a particular ellipsoidal region of the predictor variables are compared under the MVCS criterion. It is observed that whether one band is better than the other depends on the magnitude of one particular angle that determines the size of the predictor variable region. When the angle and hence the size of the predictor variable region is small, the constant width band is better than the hyperbolic band but only marginally. When the angle and hence the size of the predictor variable region is large the hyperbolic band can be substantially better than the constant width band.     

平衡单向分类模型中几种同时置信区间的比较

在平衡单向分类模型中,通过对对照的三种同时置信区间长度表达式的分析,可以看到,在不同情况下,三种区间的优劣.     

Exact inference on contrasts in means of intraclass correlation models with missing responses

Intraclass correlation models with missing data at random are considered. With a properly reduced model, a general method, which allows repeated observations with missing data in a non-monotone pattern, is proposed to construct exact test statistics and simultaneous confidence intervals for linear contrasts in the means. Simulation results are given to compare exact and asymptotic simultaneous confidence intervals. A real example is provided for the illustration of the proposed method.     

Simultaneous testing for the successive differences of exponential location parameters under heteroscedasticity

In this paper, the design-oriented two-stage and data-analysis one-stage multiple comparison procedures for successive comparisons of exponential location parameters under heteroscedasticity are proposed. One-sided and two-sided simultaneous confidence intervals are also given. We also extend these simultaneous confidence intervals for successive differences to a larger class of contrasts of the location parameters. Upper limits of critical values are obtained using the recent techniques given in Lam [Lam, K., 1987. Subset selection of normal populations under heteroscedasticity. In: Proceedings of the Second International Advanced Seminar/Workshop on Inference Procedures Associated with Statistical Ranking and Selection, Sydney, Australia; Lam, K., 1988. An improved two-stage selection procedure. Communications in Statistics Simulation and Computation. 17 (3), 995-1006]. These approximate critical values are shown to have better results than the approximate critical values using the Bonferroni inequality developed in this paper. Finally, the application of the proposed procedures is illustrated with an example.     

On Confidence Intervals in Nonparametric Binary Regression via Edgeworth Expansions

Local confidence intervals for regression function with binary response variable are constructed. These intervals are based on both theoretical and “plug-in” normal asymptotic distribution of a usual statistic. In the plug-in approach, two ways of estimating bias are proposed; for them we obtain the mean squared error and deduce an expression of an optimal bandwidth. The rate of convergence of theoretical distributions to their limits is obtained by means of Edgeworth expansions. Likewise, these expansions allow us to deduce properties about the coverage probability of the confidence intervals. Theoretic approximations to that probability are compared in a simulation study with the corresponding coverage rates.     

Likelihood-based Inference for the Ratios of Regression Coefficients in Linear Models

We consider the standard linear multiple regression model in which the parameter of interest is the ratio of two regression coefficients. Our setup includes a broad range of applications. We show that the 1− α confidence interval for the interest parameter based on the profile, conditional profile, modified profile or adjusted profile likelihood can potentially become the entire real line, while appropriately chosen integrated likelihoods do not suffer from this drawback. We further explore the asymptotic length of confidence intervals in order to compare integrated likelihood-based proposals. The analysis is facilitated by an orthogonal parameterization.