On Sample Size and Quick Simultaneous Confidence Interval Estimations for Multinomial Proportions

Abstract

Asymptotically, sample proportions from a multinomial distribution converge in distribution to a multivariate normal distribution with a singular negative product correlation structure. Based on this result, we propose a new approach to estimate the sample size requirement for constructing quick simultaneous confidence intervals (QSCI) for multinomial proportions. In addition, this new approach can be used to construct QSCI and provides a statistical justification to the reports of the opinion polling.     

Coverage of generalized confidence intervals

Generalized confidence intervals provide confidence intervals for complicated parametric functions in many common practical problems. They do not have exact frequentist coverage in general, but often provide coverage close to the nominal value and have the correct asymptotic coverage. However, in many applications generalized confidence intervals do not have satisfactory finite sample performance. We derive expansions of coverage probabilities of one-sided generalized confidence intervals and use the expansions to explain the nonuniform performance of the generalized intervals. We then show how to use these expansions to obtain improved coverage by suitable calibration. The benefits of the proposed modification are illustrated via several examples.     

On some confidence intervals for estimating the mean of a skewed population

A number of methods are available in the literature to measure confidence intervals. Here, confidence intervals for estimating the population mean of a skewed distribution are considered. This note proposes two alternative confidence intervals, namely, Median t and Mad t, which are simple adjustments to the Student's t confidence interval. In order to compare the performance of these intervals, the following criteria are considered: (i) coverage probability; (ii) average width; and (iii) ratio of coverage to width. A simulation study has been undertaken to compare the performance of the intervals. The simulation study shows that for small sample size and moderate to highly skewed distributions, the proposed Median t performs the best in the sense of higher coverage, and the Mad t performs best in the sense of smaller confidence width. The proposed methods are very easy to calculate and are not overly computer-intensive, like Bootstrap confidence intervals. Some real-life examples have been considered that support the findings of the paper to some extent.     

On confidence intervals from simulation of finite Markov chains

Consider a finite state irreducible Markov reward chain. It is shown that there exist simulation estimates and confidence intervals for the expected first passage times and rewards as well as the expected average reward, with 100% coverage probability. The length of the confidence intervals converges to zero with probability one as the sample size increases; it also satisfies a large deviations property.     

Panel模型中的精确检验和置信区间

利用广义p-值和广义置信区间的概念,研究了Panel模型中未知参数的检验和置信区间问题.对于回归系数,分别考虑了单个情形和多个线性无关情形下的检验和置信区间问题,得到了精确检验和置信区间.对于方差分量,研究了其任意线性组合的检验和置信区间问题,建立了精确检验和置信区间.基于广义p-值和广义置信区间,获取精确检验和置信区间的方法具有计算方便、易应用于小样本问题的特点.最后,分别从理论和数值上研究了这些精确检验和置信区间的统计性质.     

On the location parameter confidence intervals based on a random size sample from a partially known population

The problem of constructing confidence intervals of a fixed length for the location parameter based on a random size sample is considered. It is proposed to use the confidence interval $$\theta _p^* - u\sqrt p /\sigma< \theta< \theta _p^* + u\sqrt p /\sigma $$ , whereθ _p ^* is an adaptive estimator,σ ² is the Fisher information, and p^?1 is the mean of the sample size. Nonparametric bounds are given for the limit as p → 0 confidence probability. Bibliography: 5 titles.     

The Factor Analytic Approach to Simultaneous Inference in the General Linear Model

Abstract

Consider the general linear model (GLM) Y = Xβ + ε. Suppose Θ₁,…, Θ_k, a subset of the β's, are of interest; Θ₁,…, Θ_k may be treatment contrasts in an ANOVA setting or regression coefficients in a response surface setting. Existing simultaneous confidence intervals for Θ₁,…, Θ_k are relatively conservative or, in the case of the MEANS option in PROC GLM of SAS, possibly misleading. The difficulty is with the multidimensionality of the integration required to compute exact coverage probability when X does not follow a nice textbook design. Noting that such exact coverage probabilities can be computed if the correlation matrix R of the estimators of Θ₁, …, Θ_k has a one-factor structure in the factor analytic sense, it is proposed that approximate simultaneous confidence intervals be computed by approximating R with the closest one-factor structure correlation matrix. Computer simulations of hundreds of randomly generated designs in the settings of regression, analysis of covariance, and unbalanced block designs show that the coverage probabilities are practically exact, more so than can be anticipated by even second-order probability bounds.     

On split sample and randomized confidence intervals for binomial proportions

We study randomized confidence intervals for binomial proportions, comparing coverage, length and the impact of the randomization. It is seen that the recently proposed split sample intervals can be improved upon in various ways. Criticisms of randomized intervals are discussed.     

Asymptotic confidence intervals for Poisson regression

Let (X,Y) be a R^d×N₀-valued random vector where the conditional distribution of Y given X=x is a Poisson distribution with mean m(x). We estimate m by a local polynomial kernel estimate defined by maximizing a localized log-likelihood function. We use this estimate of m(x) to estimate the conditional distribution of Y given X=x by a corresponding Poisson distribution and to construct confidence intervals of level α of Y given X=x. Under mild regularity conditions on m(x) and on the distribution of X we show strong convergence of the integrated L₁ distance between Poisson distribution and its estimate. We also demonstrate that the corresponding confidence interval has asymptotically (i.e., for sample size tending to infinity) level α, and that the probability that the length of this confidence interval deviates from the optimal length by more than one converges to zero with the number of samples tending to infinity.     

Second-order accuracy of depth-based bootstrap confidence regions

We consider the problem of setting bootstrap confidence regions for multivariate parameters based on data depth functions. We prove, under mild regularity conditions, that depth-based bootstrap confidence regions are second-order accurate in the sense that their coverage error is of order n⁻¹, given a random sample of size n. The results hold in general for depth functions of types A and D, which cover as special cases the Tukey depth, the majority depth, and the simplicial depth. A simulation study is also provided to investigate empirically the bootstrap confidence regions constructed using these three depth functions.     

Simultaneous confidence bands for the integrated hazard function

The construction of the simultaneous confidence bands for the integrated hazard function is considered. The Nelson–Aalen estimator is used. The simultaneous confidence bands based on bootstrap methods are presented. Four methods of construction of such confidence bands are proposed. The weird and conditional bootstrap methods are used for resampling. Simulations are made to compare the actual coverage probability of the bootstrap and the asymptotic simultaneous confidence bands. It is shown that the equal-tailed bootstrap confidence band has the coverage probability closest to the nominal one. We also present application of our confidence bands to the data regarding survival after heart transplant. This research was partly supported by AGH grant No. 10.420.03.     

Unusual properties of bootstrap confidence intervals in regression problems

Summary We show that the percentile-t method, and one of the two percentile methods, have unusually good performance when employed to construct bootstrap confidence intervals in a regression setting. In the case of slope parameters, percentile-t produces two-sided intervals with coverage errorn ^-2, and one-sided intervals with coverage errorn ^-3/2, wheren is sample size. The errors are onlyn ^-1 in most other problems. One of the percentile methods produces critical points which are third-order correct for Efron's [11] relatively complex accelerated bias-corrected points.     

Bootstrap confidence interval for a correlation curve

A correlation curve measures the strength of the association between two variables locally at different values of x. The purpose of this study is to obtain point-wise confidence intervals for a correlation curve using wild bootstrap techniques. Empirical coverage probabilities are found to be close to the specified nominal level. Bootstrapping is an attractive alternative to confidence intervals based on asymptotic expressions that have slow rate of convergence.     

Determination of sample sizes for setting β-content tolerance limits controlling both tails of the normal distribution

The problem of determining sample size for two-sided-β-content tolerance intervals which control both tails of the normal distribution is investigated. The tolerance limits are defined to assure that the tail proportions do not exceed specified values p₁ and p₂. We determine the minimum sample size so that, at a given confidence level, the tail proportions will not be too small, i.e., so that the tolerance interval will not be overly conservative. Tables of the sample size n and the corresponding factors are provided.     

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.     

Simultaneous confidence intervals for several inverse Gaussian populations

In this research, we propose simultaneous confidence intervals for all pairwise comparisons of means from inverse Gaussian distribution. Our method is based on fiducial generalized pivotal quantities for vector parameters. We prove that the constructed confidence intervals have asymptotically correct coverage probabilities. Simulation results show that the simulated Type-I errors are close to the nominal level even for small samples. The proposed approach is illustrated by an example.     

The probability distribution of tolerance intervals

Tolerance intervals seem to be infrequently used outside engineering. This paper investigates tolerance intervals and relates them to confidence intervals and prediction intervals. It is believed that the introduction of these topics in introductory and intermediate statistics courses would be beneficial to the student.

Tolerance intervals are constructed using observations from the normal and the truncated Cauchy distributions. Both the expected value of the coverage and the probability distribution of the coverage is investigated. Tolerance intervals are also defined for sample data. These are compared numerically to tolerance intervals for populations.     

Controlling Type II Error While Constructing Triple Sampling Fixed Precision Confidence Intervals for the Normal Mean

The rationale and methodology for estimating a mean with a fixed width confidence interval through sampling in three stages are extended to cover the additional problem of testing hypotheses concerning shifts in the mean with controlled Type II error. The coverage probability and operating characteristic function of the confidence interval based on the integrated approach are derived and compared with those of the usual triple sampling confidence interval. The extended methodology leads to better coverage probability and uniformly better Type II error probabilities. Achieving the additional objective of controlling Type II error inevitably implies a two- to threefold increase in the required optimal sample size. Some suggestions for dealing with this apparent limitation are discussed from a practical viewpoint. It is recommended that an integrated approach to estimation and testing based on confidence intervals be incorporated in the design stage for credible inferences.     

Modeling the coverage probability of a confidence interval on the among-group variance component in the unbalanced random one-way model

The coverage probability of an approximate confidence interval on the among-group variance component, σ _α ² , in a one-way random model is modeled using generalized linear models techniques. The purpose of the proposed modeling is to derive an empirical relationship between the coverage probability on one hand, andk, n., ?, and the model’s variance components on the other hand, wherek is the number of groups, n. is the total number of observations, and ? is a measure of imbalance for the design used. The latter quantities serve as control variables in the derived model, and the coverage probability is treated as the response variable. Contour plots generated from this model can be easily used to depict the effects of the control variables on the coverage probability. In particular, the plots are utilized to compare four methods for constructing approximate confidence intervals on σ _α ² . Additional advantages of the derived model include prediction of the coverage probability for a given method using only specified values of the control variables, and the determination of operating conditions that result in improved coverage probability within the region of interest.