Comparing several exponential populations with more than one control

Suppose there are k ₁ (k ₁ ≥ 1) test treatments that we wish to compare with k ₂ (k ₂ ≥ 1) control treatments. Assume that the observations from the ith test treatment and the jth control treatment follow a two-parameter exponential distribution and , where θ is a common scale parameter and and are the location parameters of the ith test and the jth control treatment, respectively, i = 1, . . . ,k ₁; j = 1, . . . ,k ₂. In this paper, simultaneous one-sided and two-sided confidence intervals are proposed for all k ₁ k ₂ differences between the test treatment location and control treatment location parameters, namely , and the required critical points are provided. Discussions of multiple comparisons of all test treatments with the best control treatment and an optimal sample size allocation are given. Finally, it is shown that the critical points obtained can be used to construct simultaneous confidence intervals for Pareto distribution location parameters.     

Comparing confidence intervals for the logarithmic series distribution

It is assumed that a small random sample of fixed size n is drawn from a logarithmic series distribution with parameter θ and that it is desired to estimate θ by means of a two-sided confidence interval. In this note Crow's system of confidence intervals is compared, in shortness of intervals, with Clopper and Pearson's, and the corresponding randomized counterparts.     

Comparing Several Variances with a Control in Direction-mixed Families of Hypotheses

This article deals with simultaneous comparisons of k(k ? 1) populations with a control population with respect to variance in direction-mixed families of hypotheses. Computation of critical constants required for the implementation of the proposed procedures is discussed and selected values of the critical constants are tabulated. Power comparison of the proposed procedures with their existing competitors is carried out using Monte Carlo simulation. An example is given to illustrate the implementation of proposed procedures.     

Comparing multiple process overall yields from multiple manufacturing lines

A manufacturing process often uses multiple manufacturing lines to produce the same product for the productivity purpose. The capabilities of multiple manufacturing lines are often non-identical in practice. The task of process evaluation needs to measure the overall yield. This research compares the overall yield among I processes from K manufacturing lines, I ? 2, K ? 2. An unconstrained multiple comparisons with the best (UMCB) method based on the overall yield index is proposed to determine the best process with the highest overall yield. The simultaneous confidence intervals of each difference from the unknown best process in terms of the overall yield index are provided at a confidence level of at least 1 ?α. To demonstrate the effectiveness of the UMCB method, the method was applied to choose the best process producing the pre-amplifier of microelectromechanical systems (MEMS) sensors in the production environment with multiple manufacturing lines.     

Max-min multiple comparison procedure for comparing several dose levels with a zero dose control

The comparison of increasing doses of a compound to a zero dose control is of interest in medical and toxicological studies. Assume that the mean dose effects are non-decreasing among the non-zero doses of the compound. A simple procedure that modifies Dunnett's procedure is proposed to construct simultaneous confidence intervals for pairwise comparisons of each dose group with the zero dose control by utilizing the ordering of the means. The simultaneous lower bounds and upper bounds by the new procedure are monotone, which is not the case with Dunnett's procedure. This is useful to categorize dose levels. The expected gains of the new procedure over Dunnett's procedure are studied. The procedure is shown by real data to compare well with its predecessor.     

A note on successive comparisons between several ordered variances

ABSTRACT

In this article, a procedure for comparisons between k (k ? 3) successive populations with respect to the variance is proposed when it is reasonable to assume that variances satisfy simple ordering. Critical constants required for the implementation of the proposed procedure are computed numerically and selected values of the computed critical constants are tabulated. The proposed procedure for normal distribution is extended for making comparisons between successive exponential populations with respect to scale parameter. A comparison between the proposed procedure and its existing competitor procedures is carried out, using Monte Carlo simulation. Finally, a numerical example is given to illustrate the proposed procedure.     

Multistage methodologies for comparing several treatments with a control

We consider the problem of constructing a set of fixed-width simultaneous confidence intervals for the treatment-control differences of means for several independent normal populations with a common unknown variance. Taking c observations from the control population instead of the usual vector-at-a-time approach, purely sequential estimation methodology is developed and asymptotic second-order characteristics are provided. Brief remarks on the accelerated sequential and three-stage methodologies have been added. Next, with the help of simulations, performances of the purely sequential, accelerated sequential and three-stage estimation techniques are compared. Overall, the second-order asymptotics are found to provide useful approximations even for moderate sample sizes.     

Some multiple comparison procedures for comparing treatments within several groups

Multiple comparison procedures are extended to designs consisting of several groups, where the treatment means are to be compared within each group. This may arise in two-factor experiments, with a significant interaction term, when one is interested in comparing the levels of one factor at each level of the other factor. A general approach is presented for deriving the distributions and calculating critical points, following three papers which dealt with two specific procedures. These points are used for constructing simultaneous confidence intervals over some restricted set of contrasts among treatment means in each of the groups. Tables of critical values are provided for two procedures and an application is demonstrated. Some extensions are presented for the case of possible different sets of contrasts and also for unequal variances in the various groups.     

Optimal simultaneous confidence bands in simple linear regression

A simultaneous confidence band provides useful information on the plausible range of an unknown regression model. For simple linear regression models, the most frequently quoted bands in the statistical literature include the hyperbolic band and the three-segment bands. One interesting question is whether one can construct confidence bands better than the hyperbolic and three-segment bands. The optimality criteria for confidence bands include the average width criterion considered by Gafarian (1964) and Naiman (1984) among others, and the minimum area confidence set (MACS) criterion of Liu and Hayter (2007). In this paper, two families of exact 1−α$1-\alpha$ confidence bands, the inner-hyperbolic bands and the outer-hyperbolic bands, which include the hyperbolic and three-segment bands as special cases, are introduced in simple linear regression. Under the MACS criterion, the best confidence band within each family is found by numerical search and compared with the hyperbolic band, the best three-segment band and with each other. The methodologies are illustrated with a numerical example and the Matlab programs used are available upon request.     

A closed testing procedure for comparing successive exponential populations with respect to location parameter

Consider k independent random samples such that i^th sample is drawn from a two-parameter exponential population with location parameter μ_i and scale parameter θ_i,?i = 1, …, k. For simultaneously testing differences between location parameters of successive exponential populations, closed testing procedures are proposed separately for the following cases (i) when scale parameters are unknown and equal and (ii) when scale parameters are unknown and unequal. Critical constants required for the proposed procedures are obtained numerically and the selected values of the critical constants are tabulated. Simulation study revealed that the proposed procedures has better ability to detect the significant differences and has more power in comparison to exiting procedures. The illustration of the proposed procedures is given using real data.     

Simultaneous Inferences on the Cumulative Distribution Function of a Normal Distribution

This paper addresses the problem of constructing simultaneous confidence intervals for the cumulative distribution function of a normal distribution at several specified points. The procedure is based upon the observation of a random sample of independent observations from a normal distribution with an unknown mean and variance. A new methodology is proposed for obtaining confidence intervals with a specified overall simultaneous confidence level through the inversion of acceptance sets. Both one-sided and two-sided confidence intervals are considered. Some illustrations of the new method are provided, and comparisons are made with other approaches to the problem.     

Usual and Shortest Confidence Intervals on Odds Ratios from Logistic Regression

Based on the large-sample normal distribution of the sample log odds ratio and its asymptotic variance from maximum likelihood logistic regression, shortest 95% confidence intervals for the odds ratio are developed. Although the usual confidence interval on the odds ratio is unbiased, the shortest interval is not. That is, while covering the true odds ratio with the stated probability, the shortest interval covers some values below the true odds ratio with higher probability. The upper and lower limits of the shortest interval are shifted to the left of those of the usual interval, with greater shifts in the upper limits. With the log odds model γ + Xβ, in which X is binary, simulation studies showed that the approximate average percent difference in length is 7.4% for n (sample size) = 100, and 3.8% for n = 200. Precise estimates of the covering probabilities of the two types of intervals were obtained from simulation studies, and are compared graphically. For odds ratio estimates greater (less) than one, shortest intervals are more (less) likely to include one than are the usual intervals. The usual intervals are likelihood-based and the shortest intervals are not. The usual intervals have minimum expected length among the class of unbiased intervals. Shortest intervals do not provide important advantages over the usual intervals, which we recommend for practical use.     

Simultaneous confidence intervals by iteratively adjusted alpha for relative effects in the one-way layout

A bootstrap based method to construct 1−α simultaneous confidence intervals for relative effects in the one-way layout is presented. This procedure takes the stochastic correlation between the test statistics into account and results in narrower simultaneous confidence intervals than the application of the Bonferroni correction. Instead of using the bootstrap distribution of a maximum statistic, the coverage of the confidence intervals for the individual comparisons are adjusted iteratively until the overall confidence level is reached. Empirical coverage and power estimates of the introduced procedure for many-to-one comparisons are presented and compared with asymptotic procedures based on the multivariate normal distribution.     

Model robust confidence intervals

Confidence intervals are constructed for real-valued parameter estimation in a general regression model with normal errors. When the error variance is known these intervals are optimal (in the sense of minimizing length subject to guaranteed probability of coverage) among all intervals estimates which are centered at a linear estimate of the parameter. When the error variance is unknown and the regression model is an approximately linear model (a class of models which permits bounded systematic departures from an underlying ideal model) then an independent estimate of variance is found and the intervals can then be appropriately scaled.     

The Shortest Clopper–Pearson Confidence Interval for Binomial Probability

The existence of the shortest confidence interval for binomial probability is shown. The method of obtaining such an interval is presented as well.     

Exact Likelihood Inference for Two Exponential Populations Under Joint Progressive Type-II Censoring

Comparative lifetime experiments are of great importance when the interest is in ascertaining the relative merits of two competing products with regard to their reliability. In this article, we consider two exponential populations and when joint progressive Type-II censoring is implemented on the two samples. We then derive the moment generating functions and the exact distributions of the maximum likelihood estimators (MLEs) of the mean lifetimes of the two exponential populations under such a joint progressive Type-II censoring. We then discuss the exact lower confidence bounds, exact confidence intervals, and simultaneous confidence regions. Next, we discuss the corresponding approximate results based on the asymptotic normality of the MLEs as well as those based on the Bayesian method. All these confidence intervals and regions are then compared by means of Monte Carlo simulations with those obtained from bootstrap methods. Finally, an illustrative example is presented in order to illustrate all the methods of inference discussed here.     

Evaluating the Performance of Simultaneous Stepwise Confidence Intervals for the Difference between two Poisson Rates

We consider the problem of simultaneously estimating Poisson rate differences via applications of the Hsu and Berger stepwise confidence interval method (termed HBM), where comparisons to a common reference group are performed. We discuss continuity-corrected confidence intervals (CIs) and investigate the HBM performance with a moment-based CI, and uncorrected and corrected for continuity Wald and Pooled confidence intervals (CIs). Using simulations, we compare nine individual CIs in terms of coverage probability and the HBM with nine intervals in terms of family-wise error rate (FWER) and overall and local power. The simulations show that these statistical properties depend highly on parameter settings.     

A Graphical Representation for Non-Parametric Inference

A method is described for determining the sample size required for a specified precision simultaneous confidence statement about the parameters of a multinomial population. The method is based on a simultaneous confidence interval procedure due to Goodman, and the results are compared with those obtained by separately considering each cell of the multinomial population as a binomial.     

Simultaneous Confidence Intervals for the Population Cell Means,for Two-by-Two Factorial Data,that Utilize Uncertain Prior Information

Consider a two-by-two factorial experiment with more than one replicate. Suppose that we have uncertain prior information that the two-factor interaction is zero. We describe new simultaneous frequentist confidence intervals for the four population cell means, with simultaneous confidence coefficient 1 ? α, that utilize this prior information in the following sense. These simultaneous confidence intervals define a cube with expected volume that (a) is relatively small when the two-factor interaction is zero and (b) has maximum value that is not too large. Also, these intervals coincide with the standard simultaneous confidence intervals obtained by Tukey’s method, with simultaneous confidence coefficient 1 ? α, when the data strongly contradict the prior information that the two-factor interaction is zero. We illustrate the application of these new simultaneous confidence intervals to a real data set.