On the distribution-free confidence intervals and universal bounds for quantiles based on joint records

In this paper, we consider three distribution-free confidence intervals for quantiles given joint records from two independent sequences of continuous random variables with a common continuous distribution function. The coverage probabilities of these intervals are compared. We then compute the universal bounds of the expected widths of the proposed confidence intervals. These results naturally extend to any number of independent sequences instead of just two. Finally, the proposed confidence intervals are applied for a real data set to illustrate the practical usefulness of the procedures developed here.     

Exact confidence intervals for randomized response strategies

For surveys with sensitive questions, randomized response sampling strategies are often used to increase the response rate and encourage participants to provide the truth of the question while participants' privacy and confidentiality are protected. The proportion of responding ‘yes’ to the sensitive question is the parameter of interest. Asymptotic confidence intervals for this proportion are calculated from the limiting distribution of the test statistic, and are traditionally used in practice for statistical inference. It is well known that these intervals do not guarantee the coverage probability. For this reason, we apply the exact approach, adjusting the critical value as in [10 J. Frey and A. Pérez, Exact binomial confidence intervals for randomized response, Amer. Statist.66 (2012), pp. 8–15. Available at http://dx.doi.org/10.1080/00031305.2012.663680.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], to construct the exact confidence interval of the proportion based on the likelihood ratio test and three Wilson-type tests. Two randomized response sampling strategies are studied: the Warner model and the unrelated model. The exact interval based on the likelihood ratio test has shorter average length than others when the probability of the sensitive question is low. Exact Wilson intervals have good performance in other cases. A real example from a survey study is utilized to illustrate the application of these exact intervals.     

On the lengths of t-based confidence intervals

Confidence interval is a basic type of interval estimation in statistics. When dealing with samples from a normal population with the unknown mean and the variance, the traditional method to construct t-based confidence intervals for the mean parameter is to treat the n sampled units as n groups and build the intervals. Here we propose a generalized method. We first divide them into several equal-sized groups and then calculate the confidence intervals with the mean values of these groups. If we define “better” in terms of the expected length of the confidence interval, then the first method is better because the expected length of the confidence interval obtained from the first method is shorter. We prove this intuition theoretically. We also specify when the elements in each group are correlated, the first method is invalid, while the second can give us correct results in terms of the coverage probability. We illustrate this with analytical expressions. In practice, when the data set is extremely large and distributed in several data centers, the second method is a good tool to get confidence intervals, in both independent and correlated cases. Some simulations and real data analyses are presented to verify our theoretical results.     

Analytical approximations for iterated bootstrap confidence intervals

Standard algorithms for the construction of iterated bootstrap confidence intervals are computationally very demanding, requiring nested levels of bootstrap resampling. We propose an alternative approach to constructing double bootstrap confidence intervals that involves replacing the inner level of resampling by an analytical approximation. This approximation is based on saddlepoint methods and a tail probability approximation of DiCiccio and Martin (1991). Our technique significantly reduces the computational expense of iterated bootstrap calculations. A formal algorithm for the construction of our approximate iterated bootstrap confidence intervals is presented, and some crucial practical issues arising in its implementation are discussed. Our procedure is illustrated in the case of constructing confidence intervals for ratios of means using both real and simulated data. We repeat an experiment of Schenker (1985) involving the construction of bootstrap confidence intervals for a variance and demonstrate that our technique makes feasible the construction of accurate bootstrap confidence intervals in that context. Finally, we investigate the use of our technique in a more complex setting, that of constructing confidence intervals for a correlation coefficient.     

A class of fixed-width confidence intervals for a ranked normal mean

Suppose that we are given k(≥ 2) independent and normally distributed populations π₁, …, π_k where π_i has unknown mean μ_i and unknown variance σ² _i (i = 1, …, k). Let μ_[i] (i = 1, …, k) denote the i^th smallest one of μ₁, …, μ_k. A two-stage procedure is used to construct lower and upper confidence intervals for μ_[i] and then use these to obtain a class of two-sided confidence intervals on μ_[i] with fixed width. For i = k, the interval given by Chen and Dudewicz (1976) is a special case. Comparison is made between the class of two-sided intervals and a symmetric interval proposed by Chen and Dudewicz (1976) for the largest mean, and it is found that for large values of k at least one of the former intervals requires a smaller total sample size. The tables needed to actually apply the procedure are provided.     

Simultaneous confidence intervals for comparing several inverse Gaussian means under heteroscedasticity

Recently, Zhang [Simultaneous confidence intervals for several inverse Gaussian populations. Stat Probab Lett. 2014;92:125–131] proposed simultaneous pairwise confidence intervals (SPCIs) based on the fiducial generalized pivotal quantity concept to make inferences about the inverse Gaussian means under heteroscedasticity. In this paper, we propose three new methods for constructing SPCIs to make inferences on the means of several inverse Gaussian distributions when scale parameters and sample sizes are unequal. One of the methods results in a set of classic SPCIs (in the sense that it is not simulation-based inference) and the two others are based on a parametric bootstrap approach. The advantages of our proposed methods over Zhang’s (2014) method are: (i) the simulation results show that the coverage probability of the proposed parametric bootstrap approaches is fairly close to the nominal confidence coefficient while the coverage probability of Zhang’s method is smaller than the nominal confidence coefficient when the number of groups and the variance of groups are large and (ii) the proposed set of classic SPCIs is conservative in contrast to Zhang’s method.     

Confidence and prediction intervals based on interpolated records

In several statistical problems, nonparametric confidence intervals for population quantiles can be constructed and their coverage probabilities can be computed exactly, but cannot in general be rendered equal to a pre-determined level. The same difficulty arises for coverage probabilities of nonparametric prediction intervals for future observations. One solution to this difficulty is to interpolate between intervals which have the closest coverage probability from above and below to the pre-determined level. In this paper, confidence intervals for population quantiles are constructed based on interpolated upper and lower records. Subsequently, prediction intervals are obtained for future upper records based on interpolated upper records. Additionally, we derive upper bounds for the coverage error of these confidence and prediction intervals. Finally, our results are applied to some real data sets. Also, a comparison via a simulation study is done with similar classical intervals obtained before.     

Fixed width confidence intervals for the location parameter of an exponential distribution

We study confidence intervals of prescribed width for the lo-cation parameter of an exponential distribution. Asymptotic expan-sions up to terms tending to zero are obtained for the coverage probability and expected sample size. The limiting distribution of the sample size is given from which an asymptotic expression for the variance of the sample size is deduced. Sequential procedures with non-asymptotic coverage probability are also investigated     

The performance of model averaged tail area confidence intervals

We investigate the exact coverage and expected length properties of the model averaged tail area (MATA) confidence interval proposed by Turek and Fletcher, CSDA, 2012, in the context of two nested, normal linear regression models. The simpler model is obtained by applying a single linear constraint on the regression parameter vector of the full model. For given length of response vector and nominal coverage of the MATA confidence interval, we consider all possible models of this type and all possible true parameter values, together with a wide class of design matrices and parameters of interest. Our results show that, while not ideal, MATA confidence intervals perform surprisingly well in our regression scenario, provided that we use the minimum weight within the class of weights that we consider on the simpler model.     

Better approximate confidence intervals for a binomial parameter

This paper discusses five methods for constructing approximate confidence intervals for the binomial parameter Θ, based on Y successes in n Bernoulli trials. In a recent paper, Chen (1990) discusses various approximate methods and suggests a new method based on a Bayes argument, which we call method I here. Methods II and III are based on the normal approximation without and with continuity correction. Method IV uses the Poisson approximation of the binomial distribution and then exploits the fact that the exact confidence limits for the parameter of the Poisson distribution can be found through the x² distribution. The confidence limits of method IV are then provided by the Wilson-Hilferty approximation of the x². Similarly, the exact confidence limits for the binomial parameter can be expressed through the F distribution. Method V approximates these limits through a suitable version of the Wilson-Hilferty approximation. We undertake a comparison of the five methods in respect to coverage probability and expected length. The results indicate that method V has an advantage over Chen's Bayes method as well as over the other three methods.     

Applications of asymptotic confidence intervals with continuity corrections for asymmetric comparisons in noninferiority trials

A large‐sample problem of illustrating noninferiority of an experimental treatment over a referent treatment for binary outcomes is considered. The methods of illustrating noninferiority involve constructing the lower two‐sided confidence bound for the difference between binomial proportions corresponding to the experimental and referent treatments and comparing it with the negative value of the noninferiority margin. The three considered methods, Anbar, Falk–Koch, and Reduced Falk–Koch, handle the comparison in an asymmetric way, that is, only the referent proportion out of the two, experimental and referent, is directly involved in the expression for the variance of the difference between two sample proportions. Five continuity corrections (including zero) are considered with respect to each approach. The key properties of the corresponding methods are evaluated via simulations. First, the uncorrected two‐sided confidence intervals can, potentially, have smaller coverage probability than the nominal level even for moderately large sample sizes, for example, 150 per group. Next, the 15 testing methods are discussed in terms of their Type I error rate and power. In the settings with a relatively small referent proportion (about 0.4 or smaller), the Anbar approach with Yates’ continuity correction is recommended for balanced designs and the Falk–Koch method with Yates’ correction is recommended for unbalanced designs. For relatively moderate (about 0.6) and large (about 0.8 or greater) referent proportion, the uncorrected Reduced Falk–Koch method is recommended, although in this case, all methods tend to be over‐conservative. These results are expected to be used in the design stage of a noninferiority study when asymmetric comparisons are envisioned. Copyright © 2013 John Wiley & Sons, Ltd.     

New approximate confidence intervals for the difference between two Poisson means and comparison

The problem of estimating the difference between two Poisson means is considered. A new moment confidence interval (CI), and a fiducial CI for the difference between the means are proposed. The moment CI is simple to compute, and it specializes to the classical Wald CI when the sample sizes are equal. Numerical studies indicate that the moment CI offers improvement over the Wald CI when the sample sizes are different. Exact properties of the CIs based on the moment, fiducial and hybrid methods are evaluated numerically. Our numerical study indicates that the hybrid and fiducial CIs are in general comparable, and the moment CI seems to be the best when the expected total counts from both distributions are two or more. The interval estimation procedures are illustrated using two examples.     

Distribution-free confidence intervals for quantiles in small samples

Estimators for quantiles based on linear combinations of order statistics have been proposed by Harrell and Davis(1982) and kaigh and Lachenbruch (1982). Both estimators have been demonstrated to be at least as efficient for small sample point estimation as an ordinary sample quantile estimator based on one or two order statistics: Distribution-free confidence intervals for quantiles can be constructed using either of the two approaches. By means of a simulation study, these confidence intervals have been compared with several other methods of constructing confidence intervals for quantiles in small samples. For the median, the Kaigh and Lachenbruch method performed fairly well. For other quantiles, no method performed better than the method which uses pairs of order statistics.     

Design-based confidence limits for a distribution function

A method for constructing confidence limits for a distribution function is proposed. This method is a simple modification of the common method based on a normal approximation to the distribution of the estimated distribution function. The methods differ in how the estimated standard errors are used. The coverage properties of the two methods are compared in a simulation study. Coverage probabilities for the proposed method are found to be much closer to the nominal levels, particularly in the tails of the population distribution.     

A comparative study of confidence intervals for negative binomial proportion

In this paper, we investigate four existing and three new confidence interval estimators for the negative binomial proportion (i.e., proportion under inverse/negative binomial sampling). An extensive and systematic comparative study among these confidence interval estimators through Monte Carlo simulations is presented. The performance of these confidence intervals are evaluated in terms of their coverage probabilities and expected interval widths. Our simulation studies suggest that the confidence interval estimator based on saddlepoint approximation is more appealing for large coverage levels (e.g., nominal level≤1% ) whereas the score confidence interval estimator is more desirable for those commonly used coverage levels (e.g., nominal level>1% ). We illustrate these confidence interval construction methods with a real data set from a maternal congenital heart disease study.     

Parametric and nonparametric confidence intervals for estimating the difference of means of two skewed populations

In this paper, we have reviewed and proposed several interval estimators for estimating the difference of means of two skewed populations. Estimators include the ordinary-t, two versions proposed by Welch [17] and Satterthwaite [15], three versions proposed by Zhou and Dinh [18], Johnson [9], Hall [8], empirical likelihood (EL), bootstrap version of EL, median t proposed by Baklizi and Kibria [2] and bootstrap version of median t. A Monte Carlo simulation study has been conducted to compare the performance of the proposed interval estimators. Some real life health related data have been considered to illustrate the application of the paper. Based on our findings, some possible good interval estimators for estimating the mean difference of two populations have been recommended for the researchers.     

Efficiency comparison of M-estimates for scale at t-distributions

Using Fisher's information fort-distributions, the absolute asymptotic efficiency of some M-estimates for scale with known location parameter is calculated and graphically illustrated. The compared estimators are the standard deviationS _*, the mean absolute deviation, called mean deviationD _*, the median absolute deviation, called MAD_*, and some M-estimates for scale, one, which is very robust, and another one with high asymptotic efficiency fort-distributions close to the normal. The last one is considered with monotone (in the positive field) and with very late redescending χ-function too. Also the , an alternative and generalized excess measure defined as the double relative asymptotic variance of the underlying scale estimator in the previous paper, is calculated fort-distributions and graphically illustrated, because there is the relation that the higher the asymptotic efficiency of is, the lower is the corresponding .     

Mid-P confidence intervals based on the likelihood ratio for proportions estimated by group testing

Group testing has been used in many fields of study to estimate proportions. When groups are of different size, the derivation of exact confidence intervals is complicated by the lack of a unique ordering of the event space. An exact interval estimation method is described here, in which outcomes are ordered according to a likelihood ratio statistic. The method is compared with another exact method, in which outcomes are ordered by their associated MLE. Plots of the P‐value against the proportion are useful in examining the properties of the methods. Coverage provided by the intervals is assessed using several realistic grouptesting procedures. The method based on the likelihood ratio, with a mid‐P correction, is shown to give very good coverage in terms of closeness to the nominal level, and is recommended for this type of problem.     

A confidence interval for the scale parameter based on sukhtmes two-sample statistic

Sukhatme (1957) introduced a statistic which may be used to test the equality of variances in two independent samples from continuous distributions, centered at zero. It has become evident that this statistic cound be used to construct, analytically, a confidence interval for the scale parameter relating the two random variables(of.Laubscher (1968) and Noether (1967, pp. 66-69 and 1972)). In view of this additional use of the statistic, and since no tables of critical values exist, we provide such a table which makes the application of the statistic a practical proposition. In addition, a normal approximation is suggested for the use outside the range of Table III and the accuracy of this approximation is investigated. The Sukhatme test applied to sample values reduced by t n e i r medians i s studied in a small simulation exercise. It appears that this test when used in conjunction with the critical values of Sukhatme's statistic, is a very conservative one and that it is probably fairly robust with respect to the underlying population distribution.