Exact confidence region for the parameters of the gamma distribution based on two independent samples

The gamma distribution has been discussed by many authors. This article proposes an exact confidence region for the parameters of a two-parameter gamma distribution. The result is based on the fact that the percentiles of the F-distribution, with equal degrees of freedom k, are monotonic in k.     

Exact simultaneous confidence segments for all contrast comparisons

This paper provides an exact method to construct simultaneous confidence bands for all contrasts of several regression lines over a restricted explanatory variable. Due to the lack of exact methods in the literature, currently existing approaches consist mainly of simulation based approaches. Using confidence bands for regression analysis occurs ubiquitously in practice, for example, inference on the shelf-life or stability of a drug, on the reliability of an engineering system over time, on the environmental impact of a fertilizer in a field over time, to list just a few. The new method enhances currently existing approaches that are based on simulations.     

Exact confidence regions for scale and location parameters based on sample quartiles

In this paper we present relatively simple (ruler, paper, and pencil) nonparametric procedures for constructing joint confidence regions for (i) the median and the inner quartile range for the symmetric one-sample problem and (ii) the shift and ratio of scale parameters for the two-sample case. Both procedures are functions of the sample quartiles and have exact confidence levels when the populations are continuous. The one-sample case requires symmetry of first and third quartiles about the median.

The confidence regions we propose are always convex, nested for decreasing confidence levels and are compact for reasonably large sample sizes. Both exact small sample and approximate large sample distributions are given.     

Exact uniformly most powerful postselection confidence distributions

A conditioning on the event of having selected one model from a set of possibly misspecified normal linear regression models leads to the construction of uniformly optimal conditional confidence distributions. They can be used for valid postselection inference. The constructed conditional confidence distributions are finite sample exact and encompass all information regarding the focus parameter in the selected model. This includes the construction of optimal postselection confidence intervals at all significance levels and uniformly most powerful hypothesis tests.     

Exact tests and confidence regions in nonlinear regression

Nonlinear regression models with spherically symmetric error vectors and a single nonlinear parameter are considered. On the basis of a new geometric approach, exact one- and two-sided tests and confidence regions for the nonlinear parameter are derived in the cases of known and unknown error variances. A geometric measure representation formula is used to determine the power functions of the tests if the error variance is known and to derive different lower bounds for the power function of a one-sided test in the case of an unknown error variance. The latter can be done quite effectively by constructing and measuring several balls inside the critical region. A numerical study compares the results for different density generating functions of the error distribution.     

Robust confidence regions for multinomial probabilities

Generally, confidence regions for the probabilities of a multinomial population are constructed based on the Pearson χ² statistic. Morales et al. (Bootstrap confidence regions in multinomial sampling. Appl Math Comput. 2004;155:295–315) considered the bootstrap and asymptotic confidence regions based on a broader family of test statistics known as power-divergence test statistics. In this study, we extend their work and propose penalized power-divergence test statistics-based confidence regions. We only consider small sample sizes where asymptotic properties fail and alternative methods are needed. Both bootstrap and asymptotic confidence regions are constructed. We consider the percentile and the bias corrected and accelerated bootstrap confidence regions. The latter confidence region has not been studied previously for the power-divergence statistics much less for the penalized ones. Designed simulation studies are carried out to calculate average coverage probabilities. Mean absolute deviation between actual and nominal coverage probabilities is used to compare the proposed confidence regions.     

Ellipsoidal confidence regions for a normal covariance matrix

We obtain an asymptotic expansion of the confidence coefficient for an ellipsoidal confidence region on the elements of a normal covariance matrix. This leads to simultaneous confidence intervals on all linear functions of the elements of this matrix, which are compared with those of Roy (1954).     

Exact conditional inference for two uniform populations

We discuss the nature of ancillary information in the context of the continuous uniform distribution. In the one-sample problem, the existence of sufficient statistics mitigates conditioning on the ancillary configuration. In the two-sample problem, additional ancillary information becomes available when the ratio of scale parameters is known. We give exact results for conditional inferences about the common scale parameter and for the difference in location parameters of two uniform distributions. The ancillary information affects the precision of the latter through a comparison of the sample value of the ratio of scale parameters with the known population value. A limited conditional simulation compares the Type I errors and power of these exact results with approximate results using the robust pooled t-statistic.     

Simultaneous confidence bands and regions for log-location-scale distributions with censored data

In many areas of application, especially life testing and reliability, it is often of interest to estimate an unknown cumulative distribution (cdf). A simultaneous confidence band (SCB) of the cdf can be used to assess the statistical uncertainty of the estimated cdf over the entire range of the distribution. Cheng and Iles [1983. Confidence bands for cumulative distribution functions of continuous random variables. Technometrics 25 (1), 77–86] presented an approach to construct an SCB for the cdf of a continuous random variable. For the log-location-scale family of distributions, they gave explicit forms for the upper and lower boundaries of the SCB based on expected information. In this article, we extend the work of Cheng and Iles [1983. Confidence bands for cumulative distribution functions of continuous random variables. Technometrics 25 (1), 77–86] in several directions. We study the SCBs based on local information, expected information, and estimated expected information for both the “cdf method” and the “quantile method.” We also study the effects of exceptional cases where a simple SCB does not exist. We describe calibration of the bands to provide exact coverage for complete data and type II censoring and better approximate coverage for other kinds of censoring. We also discuss how to extend these procedures to regression analysis.     

Studentized bootstrap confidence intervals based on M-estimates

This article reviews and applies saddlepoint approximations to studentized confidence intervals based on robust M-estimates. The latter are known to be very accurate without needing standard theory assumptions. As examples, the classical studentized statistic, the studentized versions of Huber's M-estimate of location, of its initially MAD scaled version and of Huber's proposal 2 are considered. The aim is to know whether the studentized statistics yield robust confidence intervals with coverages close to nominal, with short intervals. The results of an extensive simulation study and the recommendations for practical use given in this article may fill gaps in the current literature and stimulate further discussion and research.     

On multivariate confidence regions and simultaneous confidence limits for ratios

Convenient general linear model computational procedures are presented for constructing multivariate confidence regions and simultaneous confidence limits for ratios of linear combinations of the parameters. The practical consequence is that a single general linear model computer program, capable of validating the underlying model and estimating the parameters, can (after slight modification) also construct the confidence regions, and even determine their precise analytic form (ellipsoid, hyperboloid, etc.). The text is deliberately factual while the appendices extend and help clarify earlier work by Henry Scheffe. As an example, a confidence ellipse and simultaneous confidence limits are constructed for several relative potencies in a classical multiple parallel line bioassay.     

Empirical likelihood confidence regions for comparison distributions and roc curves

Abstract: The authors derive empirical likelihood confidence regions for the comparison distribution of two populations whose distributions are to be tested for equality using random samples. Another application they consider is to ROC curves, which are used to compare measurements of a diagnostic test from two populations. The authors investigate the smoothed empirical likelihood method for estimation in this context, and empirical likelihood based confidence intervals are obtained by means of the Wilks theorem. A bootstrap approach allows for the construction of confidence bands. The method is illustrated with data analysis and a simulation study.     

Exact bootstrap confidence intervals for regression coefficients in small samples

ABSTRACT

Regression analysis is one of the important tools in statistics to investigate the relationships among variables. When the sample size is small, however, the assumptions for regression analysis can be violated. This research focuses on using the exact bootstrap to construct confidence intervals for regression parameters in small samples. The comparison of the exact bootstrap method with the basic bootstrap method was carried out by a simulation study. It was found that on a very small sample (n ≈ 5) under Laplace distribution with the independent variable treated as random, the exact bootstrap was more effective than the standard bootstrap confidence interval.     

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.     

A comparison of two exact confidence regions for partially nonlinear regression models

Exact confidence regions for all the parameters in nonlinear regression models can be obtained by comparing the lengths of projections of the error vector into orthogonal subspaces of the sample space. In certain partially nonlinear models an alternative exact region is obtained by replacing the linear parameters by their conditional estimates in the projection matrices. An ellipsoidal approximation to the alternative region is obtained in terms of the tangent-plane coordinates, similar to one previously obtained for the more usual region. This ellipsoid can be converted to an approximate region for the original parameters and can be used to compare the two types of exact confidence regions.     

Exact confidence bounds for an exponential parameter under hybrid censoring

Consider a life testing experiment in which n units are put on test, successive failure times are recorded, and the observation is terminated either at a specified number r of failures or a specified time T whichever is reached first. This mixture of type I and type II censoring schemes, called hybrid censoring, is of wide use. Under this censoring scheme and the assumption of an exponential life distribution, the distribution of the maximum likelihood estimator of the mean life θ is derived. It is then used to construct an exact lower confidence bound for θ.     

Exact confidence bounds for an exponential parameter under hybrid censoring

Consider a life testing experiment in which n units are put on test, successive failure times are recorded, and the observation is terminated either at a specified number r of failures or a specified time T whichever is reached first. This mixture of type I and type II censoring schemes, called hybrid censoring, is of wide use. Under this censoring scheme and the assumption of an exponential life distribution, the distribution of the maximum likelihood estimator of the mean life 6 is derived. It is then used to construct an exact lower confidence bound for θ.     

Empirical likelihood confidence regions in the single-index model with growing dimensions

This paper investigates statistical inference for the single-index model when the number of predictors grows with sample size. Empirical likelihood method for constructing confidence region for the index vector, which does not require a multivariate non parametric smoothing, is employed. However, the classical empirical likelihood ratio for this model does not remain valid because plug-in estimation of an infinite-dimensional nuisance parameter causes a non negligible bias and the diverging number of parameters/predictors makes the limit not chi-squared any more. To solve these problems, we define an empirical likelihood ratio based on newly proposed weighted estimating equations and show that it is asymptotically normal. Also we find that different weights used in the weighted residuals require, for asymptotic normality, different diverging rate of the number of predictors. However, the rate n^1/3, which is a possible fastest rate when there are no any other conditions assumed in the setting under study, is still attainable. A simulation study is carried out to assess the performance of our method.     

Joint confidence regions in the multivariate calibration problem

Consider a vector valued response variable related to a vector valued explanatory variable through a normal multivariate linear model. The multivariate calibration problem deals with statistical inference on unknown values of the explanatory variable. The problem addressed is the construction of joint confidence regions for several unknown values of the explanatory variable. The problem is investigated when the variance covariance matrix is a scalar multiple of the identity matrix and also when it is a completely unknown positive definite matrix. The problem is solved in only two cases: (i) the response and explanatory variables have the same dimensions, and (ii) the explanatory variable is a scalar. In the former case, exact joint confidence regions are derived based on a natural pivot statistic. In the latter case, the joint confidence regions are only conservative. Computational aspects and the practical implementation of the confidence regions are discussed and illustrated using an example.