Simultaneous tolerance intervals in the random one-way model with covariates

Simultaneous tolerance intervals developed by Limam and Thomas (19881, for the normal regression model, are generalized to the random one-way model with covariates. Simultaneous tolerance intervals for unit means are developed for the balanced model. A simulation study is used to estimate the exact confidence of the tolerance intervals for models with one covariate.     

The monotone boundary property and the full coverage property of confidence intervals for a binomial proportion

The methodology for deriving the exact confidence coefficient of some confidence intervals for a binomial proportion is proposed in Wang [2007. Exact confidence coefficients of confidence intervals for a binomial proportion. Statist. Sinica 17, 361–368]. The methodology requires two conditions of confidence intervals: the monotone boundary property and the full coverage property. In this paper, we show that for some confidence intervals of a binomial proportion, the two properties hold for any sample size. Based on results presented in this paper, the procedure in Wang [2007. Exact confidence coefficients of confidence intervals for a binomial proportion. Statist. Sinica 17, 361–368] can be directly used to calculate the exact confidence coefficients of these confidence intervals for any fixed sample size.     

Small-sample intervals for regression

For the general linear regression model Y = Xη + e, we construct small-sample exponentially tilted empirical confidence intervals for a linear parameter 6 = a^Tη and for nonlinear functions of η. The coverage error for the intervals is O_p(1/n), as shown in Tingley and Field (1990). The technique, though sample-based, does not require bootstrap resampling. The first step is calculation of an estimate for η. We have used a Mallows estimate. The algorithm applies whenever η is estimated as the solution of a system of equations having expected value 0. We include calculations of the relative efficiency of the estimator (compared with the classical least-squares estimate). The intervals are compared with asymptotic intervals as found, for example, in Hampel et at. (1986). We demonstrate that the procedure gives sensible intervals for small samples.     

Exact confidence limits for the probability of response in two-stage designs

In addition to point estimate for the probability of response in a two-stage design (e.g. Simon's two-stage design for binary endpoints), confidence limits should be computed and reported. The current method of inverting the p-value function to compute the confidence interval does not guarantee coverage probability in a two-stage setting. The existing exact approach to calculate one-sided limits is based on the overall number of responses to order the sample space. This approach could be conservative because many sample points have the same limits. We propose a new exact one-sided interval based on p-value for the sample space ordering. Exact intervals are computed by using binomial distributions directly, instead of a normal approximation. Both exact intervals preserve the nominal confidence level. The proposed exact interval based on the p-value generally performs better than the other exact interval with regard to expected length and simple average length of confidence intervals.     

Uncertainty Quantification in the Ensemble Kalman Filter

The ensemble Kalman filter (EnKF) provides an approximate, sequential Monte Carlo solution to the recursive data assimilation algorithm for hidden Markov chains. The challenging conditioning step is approximated by a linear updating, and the updating weights, termed Kalman weights, are inferred from the ensemble members. The EnKF scheme is known to provide unstable predictions and to underestimate the prediction intervals, and even sometimes to diverge. The underlying cause for these shortcomings is poorly understood. We find that the ensemble members couple in the conditioning procedure and that the coupling increase multiplicatively in the recursive conditioning steps. Under reasonable Gauss‐independence assumptions, exact expressions for this correlation are developed. Moreover, expressions for the precision of the predictions and the downward bias in the empirical variance introduced in one conditioning step are found. These results are confirmed by a Gauss‐linear simulation study. Furthermore, we quantitatively evaluate an alternative, improved EnKF scheme on the basis of transformations of ensemble members under the same Gauss‐independent assumptions. The scheme is compared with the frequently used ensemble inflation scheme.     

Fast Covariance Estimation for Innovations Computed from a Spatial Gibbs Point Process

In this paper, we derive an exact formula for the covariance of two innovations computed from a spatial Gibbs point process and suggest a fast method for estimating this covariance. We show how this methodology can be used to estimate the asymptotic covariance matrix of the maximum pseudo‐likelihood estimator of the parameters of a spatial Gibbs point process model. This allows us to construct asymptotic confidence intervals for the parameters. We illustrate the efficiency of our procedure in a simulation study for several classical parametric models. The procedure is implemented in the statistical software R , and it is included in spatstat , which is an R package for analyzing spatial point patterns.     

Predicting a Future Lifetime through Box-Cox Transformation

In predicting a future lifetime based on a sample of past lifetimes, the Box-Cox transformation method provides a simple and unified procedure that is shown in this article to meet or often outperform the corresponding frequentist solution in terms of coverage probability and average length of prediction intervals. Kullback-Leibler information and second-order asymptotic expansion are used to justify the Box-Cox procedure. Extensive Monte Carlo simulations are also performed to evaluate the small sample behavior of the procedure. Certain popular lifetime distributions, such as Weibull, inverse Gaussian and Birnbaum-Saunders are served as illustrative examples. One important advantage of the Box-Cox procedure lies in its easy extension to linear model predictions where the exact frequentist solutions are often not available.     

A nonparametric confidence interval for slope based on spearman's rho

An algorithm is presented for computing an exact nonparametric interval estimate of the slope parameter in a simple linear regression model. The confidence interval is obtained by inverting the hypothesis test for slope that uses Spearman's rho. This method is compared to an exact procedure based on Kendall's tau. The Spearman rho procedure will generally give exact levels of confidence closer to desired levels, especially in small samples. Monte carlo results comparing these two methods with the parametric procedure are given     

Confidence intervals for the difference in paired Youden indices

Comparison of accuracy between two diagnostic tests can be implemented by investigating the difference in paired Youden indices. However, few literature articles have discussed the inferences for the difference in paired Youden indices. In this paper, we propose an exact confidence interval for the difference in paired Youden indices based on the generalized pivotal quantities. For comparison, the maximum likelihood estimate‐based interval and a bootstrap‐based interval are also included in the study for the difference in paired Youden indices. Abundant simulation studies are conducted to compare the relative performance of these intervals by evaluating the coverage probability and average interval length. Our simulation results demonstrate that the exact confidence interval outperforms the other two intervals even with small sample size when the underlying distributions are normal. A real application is also used to illustrate the proposed intervals. Copyright © 2012 John Wiley & Sons, Ltd.     

Exact confidence intervals generated by conditional parametric bootstrapping

Conditional parametric bootstrapping is defined as the samples obtained by performing the simulations in such a way that the estimator is kept constant and equal to the estimate obtained from the data. Order statistics of the bootstrap replicates of the parameter chosen in each simulation provide exact confidence intervals, in a probabilistic sense, in models with one parameter under quite general conditions. The method is still exact in the case of nuisance parameters when these are location and scale parameters, and the bootstrapping is based on keeping the maximum-likelihood estimates constant. The method is also exact if there exists a sufficient statistic for the nuisance parameters and if the simulations are performed conditioning on this statistic. The technique may also be used to construct prediction intervals. These are generally not exact, but are likely to be good approximations.     

Prediction intervals for the random intercept linear model

Our main purpose is to give a method of constructing prediction intervals for the mean of us future observations in the random intercept linear model, A method is also indicated for constructing exact confidence intervals on the expected value of a future observation.     

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.     

Analytic computation of nonparametric Marsan–Lengliné estimates for Hawkes point processes

In 2008, Marsan and Lengliné presented a nonparametric way to estimate the triggering function of a Hawkes process. Their method requires an iterative and computationally intensive procedure which ultimately produces only approximate maximum likelihood estimates (MLEs) whose asymptotic properties are poorly understood. Here, we note a mathematical curiosity that allows one to compute, directly and extremely rapidly, exact MLEs of the nonparametric triggering function. The method here requires that the number q of intervals on which the nonparametric estimate is sought equals the number n of observed points. The resulting estimates have very high variance but may be smoothed to form more stable estimates. The performance and computational efficiency of the proposed method is verified in two disparate, highly challenging simulation scenarios: first to estimate the triggering functions, with simulation-based 95% confidence bands, for earthquakes and their aftershocks in Loma Prieta, California, and second, to characterise triggering in confirmed cases of plague in the United States over the last century. In both cases, the proposed estimator can be used to describe the rate of contagion of the processes in detail, and the computational efficiency of the estimator facilitates the construction of simulation-based confidence intervals.     

Linear calibra and conditional inference

Approximate conditional inference is developed for the linear calibration problem. It is shown that this problem can be transformed so that the primary parameter is an angle, the nuisance parameter is a radial distance, and the density is rotationally symmetric. Were the nuisance parameter known, exact location confidence intervals would be available by location of structural arguments. A confidence distribution is used to average out the nuisance parameter yielding an approximate confidence interval that involves a precision indicator derived from the radial distance. Some difficulties with the ordinary solution are avoided by the conditional procedure.     

Nearly exact tests in factorial experiments using the aligned rank transform

A procedure is studied that uses rank-transformed data to perform exact and estimated exact tests, which is an alternative to the commonly used F-ratio test procedure. First, a common parametric test statistic is computed using rank-transformed data, where two methods of ranking-ranks taken for the original observations and ranks taken after aligning the observations-are studied. Significance is then determined using either the exact permutation distribution of the statistic or an estimate of this distribution based on a random sample of all possible permutations. Simulation studies compare the performance of this method with the normal theory parametric F-test and the traditional rank transform procedure. Power and nominal type I error rates are compared under conditions when normal theory assumptions are satisfied, as well as when these assumptions are violated. The method is studied for a two-factor factorial arrangement of treatments in a completely randomized design and for a split-unit experiment. The power of the tests rivals the parametric F-test when normal theory assumptions are satisfied, and is usually superior when normal theory assumptions are not satisfied. Based on the evidence of this study, the exact aligned rank procedure appears to be the overall best choice for performing tests in a general factorial experiment.     

Confidence intervals for the regression coefficient in a simple regression model with a balanced two-fold nested error structure

ABSTRACT

In applications using a simple regression model with a balanced two-fold nested error structure, interest focuses on inferences concerning the regression coefficient. This article derives exact and approximate confidence intervals on the regression coefficient in the simple regression model with a balanced two-fold nested error structure. Eleven methods are considered for constructing the confidence intervals on the regression coefficient. Computer simulation is performed to compare the proposed confidence intervals. Recommendations are suggested for selecting an appropriate method.     

DISTRIBUTION FREE MULTIPLE COMPARISONS FOR MONOTONICALLY ORDERED TREATMENT EFFECTS

A method of calculating simultaneous one-sided confidence intervals for all ordered pairwise differences of the treatment effects_j–_i, 1 i < j k, in a one-way model without any distributional assumptions is discussed. When it is known a priori that the treatment effects satisfy the simple ordering_1k, these simultaneous confidence intervals offer the experimenter a simple way of determining which treatment effects may be declared to be unequal, and is more powerful than the usual two-sided Steel-Dwass procedure. Some exact critical points required by the confidence intervals are presented for k= 3 and small sample sizes, and other methods of critical point determination such as asymptotic approximation and simulation are discussed.     

Exact Confidence Intervals,Based on the Z Statistic,for the Difference Between Two Proportions

We show that the confidence interval version of the extended exact unconditional Z test of Suissa and Shuster (1985) for testing the equality of two binomial proportions is due to general results of Buehler (1957), Sudakov and references cited there (1974), and Harris and Soms (1984). We apply these results to obtain exact unconditional confidence intervals for the difference between two proportions, deriving an explicit solution for the “best” outcome, make some comments on Buehler's (1957) method and give a numerical example. The Appendix contains a listing of the necessary FORTRAN programs.     

CONDITIONAL INFERENCE PROCEDURES FOR THE PARETO AND POWER FUNCTION DISTRIBUTIONS BASED ON TYPE-II RIGHT CENSORED SAMPLES

In this paper we consider conditional inference procedures for the Pareto and power function distributions. We develop procedures for obtaining confidence intervals for the location and scale parameters as well as upper and lower n probability tolerance intervals for a proportion g, given a Type-II right censored sample from the corresponding distribution. The intervals are exact, and are obtained by conditioning on the observed values of the ancillary statistics. Since, for each distribution, the procedures assume that a shape parameter x is known, a sensitivity analysis is also carried out to see how the procedures are affected by changes in x.     

Small Sample Confidence Intervals for the Odds Ratio

Abstract

Numerous methods—based on exact and asymptotic distributions—can be used to obtain confidence intervals for the odds ratio in 2 × 2 tables. We examine ten methods for generating these intervals based on coverage probability, closeness of coverage probability to target, and length of confidence intervals. Based on these criteria, Cornfield’s method, without the continuity correction, performed the best of the methods examined here. A drawback to use of this method is the significant possibility that the attained coverage probability will not meet the nominal confidence level. Use of a mid-P value greatly improves methods based on the “exact” distribution. When combined with the Wilson rule for selection of a rejection set, the resulting method is a procedure that performed very well. Crow’s method, with use of a mid-P, performed well, although it was only a slight improvement over the Wilson mid-P method. Its cumbersome calculations preclude its general acceptance. Woolf's (logit) method—with the Haldane–Anscombe correction— performed well, especially with regard to length of confidence intervals, and is recommended based on ease of computation.