Computation of exact confidence intervals from discrete data using studentized test statistics

A new area of research interest is the computation of exact confidence limits or intervals for a scalar parameter of interest from discrete data by inverting a hypothesis test based on a studentized test statistic. See, for example, Chan and Zhang (1999), Agresti and Min (2001) and Agresti (2003) who deal with a difference of binomial probabilities and Agresti and Min (2002) who deal with an odds ratio. However, neither (1) a detailed analysis of the computational issues involved nor (2) a reliable method of computation that deals effectively with these issues is currently available. In this paper we solve these two problems for a very broad class of discrete data models. We suppose that the distribution of the data is determined by (,) where is a nuisance parameter vector. We also consider six different studentized test statistics. Our contributions to (1) are as follows. We show that the P-value resulting from the hypothesis test, considered as a function of the null-hypothesized value of , has both jump and drop discontinuities. Numerical examples are used to demonstrate that these discontinuities lead to the failure of simple-minded approaches to the computation of the confidence limit or interval. We also provide a new method for efficiently computing the set of all possible locations of these discontinuities. Our contribution to (2) is to provide a new and reliable method of computing the confidence limit or interval, based on the knowledge of this set.     

Importance of interpolation when constructing double-bootstrap confidence intervals

We show that, in the context of double-bootstrap confidence intervals, linear interpolation at the second level of the double bootstrap can reduce the simulation error component of coverage error by an order of magnitude. Intervals that are indistinguishable in terms of coverage error with theoretical, infinite simulation, double-bootstrap confidence intervals may be obtained at substantially less computational expense than by using the standard Monte Carlo approximation method. The intervals retain the simplicity of uniform bootstrap sampling and require no special analysis or computational techniques. Interpolation at the first level of the double bootstrap is shown to have a relatively minor effect on the simulation error.     

Tests and confidence intervals for the shape parameter of a gamma distribution

Inference based on the Central Limit Theorem has only first order accuracy. We give tests and confidence intervals (CIs) of second orderaccuracy for the shape parameter ρ of a gamma distribution for both the unscaled and scaled cases.

Tests and CIs based on moment and cumulant estimates are considered as well as those based on the maximum likelihood estimate (MLE).

For the unscaled case the MLE is the moment estimate of order zero; the most efficient moment estimate of integral order is the sample mean, having asymptotic relative efficiency (ARE) .61 when ρ= 1.

For the scaled case the most efficient moment estimate is a functionof the mean and variance. Its ARE is .39 when ρ = 1.

Our motivation for constructing these tests of ρ = 1 and CIs forρ is to provide a simple and convenient method for testing whether a distribution is exponential in situations such as rainfall models where such an assumption is commonly made.     

Randomized confidence intervals of a parameter for a family of discrete exponential type distributions

For a family of one-parameter discrete exponential type distributions, the higher order approximation of randomized confidence intervals derived from the optimum test is discussed. Indeed, it is shown that they can be asymptotically constructed by means of the Edgeworth expansion. The usefulness is seen from the numerical results in the case of Poisson and binomial distributions.     

Accurate inference for scale and location families

A great deal of inference in statistics is based on making the approximation that a statistic is normally distributed. The error in doing so is generally O(n^?1/2), where n is the sample size and can be considered when the distribution of the statistic is heavily biased or skewed. This note shows how one may reduce the error to O(n^?(j+1)/2), where j is a given integer. The case considered is when the statistic is the mean of the sample values of a continuous distribution with a scale or location change after the sample has undergone an initial transformation, which may depend on an unknown parameter. The transformation corresponding to Fisher's score function yields an asymptotically efficient procedure.     

Data-randomized confidence intervals for discrete distributions

In practice non-randomized conservative confidence intervals for the parameter of a discrete distribution are used instead of the randomized uniformly most accurate intervals. We suggest in this paper that a part of the data be used as the random mechanism to create “data-randomized” confidence intervals. A thoughtful utilization of the data leads to intervals that are shorter than the usual conservative intervals but avoids the arbitrariness of the randomized uniformly most accurate intervals. Examples are given using the binomial, Poisson, and extended hypergeometric distributions, as well as applications to a metched case-control study and a randomized clinical trial.     

Nonparametric asymptotic confidence intervals for extreme quantiles

In this paper, we propose new asymptotic confidence intervals for extreme quantiles, that is, for quantiles located outside the range of the available data. We restrict ourselves to the situation where the underlying distribution is heavy-tailed. While asymptotic confidence intervals are mostly constructed around a pivotal quantity, we consider here an alternative approach based on the distribution of order statistics sampled from a uniform distribution. The convergence of the coverage probability to the nominal one is established under a classical second-order condition. The finite sample behavior is also examined and our methodology is applied to a real dataset.     

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.     

Adaptive confidence intervals of desired length and power for normal means

In all empirical or experimental sciences, it is a standard approach to present results, additionally to point estimates, in form of confidence intervals on the parameters of interest. The length of a confidence interval characterizes the accuracy of the whole findings. Consequently, confidence intervals should be constructed to hold a desired length. Basic ideas go back to Stein (1945) and Seelbinder (1953) who proposed a two-stage procedure for hypothesis testing about a normal mean. Tukey (1953) additionally considered the probability or power a confidence interval should possess to hold its length within a desired boundary. In this paper, an adaptive multi-stage approach is presented that can be considered as an extension of Stein's concept. Concrete rules for sample size updating are provided. Following an adaptive two-stage design of O’Brien and Fleming (1979) type, a real data example is worked out in detail.     

Corrected p-values for tests based on estimated nuisance parameters

In some situations the asymptotic distribution of a random function T _n() that depends on a nuisance parameter is tractable when has known value. In that case it can be used as a test statistic, if suitably constructed, for some hypothesis. However, in practice, often needs to be replaced by an estimator S _n. In this paper general results are given concerning the asymptotic distribution of T _n(S _n) that include special cases previously dealt with. In particular, some situations are covered where the usual likelihood theory is nonregular and extreme values are employed to construct estimators and test statistics.     

A confidence region with multiple intervals in multivariate bioassay

The exact confidence region for log relative potency resulting from likelihood score methods (Williams (1988) An exact confidence interval for the relative potency estimated from a multivariate bioassay, Biometrics, 44:861-868) will very likely consist of two disjoint confidence intervals. The two methods proposed by Williams which aim to select just one (the same) confidence interval from the confidence region are nearly – but not completely – consistent. The likelihood score interval and likelihood ratio interval are asymptotically equivalent. Williams's very strong claim concerning the confidence coefficient in the second selection method is still theoretically unproved; yet, simulations show that it is true for a wide range of practical experimental situations.     

Unbiased statistical estimation functions for parameters in presence of nuisance parameters

We consider the problem of estimating a vector interesting parameter in the presence of nuisance parameters through vector unbiased statistical estimation functions (USEFs). An extension of the Cramer—Rao inequality relevant to the present problem is obtained. Three possible optimality criteria in the class of regular vector USEFs are those based on (i) the non-negative definiteness of the difference of dispersion matrices (ii) the trace of the dispersion matrix and (iii) the determinant of the dispersion matrix. We refer to these three criteria as M-optimality, T- optimality and D-optimality respectively. The equivalence of these three optimality criteria is established. By restricting the class of regular USEFs considered by Ferreira (1982), we study some interesting properties of the standardized USEFs and establish essential uniqueness of standardized M-optimal USEF in this restricted class. Finally some illustrative examples are included.     

Confidence intervals for the ratio of two variances

In this paper we consider confidence intervals for the ratio of two population variances. We propose a confidence interval for the ratio of two variances based on the t-statistic by deriving its Edgeworth expansion and considering Hall's and Johnson's transformations. Then, we consider the coverage accuracy of suggested intervals and intervals based on the F-statistic for some distributions.     

Robust Bayesian hypothesis testing in the presence of nuisance parameters

Robust Bayesian testing of point null hypotheses is considered for problems involving the presence of nuisance parameters. The robust Bayesian approach seeks answers that hold for a range of prior distributions. Three techniques for handling the nuisance parameter are studied and compared. They are (i) utilize a noninformative prior to integrate out the nuisance parameter; (ii) utilize a test statistic whose distribution does not depend on the nuisance parameter; and (iii) use a class of prior distributions for the nuisance parameter. These approaches are studied in two examples, the univariate normal model with unknown mean and variance, and a multivariate normal example.     

Properties of simultaneous confidence intervals for multinomial proportions

Inversion of Pearson's chi-square statistic yields a confidence ellipsoid that can be used for simultaneous inference concerning multinomial proportions. Because the ellipsoid is difficult to interpret, methods of simultaneous confidence interval construction have been proposed by Quesenberry and hurst,goodman,fitzpatrick and scott and sison and glaz . Based on simulation results, we discuss the performance of these methods in terms of empirical coverage probabilities and enclosed volume. None of the methods is uniformly better than all others, but the Goodman intervals control the empirical coverage probability with smaller volume than other methods when the sample size supports the large sample theory. If the expected cell counts are small and nearly equal across cells, we recommend the sison and glaz intervals.     

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.     

Charting small sample characteristics of asymptotic confidence intervals for the binomial parameterp

Small sample properties of seven confidence intervals for the binomial parameterp (based on various normal approximations) and of the Clopper-Pearson interval are compared. Coverage probabilities and expected lower and upper limits of the intervals are graphically displayed as functions of the binomial parameterp for various sample sizes.     

Construction of confidence intervals for the Laspeyres price index

In the paper, we present and discuss several methods of the construction of confidence intervals for the Laspeyres price index. We assume that prices of commodities are normally distributed and we consider both independent and dependent prices. Using Monte Carlo simulation, the paper compares the confidence interval computed from a simple econometric model with those obtained based on the Laspeyres density function. Our conclusions can be generalized to other price index formulas.