Comparison of Poisson Confidence Intervals

Abstract

The standard method of obtaining a two-sided confidence interval for the Poisson mean produces an interval which is exact but can be shortened without violating the minimum coverage requirement. We classify the intervals proposed as alternatives to the standard method interval. We carry out the classification using two desirable properties of two-sided confidence intervals.     

Confidence Intervals for Difference Between Two Poisson Rates

In this article, we develop four explicit asymptotic two-sided confidence intervals for the difference between two Poisson rates via a hybrid method. The basic idea of the proposed method is to estimate or recover the variances of the two Poisson rate estimates, which are required for constructing the confidence interval for the rate difference, from the confidence limits for the two individual Poisson rates. The basic building blocks of the approach are reliable confidence limits for the two individual Poisson rates. Four confidence interval estimators that have explicit solutions and good coverage levels are employed: the first normal with continuity correction, Rao score, Freeman and Tukey, and Jeffreys confidence intervals. Using simulation studies, we examine the performance of the four hybrid confidence intervals and compare them with three existing confidence intervals: the non-informative prior Bayes confidence interval, the t confidence interval based on Satterthwait's degrees of freedom, and the Bayes confidence interval based on Student's t confidence coefficient. Simulation results show that the proposed hybrid Freeman and Tukey, and the hybrid Jeffreys confidence intervals can be highly recommended because they outperform the others in terms of coverage probabilities and widths. The other methods tend to be too conservative and produce wider confidence intervals. The application of these confidence intervals are illustrated with three real data sets.     

Simultaneous Confidence Intervals for the Ratios of Marginal Means of a Multivariate Poisson Distribution

This article computes simultaneous confidence intervals for the ratios of marginal means of a multivariate Poisson distribution. For this, we propose a lognormal approximation technique and a bootstrap method. We demonstrate advantages of the proposed methods over existing ones through a simulation study. To illustrate their applicability to real-world problems, we apply the proposed methods to US data on infectious diseases.     

A Note on Confidence Intervals for a Linear Function of Poisson Rates

We consider three interval estimators for linear functions of Poisson rates: a Wald interval, a t interval with Satterthwaite's degrees of freedom, and a Bayes interval using noninformative priors. The differences in these intervals are illustrated using data from the Crash Records Bureau of the Texas Department of Public Safety. We then investigate the relative performance of these intervals via a simulation study. This study demonstrates that the Wald interval performs poorly when expected counts are less than 5, while the interval based on the noninformative prior performs best. It also shows that the Bayes interval and the interval based on the t distribution perform comparably well for more moderate expected counts.     

Simultaneous Confidence Intervals for the Mean of Multivariate Poisson Distribution: A Comparison

Haibing (2009) proposed a procedure for successive comparisons between ordered treatment effects in one-way layout and showed that the proposed procedure has greater power than the procedure proposed by Lee and Spurrier (1995). Critical constants required for the proposed procedure were estimated using Monte Carlo simulation and few values of the constants were tabulated which limit the applications of the proposed procedure. In this article, a numerical method, using recursive integration methodology, is discussed to compute the critical constants which work efficiently for a large number of treatments and extensive values of critical constants are tabulated for the use of practitioners. Power comparisons of Haibing's and Lee and Spurrier's procedure is also discussed.     

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.     

Confidence intervals for the correlation from a bivariate normal

Improved confidence intervals are given for the correlation coefficient of the bivariate normal distribution. These are based on Cornish–Fisher expansions for the distribution, density and quantiles of the sample correlation.     

Simultaneous Confidence Intervals for Monotone Contrasts of Normal Means

The one-way ANOVA model is considered. The variances are assumed to be either equal but unknown or unequal and unknown. Two-stage procedures for generating simultaneous confidence intervals (SCI) for the class of monotone contrasts of the means are presented. The intervals are of fixed length and independent of the unknown variances.     

Length minimization for Poisson confidence procedures

We study Poisson confidence procedures that potentially lead to short confidence intervals, investigating the class of all minimal cardinality procedures. We consider how length minimization should be properly defined, and show that Casella and Robert's (1989) criterion for comparing Poisson confidence procedures leads to a contradiction. We provide an alternative criterion for comparing length performance, identify the unique length optimal minimal cardinality procedure by this criterion, and propose a modification that eliminates an important drawback it possesses. We focus on procedures whose coverage never falls below the nominal level and discuss the case in which the nominal level represents mean coverage.     

Matching pseudocounts for interval estimation of binomial and Poisson parameters

ABSTRACT

For interval estimation of a binomial proportion and a Poisson mean, matching pseudocounts are derived, which give the one-sided Wald confidence intervals with second-order accuracy. The confidence intervals remove the bias of coverage probabilities given by the score confidence intervals. Partial poor behavior of the confidence intervals by the matching pseudocounts is corrected by hybrid methods using the score confidence interval depending on sample values.     

Confidence intervals for the comparison of two poisson distributions

Confidence interval construction the difference in mean event rates for two Index independent , Poisson samples is discussed. Intervals are derived by considering Bayes estimates of the mean event rates using a family of noninformative priors. The coverage probabilities of the proposed are compared to those of the standard Wald interval for of observed events. A compromise method of constructing interval based on the data is suggested and its properties are evaluated. The method is illustrated in several examples.     

On Least Absolute Value Estimators for Poisson Processes

We consider the problem of parameter estimation for an inhomogeneous Poisson process observed on the time interval [0, τ]. We introduce the minimum L ₁-norm estimator of the unknown parameter and study the asymptotical behaviors of the estimates when the number of observations increases. It is established that this estimator is consistent and we show that the corresponding differences converge to certain variables. These limit variables are asymptotically normal as τ tends to infinity.     

Construction of Simultaneous Confidence Intervals for Ratios of Means of Lognormal Distributions

For constructing simultaneous confidence intervals for ratios of means for lognormal distributions, two approaches using a two-step method of variance estimates recovery are proposed. The first approach proposes fiducial generalized confidence intervals (FGCIs) in the first step followed by the method of variance estimates recovery (MOVER) in the second step (FGCIs–MOVER). The second approach uses MOVER in the first and second steps (MOVER–MOVER). Performance of proposed approaches is compared with simultaneous fiducial generalized confidence intervals (SFGCIs). Monte Carlo simulation is used to evaluate the performance of these approaches in terms of coverage probability, average interval width, and time consumption.     

Confidence Intervals for Nonparametric Regression Functions with Missing Data

Suppose that we have a nonparametric regression model Y = m(X) + ε with X ∈ R^p, where X is a random design variable and is observed completely, and Y is the response variable and some Y-values are missing at random. Based on the “complete” data sets for Y after nonaprametric regression imputation and inverse probability weighted imputation, two estimators of the regression function m(x₀) for fixed x₀ ∈ R^p are proposed. Asymptotic normality of two estimators is established, which is used to construct normal approximation-based confidence intervals for m(x₀). We also construct an empirical likelihood (EL) statistic for m(x₀) with limiting distribution of χ²₁, which is used to construct an EL confidence interval for m(x₀).     

Confidence Intervals for the Scale Parameter of the Power-Law Process

The power-law process (PLP) is a two-parameter model widely used for modeling repairable system reliability. Results on exact point estimation for both parameters as well as exact interval estimation for the shape parameter are well known. In this paper, we investigate the interval estimation for the scale parameter. Asymptotic confidence intervals are derived using Fisher information matrix and theoretical results by Cocozza-Thivent (1997 Cocozza-Thivent , C. ( 1997 ). Processus Stochastiques et Fiabilité des Systèmes . Berlin : Springer-Verlag . [Google Scholar]). The accuracy of the interval estimation for finite samples is studied by simulation methods.     

Simultaneous Confidence Bands for Linear Regression with Covariates Constrained in Intervals

Abstract. The focus of this article is on simultaneous confidence bands over a rectangular covariate region for a linear regression model with k>1 covariates, for which only conservative or approximate confidence bands are available in the statistical literature stretching back to Working & Hotelling (J. Amer. Statist. Assoc. 24 , 1929; 73–85). Formulas of simultaneous confidence levels of the hyperbolic and constant width bands are provided. These involve only a k‐dimensional integral; it is unlikely that the simultaneous confidence levels can be expressed as an integral of less than k‐dimension. These formulas allow the construction for the first time of exact hyperbolic and constant width confidence bands for at least a small k(>1) by using numerical quadrature. Comparison between the hyperbolic and constant width bands is then addressed under both the average width and minimum volume confidence set criteria. It is observed that the constant width band can be drastically less efficient than the hyperbolic band when k>1. Finally it is pointed out how the methods given in this article can be applied to more general regression models such as fixed‐effect or random‐effect generalized linear regression models.     

Simulation assessments of statistical aspects of bioequivalence in the pharmaceutical industry

An Erratum has been published for this article in Pharmaceutical Statistics 2004; 3(3): 232 Since the early 1990s, average bioequivalence (ABE) has served as the international standard for demonstrating that two formulations of drug product will provide the same therapeutic benefit and safety profile. Population (PBE) and individual (IBE) bioequivalence have been the subject of intense international debate since methods for their assessment were proposed in the late 1980s. Guidance has been proposed by the Food and Drug Administration (FDA) for the implementation of these techniques in the pioneer and generic pharmaceutical industries. Hitherto no consensus among regulators, academia and industry has been established on the use of the IBE and PBE metrics. The need for more stringent bioequivalence criteria has not been demonstrated, and it is known that the PBE and IBE criteria proposed by the FDA are actually less stringent under certain conditions. The statistical properties of method of moments and restricted maximum likelihood modelling in replicate designs will be summarized, and the application of these techniques in the assessment of ABE, IBE and PBE will be considered based on a database of 51 replicate design studies and using simulation. Copyright © 2004 John Wiley & Sons, Ltd.     

Objective Bayesian Testing of a Poisson Mean

We consider testing hypotheses about a single Poisson mean. When prior information is not available, use of objective priors is of interest. We provide intrinsic priors based on the arithmetic intrinsic and fractional Bayes factors, and evaluate their characteristics.     

Hypotheses testing for a multidimensional parameter of inhomogeneous Poisson processes

We observe a realization of an inhomogeneous Poisson process whose intensity function depends on an unknown multidimensional parameter. We consider the asymptotic behaviour of the Rao score test for a simple null hypothesis against the multilateral alternative. By using the Edgeworth type expansion (under the null hypothesis) for a vector of stochastic integrals with respect to the Poisson process, we refine the (classic) threshold of the test (obtained by the central limit theorem), which improves the first type probability of error. The expansion allows us to describe the power of the test under the local alternative, i.e. a sequence of alternatives, which converge to the null hypothesis with a certain rate. The rates can be different for components of the parameter.