Exact confidence regions for species assignment based on DNA markers

Assignment of individuals to correct species or population of origin based on a comparison of allele profiles has in recent years become more accurate due to improvements in DNA marker technology. A method of assessing the error in such assignment problems is présentés. The method is based on the exact hypergeometric distributions of contingency tables conditioned on marginal totals. The result is a confidence region of fixed confidence level. This confidence level is calculable exactly in principle, and estimable very accurately by simulation, without knowledge of the true population allele frequencies. Various properties of these techniques are examined through application to several examples of actual DNA marker data and through simulation studies. Methods which may reduce computation time are discussed and illustrated.     

Confidence curves and improved exact confidence intervals for discrete distributions

The author describes a method for improving standard “exact” confidence intervals in discrete distributions with respect to size while retaining correct level. The binomial, negative binomial, hypergeometric, and Poisson distributions are considered explicitly. Contrary to other existing methods, the author's solution possesses a natural nesting condition: if α < α', the 1 ‐ α' confidence interval is included in the 1 ‐ α interval. Nonparametric confidence intervals for a quantile are also considered.     

Saddlepoint-Based Bootstrap Inference for Quadratic Estimating Equations

Abstract.  We propose an easy to implement method for making small sample parametric inference about the root of an estimating equation expressible as a quadratic form in normal random variables. It is based on saddlepoint approximations to the distribution of the estimating equation whose unique root is a parameter's maximum likelihood estimator (MLE), while substituting conditional MLEs for the remaining (nuisance) parameters. Monotoncity of the estimating equation in its parameter argument enables us to relate these approximations to those for the estimator of interest. The proposed method is equivalent to a parametric bootstrap percentile approach where Monte Carlo simulation is replaced by saddlepoint approximation. It finds applications in many areas of statistics including, nonlinear regression, time series analysis, inference on ratios of regression parameters in linear models and calibration. We demonstrate the method in the context of some classical examples from nonlinear regression models and ratios of regression parameter problems. Simulation results for these show that the proposed method, apart from being generally easier to implement, yields confidence intervals with lengths and coverage probabilities that compare favourably with those obtained from several competing methods proposed in the literature over the past half-century.     

Exact average coverage probabilities and confidence coefficients of confidence intervals for discrete distributions

For a confidence interval (L(X),U(X)) of a parameter θ in one-parameter discrete distributions, the coverage probability is a variable function of θ. The confidence coefficient is the infimum of the coverage probabilities, inf  _θ P _θ (θ∈(L(X),U(X))). Since we do not know which point in the parameter space the infimum coverage probability occurs at, the exact confidence coefficients are unknown. Beside confidence coefficients, evaluation of a confidence intervals can be based on the average coverage probability. Usually, the exact average probability is also unknown and it was approximated by taking the mean of the coverage probabilities at some randomly chosen points in the parameter space. In this article, methodologies for computing the exact average coverage probabilities as well as the exact confidence coefficients of confidence intervals for one-parameter discrete distributions are proposed. With these methodologies, both exact values can be derived.     

On construction of asymptotically correct confidence intervals

In this paper we discuss constructing confidence intervals based on asymptotic generalized pivotal quantities (AGPQs). An AGPQ associates a distribution with the corresponding parameter, and then an asymptotically correct confidence interval can be derived directly from this distribution like Bayesian or fiducial interval estimates. We provide two general procedures for constructing AGPQs. We also present several examples to show that AGPQs can yield new confidence intervals with better finite-sample behaviors than traditional methods.     

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.     

Confidence intervals for secondary parameters following a sequential test

In sequential studies, formal interim analyses are usually restricted to a consideration of a single null hypothesis concerning a single parameter of interest. Valid frequentist methods of hypothesis testing and of point and interval estimation for the primary parameter have already been devised for use at the end of such a study. However, the completed data set may warrant a more detailed analysis, involving the estimation of parameters corresponding to effects that were not used to determine when to stop, and yet correlated with those that were. This paper describes methods for setting confidence intervals for secondary parameters in a way which provides the correct coverage probability in repeated frequentist realizations of the sequential design used. The method assumes that information accumulates on the primary and secondary parameters at proportional rates. This requirement will be valid in many potential applications, but only in limited situations in survival analysis.     

Profile upper Confidence Limits from Discrete Data

When the data are discrete, standard approximate confidence limits often have coverage well below nominal for some parameter values. While ad hoc adjustments may largely solve this problem for particular cases, Kabaila & Lloyd (1997) gave a more systematic method of adjustment which leads to tight upper limits, which have coverage which is never below nominal and are as small as possible within a particular class. However, their computation for all but the simplest models is infeasible. This paper suggests modifying tight upper limits by an initial replacement of the unknown nuisance parameter vector by its profile maximum likelihood estimator. While the resulting limits no longer possess the optimal properties of tight limits exactly, the paper presents both numerical and theoretical evidence that the resulting coverage function is close to optimal. Moreover these profile upper limits are much (possibly many orders of magnitude) easier to compute than tight upper limits.