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Register a new userConstruction of Confidence Intervals
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Christoph Dalitz
Publication date
2018
fields
Mathematical Statistics
and research's language is
English
Authors
Christoph Dalitz
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Introductory texts on statistics typically only cover the classical two sigma confidence interval for the mean value and do not describe methods to obtain confidence intervals for other estimators. The present technical report fills this gap by first defining different methods for the construction of confidence intervals, and then by their application to a binomial proportion, the mean value, and to arbitrary estimators. Beside the frequentist approach, the likelihood ratio and the highest posterior density approach are explained. Two methods to estimate the variance of general maximum likelihood estimators are described (Hessian, Jackknife), and for arbitrary estimators the bootstrap is suggested. For three examples, the different methods are evaluated by means of Monte Carlo simulations with respect to their coverage probability and interval length. R code is given for all methods, and the practitioner obtains a guideline which method should be used in which cases.
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We consider a linear regression model with regression parameter beta=(beta_1,...,beta_p) and independent and identically N(0,sigma^2) distributed errors. Suppose that the parameter of interest is theta = a^T beta where a is a specified vector. Define the parameter tau=c^T beta-t where the vector c and the number t are specified and a and c are linearly independent. Also suppose that we have uncertain prior information that tau = 0. We present a new frequentist 1-alpha confidence interval for theta that utilizes this prior information. We require this confidence interval to (a) have endpoints that are continuous functions of the data and (b) coincide with the standard 1-alpha confidence interval when the data strongly contradicts this prior information. This interval is optimal in the sense that it has minimum weighted average expected length where the largest weight is given to this expected length when tau=0. This minimization leads to an interval that has the following desirable properties. This interval has expected length that (a) is relatively small when the prior information about tau is correct and (b) has a maximum value that is not too large. The following problem will be used to illustrate the application of this new confidence interval. Consider a 2-by 2 factorial experiment with 20 replicates. Suppose that the parameter of interest theta is a specified simple effect and that we have uncertain prior information that the two-factor interaction is zero. Our aim is to find a frequentist 0.95 confidence interval for theta that utilizes this prior information.
Bootstrap smoothed (bagged) parameter estimators have been proposed as an improvement on estimators found after preliminary data-based model selection. The key result of Efron (2014) is a very convenient and widely applicable formula for a delta method approximation to the standard deviation of the bootstrap smoothed estimator. This approximation provides an easily computed guide to the accuracy of this estimator. In addition, Efron (2014) proposed a confidence interval centered on the bootstrap smoothed estimator, with width proportional to the estimate of this approximation to the standard deviation. We evaluate this confidence interval in the scenario of two nested linear regression models, the full model and a simpler model, and a preliminary test of the null hypothesis that the simpler model is correct. We derive computationally convenient expressions for the ideal bootstrap smoothed estimator and the coverage probability and expected length of this confidence interval. In terms of coverage probability, this confidence interval outperforms the post-model-selection confidence interval with the same nominal coverage and based on the same preliminary test. We also compare the performance of confidence interval centered on the bootstrap smoothed estimator, in terms of expected length, to the usual confidence interval, with the same minimum coverage probablility, based on the full model.
Bootstrap smoothed (bagged) estimators have been proposed as an improvement on estimators found after preliminary data-based model selection. Efron, 2014, derived a widely applicable formula for a delta method approximation to the standard deviation of the bootstrap smoothed estimator. He also considered a confidence interval centered on the bootstrap smoothed estimator, with width proportional to the estimate of this standard deviation. Kabaila and Wijethunga, 2019, assessed the performance of this confidence interval in the scenario of two nested linear regression models, the full model and a simpler model, for the case of known error variance and preliminary model selection using a hypothesis test. They found that the performance of this confidence interval was not substantially better than the usual confidence interval based on the full model, with the same minimum coverage. We extend this assessment to the case of unknown error variance by deriving a computationally convenient exact formula for the ideal (i.e. in the limit as the number of bootstrap replications diverges to infinity) delta method approximation to the standard deviation of the bootstrap smoothed estimator. Our results show that, unlike the known error variance case, there are circumstances in which this confidence interval has attractive properties.
We present an efficient method of calculating exact confidence intervals for the hypergeometric parameter. The method inverts minimum-width acceptance intervals after shifting them to make their endpoints nondecreasing while preserving their level. The resulting set of confidence intervals achieves minimum possible average width, and even in comparison with confidence sets not required to be intervals it attains the minimum possible cardinality most of the time, and always within 1. The method compares favorably with existing methods not only in the size of the intervals but also in the time required to compute them. The available R package hyperMCI implements the proposed method.
Although parametric empirical Bayes confidence intervals of multiple normal means are fundamental tools for compound decision problems, their performance can be sensitive to the misspecification of the parametric prior distribution (typically normal distribution), especially when some strong signals are included. We suggest a simple modification of the standard confidence intervals such that the proposed interval is robust against misspecification of the prior distribution. Our main idea is using well-known Tweedies formula with robust likelihood based on $gamma$-divergence. An advantage of the new interval is that the interval lengths are always smaller than or equal to those of the parametric empirical Bayes confidence interval so that the new interval is efficient and robust. We prove asymptotic validity that the coverage probability of the proposed confidence intervals attain a nominal level even when the true underlying distribution of signals is contaminated, and the coverage accuracy is less sensitive to the contamination ratio. The numerical performance of the proposed method is demonstrated through simulation experiments and a real data application.
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