Usual Confidence Intervals Research Articles

Confidence intervals in general regression models that utilize uncertain prior information

– We consider a general regression model, either without a scale parameter or with a known scale parameter. We construct a confidence interval for a scalar parameter of interest that utilizes the uncertain prior information that a distinct scalar parameter takes a specified value. This confidence interval has excellent coverage properties. It also has local scaled expected length, where the scaling is with respect to the usual confidence interval, that (a) is substantially less than 1 when the prior information is correct, (b) has a maximum value that is not too large, and (c) is close to 1 when the data and prior information are highly discordant. An important practical application of this confidence interval is to a quantal bioassay carried out to compare two similar compounds. In this context, the uncertain prior information is that the hypothesis of “parallelism” holds. We provide extensive numerical results that illustrate the attractive properties of this confidence interval in this context.

Confidence intervals that utilize sparsity

We consider a linear regression model with orthogonal regressors and Gaussian random errors with known variance, in the low‐dimensional setting that the length of the regression parameter vector does not exceed the length of the response vector. We suppose that we have uncertain prior information that sparsity holds, that is, that many of the components of the regression parameter vector are zero. Our aim is to construct confidence intervals for the components of this vector with both the desired coverage probability and attractive expected length properties, particularly when sparsity holds. It is known that for fixed‐width confidence intervals centered on penalized maximum likelihood estimators, such as hard‐thresholding, LASSO and SCAD estimators, the achievement of the desired minimum coverage necessarily results in very unattractive expected length properties. We present confidence intervals, with data‐based widths, which achieve our aim. These confidence intervals dominate the usual confidence intervals with the same coverage probability, whenever the degree of sparsity exceeds a known value.

The significance filter, the winner's curse and the need to shrink

The “significance filter” refers to focusing exclusively on statistically significant results. Since frequentist properties such as unbiasedness and coverage are valid only before the data have been observed, there are no guarantees if we condition on significance. In fact, the significance filter leads to overestimation of the magnitude of the parameter, which has been called the “winner's curse.” It can also lead to undercoverage of the confidence interval. Moreover, these problems become more severe if the power is low. These issues clearly deserve our attention. They have been studied mostly through empirical observation and simulation, while there are relatively few mathematical results. Here we study them both from the frequentist and the Bayesian perspective. We prove that the relative bias of the magnitude is a decreasing function of the power and that the usual confidence interval undercovers when the power is less than 50%. We conclude that it is important to apply the appropriate amount of shrinkage to counter the winner's curse.

CONFIDENCE INTERVALS CONSTRUCTED BY MODEL AVERAGING AND BOOTSTRAP SMOOTHING

We assess the performance of confidence intervals constructed by model averaging and bootstrap smoothing, using a simple testbed situation of two nested linear regression models with either known error variance or unknown error variance. The assessment is carried out by deriving computationally convenient exact expressions for the coverage probability and scaled expected length where the scaling is with respect to the usual confidence interval, with the same minimum coverage, based on the full model. We assessed the properties of the confidence interval centered on the model averaged estimator proposed by Buckland et al. (1997) with half-width proportional to the standard error of the estimate put forward by Buckland et al. (1997) of their formula (9). We considered both the finite sample case and the asymptotic case. Our results show that this model averaged confidence interval cannot be generally recommended, although it performs well when the error degrees of freedom is 1. We also assessed the properties of the confidence intervals centered on the bootstrap smoothed estimator with half-width proportional to the estimate of (a) the standard deviation of this estimator and (b) the delta method approximation, derived by Efron (2014), to the standard deviation of this estimator. We show that, except for the case of unknown error variance and small error degrees of freedom, the confidence interval centered on the bootstrap smoothed estimator does not have scaled expected length that is substantially less than 1, when the simpler model is correct. For this reason, we sought a formula for the data-based width that leads to improved performance of the confidence interval centered on the bootstrap smoothed estimator. Since this search failed, it raises the following question. Is there any formula for the data-based width of such a confidence interval so that it performs well? To answer this question, we compute the performance bound due to Kabaila & Kong (2016).

When is n large enough? Looking for the right sample size to estimate proportions

ABSTRACTThe central limit theorem indicates that when the sample size goes to infinite, the sampling distribution of means tends to follow a normal distribution; it is the basis for the most usual confidence interval and sample size formulas. This study analyzes what sample size is large enough to assume that the distribution of the estimator of a proportion follows a Normal distribution. Also, we propose the use of a correction factor in sample size formulas to ensure a confidence level even when the central limit theorem does not apply for these distributions.

Confidence intervals centred on bootstrap smoothed estimators

SummaryBootstrap smoothed (bagged) parameter estimators have been proposed as an improvement on estimators found after preliminary data‐based model selection. A result of Efron in 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 considered a confidence interval centred 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 the confidence interval centred on the bootstrap smoothed estimator, in terms of expected length, to the usual confidence interval, with the same minimum coverage probability, based on the full model.

On confidence intervals centred on bootstrap smoothed estimators

Bootstrap smoothed (bagged) estimators have been proposed as an improvement on estimators found after preliminary data‐based model selection. Efron 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 centred on the bootstrap smoothed estimator, with width proportional to the estimate of this standard deviation. Recently, Kabaila and Wijethunga assessed the performance of this confidence interval in the scenario of two nested linear regression models, the full model and the 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.

Wilson Confidence Intervals for Binomial Proportions With Multiple Imputation for Missing Data

ABSTRACTWe present a Wilson interval for binomial proportions for use with multiple imputation for missing data. Using simulation studies, we show that it can have better repeated sampling properties than the usual confidence interval for binomial proportions based on Rubin’s combining rules. Further, in contrast to the usual multiple imputation confidence interval for proportions, the multiple imputation Wilson interval is always bounded by zero and one. Supplementary material is available online.

Skewness-kurtosis adjusted confidence estimators and significance tests

First and second kind modifications of usual confidence intervals for estimating the expectation and of usual local alternative parameter choices are introduced in a way such that the asymptotic behavior of the true non-covering probabilities and the covering probabilities under the modified local non-true parameter assumption can be asymptotically exactly controlled. The orders of convergence to zero of both types of probabilities are assumed to be suitably bounded below according to an Osipov-type condition and the sample distribution is assumed to satisfy a corresponding tail condition due to Linnik. Analogous considerations are presented for the power function when testing a hypothesis concerning the expectation both under the assumption of a true hypothesis as well as under a modified local alternative. A limit theorem for large deviations by S.V. Nagajev/V.V. Petrov applies to prove the results. Applications are given for exponential families.

Proportional Hazard Inverse Weibull Distribution and Associated Inference

In this study, we have considered the proportional hazard version of the inverse Weibull distribution. It has been shown that the density and hazard rate functions are unimodal and the mean residual life function is bathtub shaped. For the considered model, many functions of the parameters, such as the mean, variance, coefficient of variation and the critical points of the density, hazard rate and mean residual life functions cannot be given in explicit form. As a result, the variances of the maximum likelihood estimators of such functions cannot be given in explicit form to construct the usual asymptotic confidence interval. In this study, we use the percentile bootstrap estimation method to estimate such variances to construct the asymptotic confidence intervals. The asymptotic confidence intervals are compared with the bootstrap confidence intervals. Simulation studies are carried out to examine the performance of the maximum likelihood and bootstrap estimators. An example is provided to illustrate the procedure.

New bootstrap inference for spurious regression problems

Phillips [11] provides asymptotic theory for regressions that relate nonstationary time series including those integrated of order 1, . A practical implication of the literature on spurious regression is that one cannot trust the usual confidence intervals (CIs). In the absence of prior knowledge that two series are cointegrated, it is therefore recommended that we abandon the specification in levels and work with differenced or detrended series. For situations when the specification in levels is sacrosanct we propose new CIs based on the Maximum Entropy bootstrap explained in Vinod and López-de-Lacalle (Maximum entropy bootstrap for time series: The meboot R package, J. Statist. Softw. 29 (2009), pp. 1–19). An extensive Monte Carlo simulation shows that our proposal can provide more reliable conservative CIs than traditional and block bootstrap intervals.

On the problem of inference for inequality measures for heavy‐tailed distributions

We consider the class of heavy‐tailed income distributions and show that the shape of the income distribution has a strong effect on inference for inequality measures. In particular, we demonstrate how the severity of the inference problem responds to the exact nature of the right tail of the income distribution. It is shown that the density of the studentized inequality measure is heavily skewed to the left, and that the excessive coverage failures of the usual confidence intervals are associated with excessively low estimates of both the point measure and the variance. For further diagnostics, the coefficients of bias, skewness and kurtosis are derived and examined for both studentized and standardized inequality measures. These coefficients are also used to correct the size of confidence intervals. Exploiting the uncovered systematic relationship between the inequality estimate and its estimated variance, variance stabilizing transforms are proposed and shown to improve inference significantly.

Admissibility of the usual confidence interval for the normal mean

Suppose that X 1 , … , X n are independent and identically N ( μ , σ 2 ) distributed, where μ and σ are unknown parameters ( μ ∈ R and σ > 0 ). We prove that the usual confidence interval for μ is admissible within a broad class of confidence intervals.

New Solution to Time Series Inference in Spurious Regression Problems

Phillips (1986) provides asymptotic theory for regressions that relate nonstationary time series including those integrated of order 1, I(1). A practical implication of related literature on spurious regression is that one cannot trust the usual confidence intervals. Therefore it is recommended that after carrying out unit root tests we work with differenced series instead of original data in levels. We propose a new alternative of using confidence intervals based on Maximum Entropy bootstrap explained in Vinod and Lopez-de-Lacalle (2009, J of Statistical Software). Extensive Monte Carlo simulations show that our proposal can provide more reliable conservative confidence intervals than traditional, differencing and block bootstrap (BB) intervals. We hope to let researchers avoid differencing the variables and work with their original specifications in levels.

Admissibility of the usual confidence interval in linear regression

Consider a linear regression model with independent and identically normally distributed random errors. Suppose that the parameter of interest is a specified linear combination of the regression parameters. We prove that the usual confidence interval for this parameter is admissible within a broad class of confidence intervals.

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Constructing second‐order accurate confidence intervals for communalities in factor analysis

In an effort to find accurate alternatives to the usual confidence intervals based on normal approximations, this paper compares four methods of generating second-order accurate confidence intervals for non-standardized and standardized communalities in exploratory factor analysis under the normality assumption. The methods to generate the intervals employ, respectively, the Cornish-Fisher expansion and the approximate bootstrap confidence (ABC), and the bootstrap-t and the bias-corrected and accelerated bootstrap (BC(a)). The former two are analytical and the latter two are numerical. Explicit expressions of the asymptotic bias and skewness of the communality estimators, used in the analytical methods, are derived. A Monte Carlo experiment reveals that the performance of central intervals based on normal approximations is a consequence of imbalance of miscoverage on the left- and right-hand sides. The second-order accurate intervals do not require symmetry around the point estimates of the usual intervals and achieve better balance, even when the sample size is not large. The behaviours of the second-order accurate intervals were similar to each other, particularly for large sample sizes, and no method performed consistently better than the others.

Precision of Total Volume Estimates Resulting from Pseudo-Bayes Estimator for Stock Tables

Abstract In an earlier article we presented a novel estimator for stand tables. The estimator performed much better than the usual estimator in simulation studies. In this article we extend the method to volume estimation, and demonstrate via simulation studies that total volume estimates from the pseudo-Bayes stand table estimator are superior to estimates derived from the usual stand table estimator. We also derive estimates for the variance and mean squared error of the pseudo-Bayes total volume estimator and approximate confidence intervals. The intervals based on the mean squared error have approximately correct coverage and are narrower than the usual confidence intervals.

The Pricing Error Function

Measurement error can lead to biased parameter estimates and misleading test results. In cross-sectional asset pricing models (e.g., CAPM, Fama-French, good beta/bad beta), measurement error can easily arise in calculating portfolio betas. A common approach to the errors-in-variables problem is instrumental variables. Econometrically, conditional asset pricing models effectively serve as instrumental variables (IV) estimators of asset pricing models.A large body of econometric research has shown that IV estimation performs poorly with weak instruments. Weak instruments can lead to highly misleading parameter estimates and standard errors, so that inference is severely distorted. Biases and size distortions due to weak instruments often do not vanish in large samples. Previous work (based on econometric theory and Monte Carlo simulations) has shown that the confidence intervals can be so misleading that they do not include the true values of the model parameters. Moreover, under plausible circumstances, the usual techniques to correct these problems (including bootstrap techniques) also fail.The fundamental reason for the measurement error and weak instrument problems is weak identification. When parameters are not identifiable, it is important to allow for unbounded confidence intervals. Usual confidence intervals of the form {estimate ± asymptotic cut-off point × asymptotic standard error} are bounded by construction. When a parameter is hard to identify from the available data, the fact that any value in its parameter space is equally acceptable should be reflected in the confidence interval. Weak identification should lead to diffuse confidence sets that alert the researcher to the problem. Unfortunately, if usual confidence intervals are constructed when estimating weakly-identified parameters, the researcher may obtain very tight confidence intervals that are focused on wrong values. In traditional estimation methodology, a point estimate is found first, and confidence intervals are then constructed. Our approach proceeds in reverse: first, we build a confidence set; then we obtain a point estimate from the confidence set. The confidence set is obtained by inverting an artificial regression-based test designed to achieve type-I error control (at the desired level, say 5%) whether the instruments are weak or strong. The pricing error function given the data Y is denoted e(Y; a3B3), where a3B3 represents the parameters of the asset pricing model. Under the null hypothesis, if the pricing errors are regressed on the instruments X, the coefficients should be zero. This condition reflects the same orthogonality principle that underlies GMM. Testing this condition can be interpreted as assessing whether additional explanatory variables hold further information on returns that is not explained by the betas. This paper examines the importance of measurement error and weak instruments for existing tests of cross-sectional asset pricing models. Our paper is constructive, in the sense that: 1) we develop empirical techniques that allow researchers to determine whether measurement error or weak instruments problems are important; 2) we show that some existing asset pricing models are consistent with the data; and 3) we develop empirical techniques that are robust to measurement error and weak instruments.

Improved Exact Confidence Intervals for the Odds Ratio in Two Independent Binomial Samples

For two independent binomial samples, the usual exact confidence interval for the odds ratio based on the conditional approach can be very conservative. Recently, Agresti and Min (2002) showed that the unconditional intervals are preferable to conditional intervals with small sample sizes. We use the unconditional approach to obtain a modified interval, which has shorter length, and its coverage probability is closer to and at least the nominal confidence coefficient.