Confidence intervals for nonparametric quantile regression: an emphasis on smoothing splines approach

In this paper we consider the problem of constructing confidence intervals for nonparametric quantile regression with an emphasis on smoothing splines. The mean‐based approaches for smoothing splines of Wahba (1983) and Nychka (1988) may not be efficient for constructing confidence intervals for the underlying function when the observed data are non‐Gaussian distributed, for instance if they are skewed or heavy‐tailed. This paper proposes a method of constructing confidence intervals for the unknown τth quantile function (0<τ<1) based on smoothing splines. In this paper we investigate the extent to which the proposed estimator provides the desired coverage probability. In addition, an improvement based on a local smoothing parameter that provides more uniform pointwise coverage is developed. The results from numerical studies including a simulation study and real data analysis demonstrate the promising empirical properties of the proposed approach.     

Confidence intervals for a population proportion using group testing in the presence of misclassification

In this study, we consider the construction of the confidence interval for the population proportion while using group testing with misclassification. We propose two confidence intervals based on Cornish-Fisher expansion and a modified Wilson’s interval based on a newly developed estimator. We investigate the performance of these intervals extensively and also apply the methods to real datasets. Our newly derived methods have competitive performance compared with existing methods.     

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.     

Analytical approximations for iterated bootstrap confidence intervals

Standard algorithms for the construction of iterated bootstrap confidence intervals are computationally very demanding, requiring nested levels of bootstrap resampling. We propose an alternative approach to constructing double bootstrap confidence intervals that involves replacing the inner level of resampling by an analytical approximation. This approximation is based on saddlepoint methods and a tail probability approximation of DiCiccio and Martin (1991). Our technique significantly reduces the computational expense of iterated bootstrap calculations. A formal algorithm for the construction of our approximate iterated bootstrap confidence intervals is presented, and some crucial practical issues arising in its implementation are discussed. Our procedure is illustrated in the case of constructing confidence intervals for ratios of means using both real and simulated data. We repeat an experiment of Schenker (1985) involving the construction of bootstrap confidence intervals for a variance and demonstrate that our technique makes feasible the construction of accurate bootstrap confidence intervals in that context. Finally, we investigate the use of our technique in a more complex setting, that of constructing confidence intervals for a correlation coefficient.     

Bootstrap confidence intervals for smoothing splines and their comparison to bayesian confidence intervals

We construct bootstrap confidence intervals for smoothing spline estimates based on Gaussian data, and penalized likelihood smoothing spline estimates based on data from .exponential families. Several vari- ations of bootstrap confidence intervals are considered and compared. We find that the commonly used ootstrap percentile intervals are inferior to the T intervals and to intervals based on bootstrap estimation of mean squared errors. The best variations of the bootstrap confidence intervals behave similar to the well known Bayesian confidence intervals. These bootstrap confidence intervals have an average coverage probability across the function being estimated, as opposed to a pointwise property.     

Why High-Order Polynomials Should Not Be Used in Regression Discontinuity Designs

It is common in regression discontinuity analysis to control for third, fourth, or higher-degree polynomials of the forcing variable. There appears to be a perception that such methods are theoretically justified, even though they can lead to evidently nonsensical results. We argue that controlling for global high-order polynomials in regression discontinuity analysis is a flawed approach with three major problems: it leads to noisy estimates, sensitivity to the degree of the polynomial, and poor coverage of confidence intervals. We recommend researchers instead use estimators based on local linear or quadratic polynomials or other smooth functions.     

One stage multiple comparisons of k-1 treatment mean lifetimes with the control for exponential distributions under heteroscedasticity

Abstract

In survival or reliability data analysis, it is often useful to estimate the quantiles of the lifetime distribution, such as the median time to failure. Different nonparametric methods can construct confidence intervals for the quantiles of the lifetime distributions, some of which are implemented in commonly used statistical software packages. We here investigate the performance of different interval estimation procedures under a variety of settings with different censoring schemes. Our main objectives in this paper are to (i) evaluate the performance of confidence intervals based on the transformation approach commonly used in statistical software, (ii) introduce a new density-estimation-based approach to obtain confidence intervals for survival quantiles, and (iii) compare it with the transformation approach. We provide a comprehensive comparative study and offer some useful practical recommendations based on our results. Some numerical examples are presented to illustrate the methodologies developed.     

Asymptotic confidence interval of power spectrum of a continuous time process through progressively faster sampling

If the power spectral density of a continuous time stationary stochastic process is not limited to a finite bandwidth, data sampled from that process at any uniform sampling rate leads to biased and inconsistent spectrum estimators, which are unsuitable for constructing confidence intervals. In this paper, we use the smoothed periodogram estimator to construct asymptotic confidence intervals shrinking to the true spectra, by allowing the sampling rate to go to infinity suitably fast as the sample size goes to infinity. The proposed method requires minimal computation, as it does not involve bootstrap or other resampling. The method is illustrated through a Monte-Carlo simulation study, and its performance is compared with that of the corresponding method based on uniform sampling at a fixed rate.     

Simultaneous confidexce intervals in an extended crow iii curve model

This paper deals with estimation problems under an extended growth curve model with two hierarchical within-individuals design matrices. The model in cludes the one whose mean structure consists of polynomial growth curves with two different degrees. First we propose certain simple estimators of the mean and covariance parameters which are closely related to the MEE's. Some basic properties of the estimators are given. Simultaneous confidence intervals are constructed, based on the estimators, for each and both of two growth curves. We give asymptotic approximations for the corresponding critical points. A numerical example is also given.     

Free-knot polynomial splines with confidence intervals

Summary.  We construct approximate confidence intervals for a nonparametric regression function, using polynomial splines with free-knot locations. The number of knots is determined by generalized cross-validation. The estimates of knot locations and coefficients are obtained through a non-linear least squares solution that corresponds to the maximum likelihood estimate. Confidence intervals are then constructed based on the asymptotic distribution of the maximum likelihood estimator. Average coverage probabilities and the accuracy of the estimate are examined via simulation. This includes comparisons between our method and some existing methods such as smoothing spline and variable knots selection as well as a Bayesian version of the variable knots method. Simulation results indicate that our method works well for smooth underlying functions and also reasonably well for discontinuous functions. It also performs well for fairly small sample sizes.     

Local Likelihood Estimation in Generalized Additive Models

ABSTRACT.  Generalized additive models are a popular class of multivariate non-parametric regression models, due in large part to the ease of use of the local scoring estimation algorithm. However, the theoretical properties of the local scoring estimator are poorly understood. In this article, we propose a local likelihood estimator for generalized additive models that is closely related to the local scoring estimator fitted by local polynomial regression. We derive the statistical properties of the estimator and show that it achieves the same asymptotic convergence rate as a one-dimensional local polynomial regression estimator. We also propose a wild bootstrap estimator for calculating point-wise confidence intervals for the additive component functions. The practical behaviour of the proposed estimator is illustrated through a simulation experiment.     

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 the Current Status Model

We discuss a new way of constructing pointwise confidence intervals for the distribution function in the current status model. The confidence intervals are based on the smoothed maximum likelihood estimator, using local smooth functional theory and normal limit distributions. Bootstrap methods for constructing these intervals are considered. Other methods to construct confidence intervals, using the non‐standard limit distribution of the (restricted) maximum likelihood estimator, are compared with our approach via simulations and real data applications.     

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.     

Statistical inference for multiple change-point models

In this article, we propose a new technique for constructing confidence intervals for the mean of a noisy sequence with multiple change-points. We use the weighted bootstrap to generalize the bootstrap aggregating or bagging estimator. A standard deviation formula for the bagging estimator is introduced, based on which smoothed confidence intervals are constructed. To further improve the performance of the smoothed interval for weak signals, we suggest a strategy of adaptively choosing between the percentile intervals and the smoothed intervals. A new intensity plot is proposed to visualize the pattern of the change-points. We also propose a new change-point estimator based on the intensity plot, which has superior performance in comparison with the state-of-the-art segmentation methods. The finite sample performance of the confidence intervals and the change-point estimator are evaluated through Monte Carlo studies and illustrated with a real data example.     

Robust simultaneous confidence interval estimation of principal component loadings

We propose a method that integrates bootstrap into the forward search algorithm in the construction of robust confidence intervals for elements of the eigenvectors of the correlation matrix in the presence of outliers. Coverage probability of the bootstrap simultaneous confidence intervals was compared to the coverage probabilities of regular asymptotic confidence region and asymptotic confidence region based on the minimum covariance determinant (MCD) approach through a simulation study. The method produced more stable coverage probabilities for datasets with or without outliers and across several sample sizes compared to approaches based on asymptotic confidence regions.     

Likelihood Ratio Confidence Bands in Non–parametric Regression with Censored Data

Let ( X , Y ) be a random vector, where Y denotes the variable of interest possibly subject to random right censoring, and X is a covariate. We construct confidence intervals and bands for the conditional survival and quantile function of Y given X using a non-parametric likelihood ratio approach. This approach was introduced by Thomas & Grunkemeier (1975 ), who estimated confidence intervals of survival probabilities based on right censored data. The method is appealing for several reasons: it always produces intervals inside [0, 1], it does not involve variance estimation, and can produce asymmetric intervals. Asymptotic results for the confidence intervals and bands are obtained, as well as simulation results, in which the performance of the likelihood ratio intervals and bands is compared with that of the normal approximation method. We also propose a bandwidth selection procedure based on the bootstrap and apply the technique on a real data set.     

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.     

Confidence Intervals for the Weighted Sum of Two Independent Binomial Proportions

Confidence intervals for the difference of two binomial proportions are well known, however, confidence intervals for the weighted sum of two binomial proportions are less studied. We develop and compare seven methods for constructing confidence intervals for the weighted sum of two independent binomial proportions. The interval estimates are constructed by inverting the Wald test, the score test and the Likelihood ratio test. The weights can be negative, so our results generalize those for the difference between two independent proportions. We provide a numerical study that shows that these confidence intervals based on large‐sample approximations perform very well, even when a relatively small amount of data is available. The intervals based on the inversion of the score test showed the best performance. Finally, we show that as for the difference of two binomial proportions, adding four pseudo‐outcomes to the Wald interval for the weighted sum of two binomial proportions improves its coverage significantly, and we provide a justification for this correction.     

A procedure for approximate negative binomial tolerance intervals

In this article, we present a procedure for approximate negative binomial tolerance intervals. We utilize an approach that has been well-studied to approximate tolerance intervals for the binomial and Poisson settings, which is based on the confidence interval for the parameter in the respective distribution. A simulation study is performed to assess the coverage probabilities and expected widths of the tolerance intervals. The simulation study also compares eight different confidence interval approaches for the negative binomial proportions. We recommend using those in practice that perform the best based on our simulation results. The method is also illustrated using two real data examples.