Optimal Bootstrap Sample Size in Construction of Percentile Confidence Bounds

In traditional bootstrap applications the size of a bootstrap sample equals the parent sample size, n say. Recent studies have shown that using a bootstrap sample size different from n may sometimes provide a more satisfactory solution. In this paper we apply the latter approach to correct for coverage error in construction of bootstrap confidence bounds. We show that the coverage error of a bootstrap percentile method confidence bound, which is of order O ( n ^−2/2) typically, can be reduced to O ( n ⁻¹) by use of an optimal bootstrap sample size. A simulation study is conducted to illustrate our findings, which also suggest that the new method yields intervals of shorter length and greater stability compared to competitors of similar coverage accuracy.     

Test inversion bootstrap confidence intervals

In this paper we explore the theoretical and practical implications of using bootstrap test inversion to construct confidence intervals. In the presence of nuisance parameters, we show that the coverage error of such intervals is O ( n ^−1/2) which may be reduced to O ( n ⁻¹) if a Studentized statistic is used. We present three simulation studies and compare the performance of test inversion methods with established methods on the problem of estimating a confidence interval for the dose–response parameter in models of the Japanese atomic bomb survivors data.     

Importance resampling for the smoothed bootstrap

Alternative methods of estimating properties of unknown distributions include the bootstrap and the smoothed bootstrap. In the standard bootstrap setting, Johns (1988) introduced an importance resam¬pling procedure that results in more accurate approximation to the bootstrap estimate of a distribution function or a quantile. With a suitable “exponential tilting” similar to that used by Johns, we derived a smoothed version of importance resampling in the framework of the smoothed bootstrap. Smoothed importance resampling procedures were developed for the estimation of distribution functions of the Studentized mean, the Studentized variance, and the correlation coefficient. Implementation of these procedures are presented via simulation results which concentrate on the problem of estimation of distribution functions of the Studentized mean and Studentized variance for different sample sizes and various pre-specified smoothing bandwidths for the normal data; additional simulations were conducted for the estimation of quantiles of the distribution of the Studentized mean under an optimal smoothing bandwidth when the original data were simulated from three different parent populations: lognormal, t(3) and t(10). These results suggest that in cases where it is advantageous to use the smoothed bootstrap rather than the standard bootstrap, the amount of resampling necessary might be substantially reduced by the use of importance resampling methods and the efficiency gains depend on the bandwidth used in the kernel density estimation.     

Zero-Bias Locally Adaptive Density Estimators

Strategies for improving fixed non-negative kernel estimators have focused on reducing the bias, either by employing higher-order kernels or by adjusting the bandwidth locally. Intuitively, bandwidths in the tails should be relatively larger in order to reduce wiggles since there is less data available in the tails. We show that in regions where the density function is convex, it is theoretically possible to find local bandwidths such that the pointwise bias is exactly zero. The corresponding pointwise mean squared error converges at the parametric rate of O ( n ⁻¹ ) rather than the slower O ( n ^−4/5). These so-called zero-bias bandwidths are constant and are usually orders of magnitude larger than the optimal locally adaptive bandwidths predicted by asymptotic mean squared error analysis. We describe data-based algorithms for estimating zero-bias bandwidths over intervals where the density is convex. We find that our particular density estimator attains the usual O ( n ^−4/5) rate. However, we demonstrate that the algorithms can provide significant improvement in mean squared error, often clearly visually superior curves, and a new operating point in the usual bias-variance tradeoff.     

New bootstrap confidence intervals for means of positively skewed distributions

ABSTRACT

In this paper, we consider the problem of constructing non parametric confidence intervals for the mean of a positively skewed distribution. We suggest calibrated, smoothed bootstrap upper and lower percentile confidence intervals. For the theoretical properties, we show that the proposed one-sided confidence intervals have coverage probability α + O(n^{? 3/2}). This is an improvement upon the traditional bootstrap confidence intervals in terms of coverage probability. A version smoothed approach is also considered for constructing a two-sided confidence interval and its theoretical properties are also studied. A simulation study is performed to illustrate the performance of our confidence interval methods. We then apply the methods to a real data set.     

Improving coverage accuracy of nonparametric prediction intervals

Methods are suggested for improving the coverage accuracy of intervals for predicting future values of a random variable drawn from a sampled distribution. It is shown that properties of solutions to such problems may be quite unexpected. For example, the bootstrap and the jackknife perform very poorly when used to calibrate coverage, although the jackknife estimator of the true coverage is virtually unbiased. A version of the smoothed bootstrap can be employed for successful calibration, however. Interpolation among adjacent order statistics can also be an effective way of calibrating, although even there the results are unexpected. In particular, whereas the coverage error can be reduced from O ( n ^-1) to orders O ( n ^-2) and O ( n ^-3) (where n denotes the sample size) by interpolating among two and three order statistics respectively, the next two orders of reduction require interpolation among five and eight order statistics respectively.     

Estimation of the Quantile Function of an IFRA Distribution

Let F and G be lifetime distributions and consider the problem of estimating F ⁻¹ when it is known that G ⁻¹ F is star-shaped. Estimators of F ⁻¹ are considered here which are shown to be uniformly strongly consistent. The case of censored data is also presented. Asymptotic confidence intervals and bands for F ⁻¹ are provided. The result are applicable, for example, to the estimation of quantile functions of k -out-of- n systems in reliability. The special case of an IFRA distribution follows immediately from the more general case presented here     

Bootstrap confidence intervals in mixtures of discrete distributions

The problem of building bootstrap confidence intervals for small probabilities with count data is addressed. The law of the independent observations is assumed to be a mixture of a given family of power series distributions. The mixing distribution is estimated by nonparametric maximum likelihood and the corresponding mixture is used for resampling. We build percentile-t and Efron percentile bootstrap confidence intervals for the probabilities and we prove their consistency in probability. The new theoretical results are supported by simulation experiments for Poisson and geometric mixtures. We compare percentile-t and Efron percentile bootstrap intervals with eight other bootstrap or asymptotic theory based intervals. It appears that Efron percentile bootstrap intervals outperform the competitors in terms of coverage probability and length.     

On the bootstrap in cube root asymptotics

The authors study the application of the bootstrap to a class of estimators which converge at a nonstandard rate to a nonstandard distribution. They provide a theoretical framework to study its asymptotic behaviour. A simulation study shows that in the case of an estimator such as Chernoff's estimator of the mode, usually the basic bootstrap confidence intervals drastically undercover while the percentile bootstrap intervals overcover. This is a rare instance where basic and percentile confidence intervals, which have exactly the same length, behave in a very different way. In the case of Chernoff's estimator, if the distribution is symmetric, it is possible to bootstrap from a smooth symmetric estimator of the distribution for which the basic bootstrap confidence intervals will have the claimed coverage probability while the percentile bootstrap interval will have an asymptotic coverage of 1!     

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.     

A New Characterization and Estimation of the Zero–bias Bandwidth

It is well established that bandwidths exist that can yield an unbiased non–parametric kernel density estimate at points in particular regions (e.g. convex regions) of the underlying density. These zero–bias bandwidths have superior theoretical properties, including a 1/n convergence rate of the mean squared error. However, the explicit functional form of the zero–bias bandwidth has remained elusive. It is difficult to estimate these bandwidths and virtually impossible to achieve the higher–order rate in practice. This paper addresses these issues by taking a fundamentally different approach to the asymptotics of the kernel density estimator to derive a functional approximation to the zero–bias bandwidth. It develops a simple approximation algorithm that focuses on estimating these zero–bias bandwidths in the tails of densities where the convexity conditions favourable to the existence of the zerobias bandwidths are more natural. The estimated bandwidths yield density estimates with mean squared error that is O(n^–4/5), the same rate as the mean squared error of density estimates with other choices of local bandwidths. Simulation studies and an illustrative example with air pollution data show that these estimated zero–bias bandwidths outperform other global and local bandwidth estimators in estimating points in the tails of densities.     

On multivariate smoothed bootstrap consistency

This paper deals with the convergence in Mallows metric for classical multivariate kernel distribution function estimators. We prove the convergence in Mallows metric of a locally orientated kernel smooth estimator belonging to the class of sample smoothing estimators. The consistency follows for the smoothed bootstrap for regular functions of the marginal means. Two simple simulation studies show how the smoothed versions of the bootstrap give better results than the classical technique.     

On the Distance Between Cumulative Sum Diagram and Its Greatest Convex Minorant for Unequally Spaced Design Points

Abstract.  The supremum difference between the cumulative sum diagram, and its greatest convex minorant (GCM), in case of non-parametric isotonic regression is considered. When the regression function is strictly increasing, and the design points are unequally spaced, but approximate a positive density in even a slow rate ( n ^−1/3), then the difference is shown to shrink in a very rapid (close to n ^−2/3) rate. The result is analogous to the corresponding result in case of a monotone density estimation established by Kiefer and Wolfowitz, but uses entirely different representation. The limit distribution of the GCM as a process on the unit interval is obtained when the design variables are i.i.d. with a positive density. Finally, a pointwise asymptotic normality result is proved for the smooth monotone estimator, obtained by the convolution of a kernel with the classical monotone estimator.     

Consistent Smooth Bootstrap Kernel Intensity Estimation for Inhomogeneous Spatial Poisson Point Processes

Non‐parametric estimation and bootstrap techniques play an important role in many areas of Statistics. In the point process context, kernel intensity estimation has been limited to exploratory analysis because of its inconsistency, and some consistent alternatives have been proposed. Furthermore, most authors have considered kernel intensity estimators with scalar bandwidths, which can be very restrictive. This work focuses on a consistent kernel intensity estimator with unconstrained bandwidth matrix. We propose a smooth bootstrap for inhomogeneous spatial point processes. The consistency of the bootstrap mean integrated squared error (MISE) as an estimator of the MISE of the consistent kernel intensity estimator proves the validity of the resampling procedure. Finally, we propose a plug‐in bandwidth selection procedure based on the bootstrap MISE and compare its performance with several methods currently used through both as a simulation study and an application to the spatial pattern of wildfires registered in Galicia (Spain) during 2006.     

Iterated Bootstrap-t Confidence Intervals for Density Functions

Abstract.  Conventional bootstrap- t intervals for density functions based on kernel density estimators exhibit poor coverages due to failure of the bootstrap to estimate the bias correctly. The problem can be resolved by either estimating the bias explicitly or undersmoothing the kernel density estimate to undermine its bias asymptotically. The resulting bias-corrected intervals have an optimal coverage error of order arbitrarily close to second order for a sufficiently smooth density function. We investigated the effects on coverage error of both bias-corrected intervals when the nominal coverage level is calibrated by the iterated bootstrap. In either case, an asymptotic reduction of coverage error is possible provided that the bias terms are handled using an extra round of smoothed bootstrapping. Under appropriate smoothness conditions, the optimal coverage error of the iterated bootstrap- t intervals has order arbitrarily close to third order. Examples of both simulated and real data are reported to illustrate the iterated bootstrap procedures.     

Saddlepoint Approximations for the Difference of Order Statistics and Studentized Sample Quantiles

For a sample from a given distribution the difference of two order statistics and the Studentized quantile are statistics whose distribution is needed to obtain tests and confidence intervals for quantiles and quantile differences. This paper gives saddlepoint approximations for densities and saddlepoint approximations of the Lugannani–Rice form for tail probabilities of these statistics. The relative errors of the approximations are n ⁻¹ uniformly in a neighbourhood of the parameters and this uniformity is global if the densities are log-concave.     

Bootstrap confidence intervals of generalized process capability index Cpyk for Lindley and power Lindley distributions

One of the indicators for evaluating the capability of a process is the process capability index. In this article, bootstrap confidence intervals of the generalized process capability index (GPCI) proposed by Maiti et al. are studied through simulation, when the underlying distributions are Lindley and Power Lindley distributions. The maximum likelihood method is used to estimate the parameters of the models. Three bootstrap confidence intervals namely, standard bootstrap (SB), percentile bootstrap (PB), and bias-corrected percentile bootstrap (BCPB) are considered for obtaining confidence intervals of GPCI. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average width of the bootstrap confidence intervals. Simulation results show that the estimated coverage probabilities of the percentile bootstrap confidence interval and the bias-corrected percentile bootstrap confidence interval get closer to the nominal confidence level than those of the standard bootstrap confidence interval. Finally, three real datasets are analyzed for illustrative purposes.     

Smoothed bootstrap bandwidth selection for nonparametric hazard rate estimation

A smoothed bootstrap method is presented for the purpose of bandwidth selection in nonparametric hazard rate estimation for iid data. In this context, two new bootstrap bandwidth selectors are established based on the exact expression of the bootstrap version of the mean integrated squared error of some approximations of the kernel hazard rate estimator. This is very useful since Monte Carlo approximation is no longer needed for the implementation of the two bootstrap selectors. A simulation study is carried out in order to show the empirical performance of the new bootstrap bandwidths and to compare them with other existing selectors. The methods are illustrated by applying them to a diabetes data set.     

An Adaptive Two-stage Estimation Method for Additive Models

Abstract.  In this paper, a two-stage estimation method for non-parametric additive models is investigated. Differing from Horowitz and Mammen's two-stage estimation, our first-stage estimators are designed not only for dimension reduction but also as initial approximations to all of the additive components. The second-stage estimators are obtained by using one-dimensional non-parametric techniques to refine the first-stage ones. From this procedure, we can reveal a relationship between the regression function spaces and convergence rate, and then provide estimators that are optimal in the sense that, better than the usual one-dimensional mean-squared error (MSE) of the order n ^−4/5 , the MSE of the order n ^{− 1} can be achieved when the underlying models are actually parametric. This shows that our estimation procedure is adaptive in a certain sense. Also it is proved that the bandwidth that is selected by cross-validation depends only on one-dimensional kernel estimation and maintains the asymptotic optimality. Simulation studies show that the new estimators of the regression function and all components outperform the existing estimators, and their behaviours are often similar to that of the oracle estimator.     

Bootstrap methods for bias correction and confidence interval estimation for nonlinear quantile regression of longitudinal data

This paper examines the use of bootstrapping for bias correction and calculation of confidence intervals (CIs) for a weighted nonlinear quantile regression estimator adjusted to the case of longitudinal data. Different weights and types of CIs are used and compared by computer simulation using a logistic growth function and error terms following an AR(1) model. The results indicate that bias correction reduces the bias of a point estimator but fails for CI calculations. A bootstrap percentile method and a normal approximation method perform well for two weights when used without bias correction. Taking both coverage and lengths of CIs into consideration, a non-bias-corrected percentile method with an unweighted estimator performs best.