IMPORTANCE BOOTSTRAP RESAMPLING FOR PROPORTIONAL HAZARDS REGRESSION

Importance resampling is an approach that uses exponential tilting to reduce the resampling necessary for the construction of nonparametric bootstrap confidence intervals. The properties of bootstrap importance confidence intervals are well established when the data is a smooth function of means and when there is no censoring. However, in the framework of survival or time-to-event data, the asymptotic properties of importance resampling have not been rigorously studied, mainly because of the unduly complicated theory incurred when data is censored. This paper uses extensive simulation to show that, for parameter estimates arising from fitting Cox proportional hazards models, importance bootstrap confidence intervals can be constructed if the importance resampling probabilities of the records for the n individuals in the study are determined by the empirical influence function for the parameter of interest. Our results show that, compared to uniform resampling, importance resampling improves the relative mean-squared-error (MSE) efficiency by a factor of nine (for n = 200). The efficiency increases significantly with sample size, is mildly associated with the amount of censoring, but decreases slightly as the number of bootstrap resamples increases. The extra CPU time requirement for calculating importance resamples is negligible when compared to the large improvement in MSE efficiency. The method is illustrated through an application to data on chronic lymphocytic leukemia, which highlights that the bootstrap confidence interval is the preferred alternative to large sample inferences when the distribution of a specific covariate deviates from normality. Our results imply that, because of its computational efficiency, importance resampling is recommended whenever bootstrap methodology is implemented in a survival framework. Its use is particularly important when complex covariates are involved or the survival problem to be solved is part of a larger problem; for instance, when determining confidence bounds for models linking survival time with clusters identified in gene expression microarray data.     

Goodness of fit tests with interval censored data

Cramér–von Mises type goodness of fit tests for interval censored data case 2 are proposed based on a resampling method called the leveraged bootstrap, and their asymptotic consistency is shown. The proposed tests are computationally efficient, and in fact can be applied to other types of censored data, including right censored data, doubly censored data and (mixture of) case k interval censored data. Some simulation results and an example from AIDS research are presented.     

On progressive hybrid censored exponential distribution

In this paper, we introduce a new adaptive Type-I progressive hybrid censoring scheme, which has some advantages over the progressive hybrid censoring schemes already discussed in the literature. Based on an adaptive Type-I progressively hybrid censored sample, we derive the exact distribution of the maximum-likelihood estimator (MLE) of the mean lifetime of an exponential distribution as well as confidence intervals for the failure rate using exact distribution, asymptotic distribution, and three parametric bootstrap resampling methods. Furthermore, we provide computational formula for the expected number of failures and investigate the performance of the point and interval estimation for the failure rate in this case. An alternative simple form for the distribution of the MLE under adaptive Type-II progressive hybrid censoring scheme proposed by Ng et al. [Statistical analysis of exponential lifetimes under an adaptive Type-II progressive censoring scheme, Naval Res. Logist. 56 (2009), pp. 687–698] is obtained. Finally, from the exact distribution of the MLE, we establish the explicit expression for the Bayes risk of a sampling plan under adaptive Type-II progressive hybrid censoring scheme when a general loss function is used, and present some optimal Bayes solutions under four different progressive hybrid censoring schemes to illustrate the effectiveness of the proposed method.     

A weighted bootstrap approach to bootstrap iteration

The operation of resampling from a bootstrap resample, encountered in applications of the double bootstrap, maybe viewed as resampling directly from the sample but using probability weights that are proportional to the numbers of times that sample values appear in the resample. This suggests an approximate approach to double-bootstrap Monte Carlo simulation, where weighted bootstrap methods are used to circumvent much of the labour involved in compounded Monte Carlo approximation. In the case of distribution estimation or, equivalently, confidence interval calibration, the new method may be used to reduce the computational labour. Moreover, the method produces the same order of magnitude of coverage error for confidence intervals, or level error for hypothesis tests, as a full application of the double bootstrap.     

Methods of Statistical Inference for Median Regression Models with Doubly Censored Data

Recently, least absolute deviations (LAD) estimator for median regression models with doubly censored data was proposed and the asymptotic normality of the estimator was established. However, it is invalid to make inference on the regression parameter vectors, because the asymptotic covariance matrices are difficult to estimate reliably since they involve conditional densities of error terms. In this article, three methods, which are based on bootstrap, random weighting, and empirical likelihood, respectively, and do not require density estimation, are proposed for making inference for the doubly censored median regression models. Simulations are also done to assess the performance of the proposed methods.     

Bootstrapping survival times in stochastic systems by using saddlepoint approximations

The single bootstrap is implemented by using a saddlepoint approximation to determine estimates for the survival and hazard functions of first-passage times in complicated semi-Markov processes. The double bootstrap is also implemented by resampling saddlepoint inversions and provides BC_a confidence bands for these functions. Confidence intervals for the mean and variance of first-passage times are easily computed. A new characterization of the asymptotic hazard rate for survival times is presented and leads to an indirect method for constructing its bootstrap confidence interval.     

Estimating kurtosis and confidence intervals for the variance under nonnormality

Exact confidence intervals for variances rely on normal distribution assumptions. Alternatively, large-sample confidence intervals for the variance can be attained if one estimates the kurtosis of the underlying distribution. The method used to estimate the kurtosis has a direct impact on the performance of the interval and thus the quality of statistical inferences. In this paper the author considers a number of kurtosis estimators combined with large-sample theory to construct approximate confidence intervals for the variance. In addition, a nonparametric bootstrap resampling procedure is used to build bootstrap confidence intervals for the variance. Simulated coverage probabilities using different confidence interval methods are computed for a variety of sample sizes and distributions. A modification to a conventional estimator of the kurtosis, in conjunction with adjustments to the mean and variance of the asymptotic distribution of a function of the sample variance, improves the resulting coverage values for leptokurtically distributed populations.     

Nonparametric Quantile Estimation with Correlated Failure Time Data

In biomedical studies, correlated failure time data arise often. Although point and confidence interval estimation for quantiles with independent censored failure time data have been extensively studied, estimation for quantiles with correlated failure time data has not been developed. In this article, we propose a nonparametric estimation method for quantiles with correlated failure time data. We derive the asymptotic properties of the quantile estimator and propose confidence interval estimators based on the bootstrap and kernel smoothing methods. Simulation studies are carried out to investigate the finite sample properties of the proposed estimators. Finally, we illustrate the proposed method with a data set from a study of patients with otitis media.     

Hybrid regions for post-change mean in a sequence of normal variables

The hybrid bootstrap uses resampling ideas to extend the duality approach to the interval estimation for a parameter of interest when there are nuisance parameters. The confidence region constructed by the hybrid bootstrap may perform much better than the ordinary bootstrap region in a situation where the data provide substantial information about the nuisance parameter, but limited information about the parameter of interest. We apply this method to estimate the post-change mean after a change is detected by a stopping procedure in a sequence of independent normal variables. Since distribution theory in change point problems is generally a challenge, we use bootstrap simulation to find empirical distributions of test statistics and calculate critical thresholds. Both likelihood ratio and Bayesian test statistics are considered to set confidence regions for post-change means in the normal model. In the simulation studies, the performance of hybrid regions are compared with that of ordinary bootstrap regions in terms of the widths and coverage probabilities of confidence intervals.     

Comparisons of approximate confidence intervals for the smallest extreme value distribution simple linear regression model under time censoring

Exact confidence interval estimation for accelerated life regression models with censored smallest extreme value (or Weibull) data is often impractical. This paper evaluates the accuracy of approximate confidence intervals based on the asymptotic normality of the maximum likelihood estimator, the asymptotic X²distribution of the likelihood ratio statistic, mean and variance correction to the likelihood ratio statistic, and the so-called Bartlett correction to the likelihood ratio statistic. The Monte Carlo evaluations under various degrees of time censoring show that uncorrected likelihood ratio intervals are very accurate in situations with heavy censoring. The benefits of mean and variance correction to the likelihood ratio statistic are only realized with light or no censoring. Bartlett correction tends to result in conservative intervals. Intervals based on the asymptotic normality of maximum likelihood estimators are anticonservative and should be used with much caution.     

Semiparametric Inference for ROC Curves with Censoring

Abstract.  Comparison of two samples can sometimes be conducted on the basis of analysis of receiver operating characteristic (ROC) curves. A variety of methods of point estimation and confidence intervals for ROC curves have been proposed and well studied. We develop smoothed empirical likelihood-based confidence intervals for ROC curves when the samples are censored and generated from semiparametric models. The resulting empirical log-likelihood function is shown to be asymptotically chi-squared. Simulation studies illustrate that the proposed empirical likelihood confidence interval is advantageous over the normal approximation-based confidence interval. A real data set is analysed using the proposed method.     

Estimation of P(Y < X) for Weibull Distribution Under Progressive Type-II Censoring

Based on progressively Type II censored samples, we consider the estimation of R = P(Y < X) when X and Y are two independent Weibull distributions with different shape parameters, but having the same scale parameter. The maximum likelihood estimator, approximate maximum likelihood estimator, and Bayes estimator of R are obtained. Based on the asymptotic distribution of R, the confidence interval of R are obtained. Two bootstrap confidence intervals are also proposed. Analysis of a real data set is given for illustrative purposes. Monte Carlo simulations are also performed to compare the different proposed methods.     

Censored median regression and profile empirical likelihood

We implement profile empirical likelihood-based inference for censored median regression models. Inference for any specified subvector is carried out by profiling out the nuisance parameters from the “plug-in” empirical likelihood ratio function proposed by Qin and Tsao. To obtain the critical value of the profile empirical likelihood ratio statistic, we first investigate its asymptotic distribution. The limiting distribution is a sum of weighted chi square distributions. Unlike for the full empirical likelihood, however, the derived asymptotic distribution has intractable covariance structure. Therefore, we employ the bootstrap to obtain the critical value, and compare the resulting confidence intervals with the ones obtained through Basawa and Koul’s minimum dispersion statistic. Furthermore, we obtain confidence intervals for the age and treatment effects in a lung cancer data set.     

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.     

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.     

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.     

Efficient mean estimation in log-normal linear models

Log-normal linear models are widely used in applications, and many times it is of interest to predict the response variable or to estimate the mean of the response variable at the original scale for a new set of covariate values. In this paper we consider the problem of efficient estimation of the conditional mean of the response variable at the original scale for log-normal linear models. Several existing estimators are reviewed first, including the maximum likelihood (ML) estimator, the restricted ML (REML) estimator, the uniformly minimum variance unbiased (UMVU) estimator, and a bias-corrected REML estimator. We then propose two estimators that minimize the asymptotic mean squared error and the asymptotic bias, respectively. A parametric bootstrap procedure is also described to obtain confidence intervals for the proposed estimators. Both the new estimators and the bootstrap procedure are very easy to implement. Comparisons of the estimators using simulation studies suggest that our estimators perform better than the existing ones, and the bootstrap procedure yields confidence intervals with good coverage properties. A real application of estimating the mean sediment discharge is used to illustrate the methodology.     

A general bootstrap algorithm for hypothesis testing

The bootstrap is a intensive computer-based method originally mainly devoted to estimate the standard deviations, confidence intervals and bias of the studied statistic. This technique is useful in a wide variety of statistical procedures, however, its use for hypothesis testing, when the data structure is complex, is not straightforward and each case must be particularly treated. A general bootstrap method for hypothesis testing is studied. The considered method preserves the data structure of each group independently and the null hypothesis is only used in order to compute the bootstrap statistic values (not at the resampling, as usual). The asymptotic distribution is developed and several case studies are discussed.     

Estimation procedures for Maxwell distribution under type-I progressive hybrid censoring scheme

The Maxwell (or Maxwell–Boltzmann) distribution was invented to solve the problems relating to physics and chemistry. It has also proved its strength of analysing the lifetime data. For this distribution, we consider point and interval estimation procedures in the presence of type-I progressively hybrid censored data. We obtain maximum likelihood estimator of the parameter and provide asymptotic and bootstrap confidence intervals of it. The Bayes estimates and Bayesian credible and highest posterior density intervals are obtained using inverted gamma prior. The expression of the expected number of failures in life testing experiment is also derived. The results are illustrated through the simulation study and analysis of a real data set is presented.     

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.