Bootstrap confidence intervals for the mode of the hazard function

In many applications of lifetime data analysis, it is important to perform inferences about the mode of the hazard function in situations of lifetime data modeling with unimodal hazard functions. For lifetime distributions where the mode of the hazard function can be analytically calculated, its maximum likelihood estimator is easily obtained from the invariance properties of the maximum likelihood estimators. From the asymptotical normality of the maximum likelihood estimators, confidence intervals can be obtained. However, these results might not be very accurate for small sample sizes and/or large proportion of censored observations. Considering the log-logistic distribution for the lifetime data with shape parameter beta>1, we present and compare the accuracy of asymptotical confidence intervals with two confidence intervals based on bootstrap simulation. The alternative methodology of confidence intervals for the mode of the log-logistic hazard function are illustrated in three numerical examples.     

A comparative study of nonparametric estimation in Weibull regression: A penalized likelihood approach

The Weibull distribution is popularly used to model lifetime distributions in many areas of applied statistics. This paper employs a penalized likelihood method to estimate the shape parameter and an unknown regression function simultaneously in a nonparametric Weibull regression. Four methods were considered: two cross-validation methods, a corrected Akaike information criterion, and a Bayesian information criterion. Each method was evaluated based on shape parameter estimation as well as selecting the smoothing parameter in a penalized likelihood model through a simulation study. Adapting a lower-dimensional approximation and deriving confidence intervals from Bayes models of the penalized likelihood, the comparative performances of methods using both censored and uncensored data were examined for various censoring rates. The methods are applied to a real data example of lung cancer.     

Estimation in step-stress accelerated life tests for the exponentiated exponential distribution with type-I censoring

The step-stress accelerated life tests allow the experimenter to increase the stress levels at fixed times during the experiment. The lifetime of a product at any level of stress is assumed to have an exponentiated distribution, whose baseline distribution is a general class of distributions which includes, among others, Weibull, compound Weibull, Pareto, Gompertz, normal and logistic distributions. The scale parameter of the baseline distribution is assumed to be a log-linear function of the stress and a cumulative exposure model holds. Special attention is paid to an exponentiated exponential distribution. Based on type-I censoring, the maximum likelihood estimates of the parameters under consideration are obtained. A Monte Carlo simulation study is carried out to investigate the precision of the maximum likelihood estimates and to obtain the coverage probabilities of the bootstrap confidence intervals for the parameters involved. Finally, an example is presented to illustrate the two discussed methods of bootstrap confidence intervals.     

The exponentiated exponential mixture and non-mixture cure rate model in the presence of covariates

This paper presents estimates for the parameters included in long-term mixture and non-mixture lifetime models, applied to analyze survival data when some individuals may never experience the event of interest. We consider the case where the lifetime data have a two-parameters exponentiated exponential distribution. The two-parameter exponentiated exponential or the generalized exponential distribution is a particular member of the exponentiated Weibull distribution introduced by [31]. Classical and Bayesian procedures are used to get point and confidence intervals of the unknown parameters. We consider a general survival model where the scale, shape and cured fraction parameters of the exponentiated exponential distribution depends on covariates.     

A Simulation Tool for Efficient Analogy Based Cost Estimation

Estimation of a software project effort, based on project analogies, is a promising method in the area of software cost estimation. Projects in a historical database, that are analogous (similar) to the project under examination, are detected, and their effort data are used to produce estimates. As in all software cost estimation approaches, important decisions must be made regarding certain parameters, in order to calibrate with local data and obtain reliable estimates. In this paper, we present a statistical simulation tool, namely the bootstrap method, which helps the user in tuning the analogy approach before application to real projects. This is an essential step of the method, because if inappropriate values for the parameters are selected in the first place, the estimate will be inevitably wrong. Additionally, we show how measures of accuracy and in particular, confidence intervals, may be computed for the analogy-based estimates, using the bootstrap method with different assumptions about the population distribution of the data set. Estimate confidence intervals are necessary in order to assess point estimate accuracy and assist risk analysis and project planning. Examples of bootstrap confidence intervals and a comparison with regression models are presented on well-known cost data sets.     

Quantifying effects in two-sample environmental experiments using bootstrap confidence intervals

Two-sample experiments (paired or unpaired) are often used to analyze treatment effects in life and environmental sciences. Quantifying an effect can be achieved by estimating the difference in center of location between a treated and a control sample. In unpaired experiments, a shift in scale is also of interest. Non-normal data distributions can thereby impose a serious challenge for obtaining accurate confidence intervals for treatment effects. To study the effects of non-normality we analyzed robust and non-robust measures of treatment effects: differences of averages, medians, standard deviations, and normalized median absolute deviations in case of unpaired experiments, and average of differences and median of differences in case of paired experiments. A Monte Carlo study using bivariate lognormal distributions was carried out to evaluate coverage performances and lengths of four types of nonparametric bootstrap confidence intervals, namely normal, Student's t, percentile, and BCa for the estimated measures. The robust measures produced smaller coverage errors than their non-robust counterparts. On the other hand, the robust versions gave average confidence interval lengths approximately 1.5 times larger. In unpaired experiments, BCa confidence intervals performed best, while in paired experiments, Student's t was as good as BCa intervals. Monte Carlo results are discussed and recommendations on data sizes are presented. In an application to physiological source–sink manipulation experiments with sunflower, we quantify the effect of an increased or decreased source–sink ratio on the percentage of unfilled grains and the dry mass of a grain. In an application to laboratory experiments with wastewater, we quantify the disinfection effect of predatory microorganisms. The presented bootstrap method to compare two samples is broadly applicable to measured or modeled data from the entire range of environmental research and beyond.     

Confidence intervals for median survival time with recurrent event data

Several methods of constructing confidence intervals for the median survival time of a recurrent event data are developed. One of them is based on asymptotic variances estimated using some transformations. Others are based on bootstrap techniques. Two types of recurrent event models are considered: the first one is a model where the inter-event times are independent and identically distributed, and the second one is a model where the inter-event times are associated, with the association arising from a gamma frailty model. Bootstrap and asymptotic confidence intervals are studied through simulation. These methods are applied and compared using two real data sets arising in the biomedical and public health settings, using an available R package. The first example belongs to data from a study concerning small bowel motility where an independent model may be assumed. The second example involves hospital readmissions in patients diagnosed with colorectal cancer. In this example the interoccurrence times are correlated.     

Evaluation of several nonparametric bootstrap methods to estimate confidence intervals for software metrics

Sample statistics and model parameters can be used to infer the properties, or characteristics, of the underlying population in typical data-analytic situations. Confidence intervals can provide an estimate of the range within which the true value of the statistic lies. A narrow confidence interval implies low variability of the statistic, justifying a strong conclusion made from the analysis. Many statistics used in software metrics analysis do not come with theoretical formulas to allow such accuracy assessment. The Efron bootstrap statistical analysis appears to address this weakness. In this paper, we present an empirical analysis of the reliability of several Efron nonparametric bootstrap methods in assessing the accuracy of sample statistics in the context of software metrics. A brief review on the basic concept of various methods available for the estimation of statistical errors is provided, with the stated advantages of the Efron bootstrap discussed. Validations of several different bootstrap algorithms are performed across basic software metrics in both simulated and industrial software engineering contexts. It was found that the 90 percent confidence intervals for mean, median, and Spearman correlation coefficients were accurately predicted. The 90 percent confidence intervals for the variance and Pearson correlation coefficients were typically underestimated (60-70 percent confidence interval), and those for skewness and kurtosis overestimated (98-100 percent confidence interval). It was found that the Bias-corrected and accelerated bootstrap approach gave the most consistent confidence intervals, but its accuracy depended on the metric examined. A method for correcting the under-/ overestimation of bootstrap confidence intervals for small data sets is suggested, but the success of the approach was found to be inconsistent across the tested metrics.     

Model based bootstrap methods for interval censored data

The performance of model based bootstrap methods for constructing point-wise confidence intervals around the survival function with interval censored data is investigated. It is shown that bootstrapping from the nonparametric maximum likelihood estimator of the survival function is inconsistent for the current status model. A model based smoothed bootstrap procedure is proposed and proved to be consistent. In fact, a general framework for proving the consistency of any model based bootstrap scheme in the current status model is established. In addition, simulation studies are conducted to illustrate the (in)-consistency of different bootstrap methods in mixed case interval censoring. The conclusions in the interval censoring model would extend more generally to estimators in regression models that exhibit non-standard rates of convergence.     

On the efficiency of bootstrap method into the analysis contingency table

The bootstrap method is a computer intensive statistical method that is widely used in performing nonparametric inference. Categorical data analysis, in particular the analysis of contingency tables, is commonly used in applied field. This work considers nonparametric bootstrap tests for the analysis of contingency tables. There are only a few research papers which exploit this field. The p-values of tests in contingency tables are discrete and should be uniformly distributed under the null hypothesis. The results of this article show that corresponding bootstrap versions work better than the standard tests. Properties of the proposed tests are illustrated and discussed using Monte Carlo simulations. This article concludes with an analytical example that examines the performance of the proposed tests and the confidence interval of the association coefficient.     

Bootstrap hypothesis testing for some common statistical problems: A critical evaluation of size and power properties

The construction of bootstrap hypothesis tests can differ from that of bootstrap confidence intervals because of the need to generate the bootstrap distribution of test statistics under a specific null hypothesis. Similarly, bootstrap power calculations rely on resampling being carried out under specific alternatives. We describe and develop null and alternative resampling schemes for common scenarios, constructing bootstrap tests for the correlation coefficient, variance, and regression/ANOVA models. Bootstrap power calculations for these scenarios are described. In some cases, null-resampling bootstrap tests are equivalent to tests based on appropriately constructed bootstrap confidence intervals. In other cases, particularly those for which simple percentile-method bootstrap intervals are in routine use such as the correlation coefficient, null-resampling tests differ from interval-based tests. We critically assess the performance of bootstrap tests, examining size and power properties of the tests numerically using both real and simulated data. Where they differ from tests based on bootstrap confidence intervals, null-resampling tests have reasonable size properties, outperforming tests based on bootstrapping without regard to the null hypothesis. The bootstrap tests also have reasonable power properties.     

Bootstrap hypothesis testing for some common statistical problems: A critical evaluation of size and power properties

The construction of bootstrap hypothesis tests can differ from that of bootstrap confidence intervals because of the need to generate the bootstrap distribution of test statistics under a specific null hypothesis. Similarly, bootstrap power calculations rely on resampling being carried out under specific alternatives. We describe and develop null and alternative resampling schemes for common scenarios, constructing bootstrap tests for the correlation coefficient, variance, and regression/ANOVA models. Bootstrap power calculations for these scenarios are described. In some cases, null-resampling bootstrap tests are equivalent to tests based on appropriately constructed bootstrap confidence intervals. In other cases, particularly those for which simple percentile-method bootstrap intervals are in routine use such as the correlation coefficient, null-resampling tests differ from interval-based tests. We critically assess the performance of bootstrap tests, examining size and power properties of the tests numerically using both real and simulated data. Where they differ from tests based on bootstrap confidence intervals, null-resampling tests have reasonable size properties, outperforming tests based on bootstrapping without regard to the null hypothesis. The bootstrap tests also have reasonable power properties.     

Confidence Interval Estimation for Inequality Indices of the Gini Family

In this paper we present some nonparametric bootstrap methods to constructdistribution-free confidence intervals for inequality indices belonging to theGini family. These methods have a coverage accuracy better than that obtainedwith the asymptotic distribution of their natural estimators, typically thestandard normal. The coverage performances of these methods are evaluated forthe index R by Gini with a Monte Carlo experiment on samples simulated fromthe Dagum income model (Type I), which is usually used to describe the incomedistribution.     

Estimating reliability in proportional odds ratio models

In this paper, we are mainly interested in inference on the reliability coefficient, R=P(X<Y), in proportional odds ratio models based on the new family of tilted survival functions introduced by Marshall and Olkin [Marshall, A.W., Olkin, I., 1997. A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84 (3), 641-652]. We also present some results on stochastic comparison between the survival distribution functions. Asymptotic and various bootstrap confidence intervals of R are investigated. The performance of asymptotic and bootstrap confidence intervals is studied through a simulation. A numerical example based on real-life data is presented to illustrate the implementation of the proposed procedure.     

Simultaneous confidence intervals for ratios of means of several lognormal distributions: A parametric bootstrap approach

For constructing simultaneous confidence intervals for the ratios of means of several lognormal distributions, we propose a new parametric bootstrap method, which is different from an inaccurate parametric bootstrap method previously considered in the literature. Our proposed method is conceptually simpler than other proposed methods, which are based on the concepts of generalized pivotal quantities and fiducial generalized pivotal quantities. Also, our extensive simulation results indicate that our proposed method consistently performs better than other methods: its coverage probability is close to the nominal confidence level and the resulting intervals are typically shorter than the intervals produced by other methods.     

Constant-partially accelerated life tests for Burr type-XII distribution with progressive type-II censoring

Based on progressively type-II censored samples, constant-partially accelerated life tests (PALTs) when the lifetime of items under use condition follow the two-parameter Burr type-XII (Burr(c,k)) distribution are considered. The likelihood equations of the involved parameters are derived and then reduced to a single nonlinear equation to be solved numerically to obtain the maximum likelihood estimates (MLEs) of the parameters. The observed Fisher information matrix, as well as the asymptotic variance-covariance matrix of the MLEs are derived. Approximate confidence intervals (CIs) for the parameters, based on normal approximation to the asymptotic distribution of MLEs, studentized-t and percentile bootstrap CIs are derived. A Monte Carlo simulation study is carried out to investigate the precision of the MLEs and to compare the performance of the CIs considered. Finally, two examples presented to illustrate our results are followed by conclusions.     

Weighted bootstrap method for fuzzy data

The weighted bootstrap contained in the monograph by Barbe and Bertail in Lecture Notes in Statist, Springer (1995) is a simple and straight-forward method for calculating approximated biases, standard deviations, confidence intervals, and so forth, in almost any nonparametric estimation problem. In this paper, we consider another example, namely, fuzzy data, and use the weighted bootstrap to answer several questions concerning the minimum inaccuracy estimator (Corral and Gil in Stochastica 8:63–81, 1984): (a) What is the standard error of this estimator? (b) What is a reasonable confidence interval for such a estimate? The validity of weighted bootstrap method is investigated using a real data and computer simulation.     

A semiparametric bootstrap approach to correlated data analysis problems

In this note, we outline a simple to use yet powerful bootstrap algorithm for handling correlated outcome variables in terms of either hypothesis testing or confidence intervals using only the marginal models. This new method can handle combinations of continuous and discrete data and can be used in conjunction with other covariates in a model. The procedure is based upon estimating the family-wise error (FWE) rate and then making a Bonferroni-type correction. A simulation study illustrates the accuracy of the algorithm over a variety of correlation structures.     

Inference for a simple step-stress model with competing risks for failure from the exponential distribution under time constraint

In reliability analysis, accelerated life-testing allows for gradual increment of stress levels on test units during an experiment. In a special class of accelerated life tests known as step-stress tests, the stress levels increase discretely at pre-fixed time points, and this allows the experimenter to obtain information on the parameters of the lifetime distributions more quickly than under normal operating conditions. Moreover, when a test unit fails, there are often more than one fatal cause for the failure, such as mechanical or electrical. In this article, we consider the simple step-stress model under time constraint when the lifetime distributions of the different risk factors are independently exponentially distributed. Under this setup, we derive the maximum likelihood estimators (MLEs) of the unknown mean parameters of the different causes under the assumption of a cumulative exposure model. Since it is found that the MLEs do not exist when there is no failure by any particular risk factor within the specified time frame, the exact sampling distributions of the MLEs are derived through the use of conditional moment generating functions. Using these exact distributions as well as the asymptotic distributions, the parametric bootstrap method, and the Bayesian posterior distribution, we discuss the construction of confidence intervals and credible intervals for the parameters. Their performance is assessed through Monte Carlo simulations and finally, we illustrate the methods of inference discussed here with an example.