Nonparametric confidence intervals for population variance of one sample and the difference of variances of two samples

Confidence intervals for the population variance and the difference in variances of two populations based on the ordinary t-statistics combined with the bootstrap method are suggested. Theoretical and practical aspects of the suggested techniques are presented, as well as their comparison with existing methods (methods based on Chi-square statistics and F-statistics). In addition, application of presented methods in domain of insurance property data set is described and analyzed. For data from exponential distribution confidence intervals, which are calculated using described methods (based on transformation of the t-statistics and bootstrap technique), give consistent and best coverage in comparison with other methods.     

Estimation of reliability in a series system with random sample size

In this paper, we are interested in the estimation of the reliability coefficient R=P(X>Y), when the data on the minimum of two exponential samples, with random sample size, are available. The confidence intervals of R, based on maximum likelihood and bootstrap methods, are developed. The performance of these confidence intervals is studied through extensive simulation. A numerical example, based on a real data, is presented to illustrate the implementation of the proposed procedure.     

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 based on the percentile bootstrap approach

This note concerns the construction of bootstrap simultaneous confidence intervals (SCI) for m parameters. Given B bootstrap samples, we suggest an algorithm with complexity of O(mBlog(B)). We apply our algorithm to construct a confidence region for time dependent probabilities of progression in multiple sclerosis and for coefficients in a logistic regression analysis. Alternative normal based simultaneous confidence intervals are presented and compared to the bootstrap intervals.     

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.     

Estimation comparison on busy period for a controllable M/G/1 system with bicriterion policy

We propose an estimator for the expected busy period (denoted by β) of a controllable M/G/1 queueing system in which the server applies a bicriterion 〈p, N〉 policy during his idle period. Using this estimator, we construct new confidence intervals for β, which are based on five bootstrap methods; standard bootstrap (SB), percentile bootstrap (PB), bootstrap pivotal (BP), bias-corrected percentile bootstrap (BCPB), bias-corrected and accelerated bootstrap (BCa). A numerical simulation study is conducted in order to demonstrate performance of the proposed estimator $\hat{\beta }$ and bootstrap confidence intervals for β. From the simulation results, we show that $\hat{\beta }$ is a consistent estimator for β, which agrees with the theoretical results. In addition, we also investigate the accuracy of the five bootstrap confidence intervals by calculating the coverage percentage and the relative coverage (defined as the ratio of coverage percentage to average length of confidence interval). Detailed discussions of simulation results for three queueing models are presented.     

A goodness of fit test for copulas based on Rosenblatt's transformation

A goodness of fit test for copulas based on Rosenblatt's transformation is investigated. This test performs well if the marginal distribution functions are known and are used in the test statistic. If the marginal distribution functions are unknown and are replaced by their empirical estimates, then the test's properties change significantly. This is shown in detail by simulation for special cases. A bootstrap version of the test is suggested and it is shown by simulation that it performs well. An empirical application of this test to daily returns of German assets reveals that a Gaussian copula is unsuitable to describe their dependence structure. A t_ν-copula with low degrees of freedom such as ν=4 or 5 fits the data in some cases.     

T-sum of sigmoid-shaped fuzzy intervals

The usual arithmetic operations on real numbers can be extended to arithmetical operations on fuzzy intervals by means of Zadeh’s extension principle based on a t-norm T. A t-norm is called consistent with respect to a class of fuzzy intervals for some arithmetic operation, if this arithmetic operation is closed for this class. It is important to know which t-norms are consistent with particular types of fuzzy intervals. Recently, Dombi and Gy?rbíró [J. Dombi, N. Gy?rbíró, Additions of sigmoid-shaped fuzzy intervals using the Dombi operator and infinite sum theorems, Fuzzy Sets and Systems 157 (2006) 952-963] proved that addition is closed if the Dombi t-norm is used with sigmoid-shaped fuzzy intervals. In this paper, we define a broader class of sigmoid-shaped fuzzy intervals. Then, we study t-norms that are consistent with these particular types of fuzzy intervals. Dombi and Gy?rbíró’s results are special cases of the results described in this paper.     

Type I multivariate zero-inflated Poisson distribution with applications

Motivated from the stochastic representation of the univariate zero-inflated Poisson (ZIP) random variable, the authors propose a multivariate ZIP distribution, called as Type I multivariate ZIP distribution, to model correlated multivariate count data with extra zeros. The distributional theory and associated properties are developed. Maximum likelihood estimates for parameters of interest are obtained by Fisher’s scoring algorithm and the expectation–maximization (EM) algorithm, respectively. Asymptotic and bootstrap confidence intervals of parameters are provided. Likelihood ratio test and score test are derived and are compared via simulation studies. Bayesian methods are also presented if prior information on parameters is available. Two real data sets are used to illustrate the proposed methods. Under both AIC and BIC, our analysis of the two data sets supports the Type I multivariate zero-inflated Poisson model as a much less complex alternative with feasibility to the existing multivariate ZIP models proposed by Li et al. (Technometrics, 29–38, Vol 41, 1999).     

Bootstrap confidence intervals for principal response curves

The principal response curve (PRC) model is of use to analyse multivariate data resulting from experiments involving repeated sampling in time. The time-dependent treatment effects are represented by PRCs, which are functional in nature. The sample PRCs can be estimated using a raw approach, or the newly proposed smooth approach. The generalisability of the sample PRCs can be judged using confidence bands. The quality of various bootstrap strategies to estimate such confidence bands for PRCs is evaluated. The best coverage was obtained with BC_a intervals using a non-parametric bootstrap. The coverage appeared to be generally good, except for the case of exactly zero population PRCs for all conditions. Then, the behaviour is irregular, which is caused by the sign indeterminacy of the PRCs. The insights obtained into the optimal bootstrap strategy are useful to apply in the PRC model, and more generally for estimating confidence intervals in singular value decomposition based methods.     

An empirical study of a test for polynomial relationships in randomly right censored regression models

In this paper, a test statistic is constructed to test polynomial relationships in randomly right censored regression models based on the local polynomial smoothing technique. Two bootstrap procedures, namely the residual-based bootstrap and the naive bootstrap procedures, are suggested to derive the p-value of the test. Some simulations are conducted to empirically assess the performance of the two bootstrap procedures. The results demonstrate that the residual-based bootstrap performs much better than the naive bootstrap and the test method with the residual-based bootstrap to derive the p-value works satisfactorily. Although the limiting distribution of the test statistic and the consistency of the bootstrap approximations remain to be investigated, simulation results indicate that the proposed test method may be of some practical use. As a real example, the proposed test is applied to the Stanford heart transplant data.     

L-class enumeration algorithms for a discrete production planning problem with interval resource quantities

A discrete production planning problem which may be formulated as the multidimensional knapsack problem is considered, while resource quantities of the problem are supposed to be given as intervals. An approach for solving this problem based on using its relaxation set is suggested. Some L-class enumeration algorithms for the problem are described. Results of computational experiments are presented.     

Comparing medians

Many methods have been proposed for comparing the medians of J independent groups. Generally, however, extant techniques require very restrictive assumptions or they are known to perform in an unsatisfactory manner in simulations. Included are many well-known rank-based methods plus certain types of bootstrap techniques. One goal here is to point out that two recently proposed methods also perform poorly when there are tied values. Another goal is to examine the small sample properties of several alternative methods that have not been previously studied. The main result is that a multiple comparison technique (called method R), is the only method to perform well in all the situations considered here. For an omnibus test with J>2 groups and no tied values, two methods are found that control Type I error probabilities reasonably well, one of which is based in part on results in Liu and Singh [1997. Notions of limiting p-values based on data depth and bootstrap. J. Amer. Statist. Assoc. 92, 266-277]. With tied values, the second method is found to be more satisfactory, but even it can perform poorly.     

Local bootstrap approaches for fractional differential parameter estimation in ARFIMA models

In this paper we investigate bootstrap techniques applied to the estimation of the fractional differential parameter in ARFIMA models, d. The novelty is the focus on the local bootstrap of the periodogram function. The approach is then applied to three different semiparametric estimators of d, known from the literature, based upon the periodogram function. By means of an extensive set of simulation experiments, the bias and mean square errors are quantified for each estimator and the efficacy of the local bootstrap is stated in terms of low bias, short confidence intervals, and low CPU times. Finally, a real data set is analyzed to demonstrate that the methodology may be quite effective in solving real problems.     

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.     

A flexible approximate likelihood ratio test for detecting differential expression in microarray data

Identifying differentially expressed genes in microarray data has been studied extensively and several methods have been proposed. Most popular methods in the study of gene expression microarray data analysis rely on normal distribution assumption and are based on a Wald statistic. These methods may be inefficient when expression levels follow a skewed distribution. To deal with possible violations of the normality assumption, we propose a method based on Generalized Logistic Distribution of Type II (GLDII). The motivation behind this distributional assumption is to allow longer tails than normal distribution. This is important in analyzing gene expression data since extreme values are common in such experiments. The shape parameter for GLDII allows flexibility in modeling a wide range of distributions. To simplify the computational complexity involved in carrying out Likelihood Ratio (LR) tests for several thousands of genes, an Approximate LR Test (ALRT) is proposed. We also generalize the two-class ALRT method to multi-class microarray data. The performance of the ALRT method under the GLDII assumption is compared to methods based on Wald-type statistics using simulation. The results from the simulations show that our method performs quite well compared to the significance analysis of microarrays (SAM) approach using standardized Wilcoxon rank statistics and the empirical Bayes (E-B) t-statistics. Our method is also less sensitive to extreme values. We illustrate our method using two publicly available gene expression data sets.     

Approximately calibrated small sample inference about means from bivariate normal data with missing values

Frequency properties of approximate Bayesian posterior probability intervals are considered for a small bivariate sample with data missing on one variable. For a class of priors, approximations to the posterior distribution based on matching moments of the t distribution are derived, and compared with the true distributions computed numerically. Coverage properties of highest posterior density intervals for three choices of prior are evaluated by simulation, and compared with other solutions. The simulations suggest that a second moment t approximation combined with the Jeffreys' prior for the bivariate distribution provides intervals that are quite well calibrated, in the sense of having approximate or slightly conservative coverage for a wide range of values of the underlying parameters. The use of calibration to select a suitable reference prior seems to have potential for a large number of problems.     

Using the bootstrap for finite sample confidence intervals of the log periodogram regression

Log periodogram regression is widely applied in empirical applications to estimate the memory parameter, d, of long memory time series. This estimator is consistent for d<1 and pivotal asymptotically normal for d<3/4. However, the asymptotic distribution is a poor approximation of the (unknown) finite sample distribution if the sample size is small. Finite sample improvements in the construction of confidence intervals can be achieved by different nonparametric bootstrap procedures based on the residuals of log periodogram regression. In addition to the basic residual bootstrap, the local and block bootstraps seem adequate for replicating the structure that may arise in the errors of the regression when the series shows weak dependence in addition to long memory. The performances of different bias correcting bootstrap techniques and a bias reduced log periodogram regression are also analyzed with a view to adjusting the bias caused by that structure. Finally, an application to the Nelson and Plosser US macroeconomic data is included.     

Model selection by bootstrap penalization for classification

We consider the binary classification problem. Given an i.i.d. sample drawn from the distribution of an χ×{0,1}?valued random pair, we propose to estimate the so-called Bayes classifier by minimizing the sum of the empirical classification error and a penalty term based on Efron’s or i.i.d. weighted bootstrap samples of the data. We obtain exponential inequalities for such bootstrap type penalties, which allow us to derive non-asymptotic properties for the corresponding estimators. In particular, we prove that these estimators achieve the global minimax risk over sets of functions built from Vapnik-Chervonenkis classes. The obtained results generalize Koltchinskii (2001) and Bartlett et al.’s (2002) ones for Rademacher penalties that can thus be seen as special examples of bootstrap type penalties. To illustrate this, we carry out an experimental study in which we compare the different methods for an intervals model selection problem.