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.     

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.     

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.     

Fuzzy estimation for process capability indices

Process capability indices are summary statistics which measure the actual or the potential performance of process characteristics relative to the target and specification limits. In most traditional methods, precise estimation is used to assess the capability of manufacturing processes. In this paper we introduce an algorithm based on Buckley’s estimation approach, and use a family of confidence intervals to estimate process capability indices C_p, C_pk and C_pm. The estimators of these indices thus obtained are triangular shaped fuzzy numbers. We also present and illustrate method for the comparison of estimated process capability indices. Numerical examples are given to show the performance of the method.     

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.     

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.     

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.     

Optimal interval estimation fusion based on sensor interval estimates with confidence degrees

The interval estimation fusion method based on sensor interval estimates and their confidence degrees is developed. When sensor estimates are independent of each other, a combination rule to merge sensor estimates and their confidence degrees is proposed. Moreover, two optimization criteria: minimizing interval length with an allowable minimum confidence degree, or maximizing confidence degree with an allowable maximum interval length are suggested. In terms of the two criteria, an optimal interval estimation fusion can be obtained based on the combined intervals and their confidence degrees. Then we can extend the results on the combined interval outputs and their confidence degrees to obtain a conditional combination rule and the corresponding optimal fault-tolerant interval estimation fusion in terms of the two criteria. It is easy to see that Marzullo's fault-tolerant interval estimation fusion [Marzullo, (1990). Tolerating failures of continuous-valued sensors. ACM Transactions on Computer System, 8(4), 284-304] is a special case of our method.     

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.     

Comparing cost prediction models by resampling techniques

The accurate software cost prediction is a research topic that has attracted much of the interest of the software engineering community during the latest decades. A large part of the research efforts involves the development of statistical models based on historical data. Since there are a lot of models that can be fitted to certain data, a crucial issue is the selection of the most efficient prediction model. Most often this selection is based on comparisons of various accuracy measures that are functions of the model’s relative errors. However, the usual practice is to consider as the most accurate prediction model the one providing the best accuracy measure without testing if this superiority is in fact statistically significant. This policy can lead to unstable and erroneous conclusions since a small change in the data is able to turn over the best model selection. On the other hand, the accuracy measures used in practice are statistics with unknown probability distributions, making the testing of any hypothesis, by the traditional parametric methods, problematic. In this paper, the use of statistical simulation tools is proposed in order to test the significance of the difference between the accuracy of two prediction methods: regression and estimation by analogy. The statistical simulation procedures involve permutation tests and bootstrap techniques for the construction of confidence intervals for the difference of measures. Four known datasets are used for experimentation in order to validate the results and make comparisons between the simulation methods and the traditional parametric and non-parametric procedures.     

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.     

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.     

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.     

Half-life estimation based on the bias-corrected bootstrap: A highest density region approach

The half-life is defined as the number of periods required for the impulse response to a unit shock to a time series to dissipate by half. It is widely used as a measure of persistence, especially in international economics to quantify the degree of mean-reversion of the deviation from an international parity condition. Several studies have proposed bias-corrected point and interval estimation methods. However, they have found that the confidence intervals are rather uninformative with their upper bound being either extremely large or infinite. This is largely due to the distribution of the half-life estimator being heavily skewed and multi-modal. A bias-corrected bootstrap procedure for the estimation of half-life is proposed, adopting the highest density region (HDR) approach to point and interval estimation. The Monte Carlo simulation results reveal that the bias-corrected bootstrap HDR method provides an accurate point estimator, as well as tight confidence intervals with superior coverage properties to those of its alternatives. As an application, the proposed method is employed for half-life estimation of the real exchange rates of 17 industrialized countries. The results indicate much faster rates of mean-reversion than those reported in previous studies.     

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.