Smoothed Bootstrap Method Research Articles

Smoothed Bootstrap Methods for Hypothesis Testing

This paper demonstrates the application of smoothed bootstrap methods and Efron’s methods for hypothesis testing on real-valued data, right-censored data and bivariate data. The tests include quartile hypothesis tests, two sample medians and Pearson and Kendall correlation tests. Simulation studies indicate that the smoothed bootstrap methods outperform Efron’s methods in most scenarios, particularly for small datasets. The smoothed bootstrap methods provide smaller discrepancies between the actual and nominal error rates, which makes them more reliable for testing hypotheses.

Smoothed bootstrap for right-censored data

A smoothed bootstrap method is introduced for right-censored data based on the right-censoring-A(n) assumption introduced by Coolen and Yan, which is a generalization of Hill’s A(n) assumption for right-censored data. The smoothed bootstrap method is compared to Efron’s method for right-censored data through simulations. The comparison is conducted in terms of the coverage of percentile confidence intervals for the quartiles. From the study, it is found that the smoothed bootstrap method mostly performs better than Efron’s method, in particular for small data sets. We also illustrate the use of the method for survival function inference and compare it to a smoothed Kaplan-Meier bootstrap method through simulations.

Nonparametric quantile regression with missing data using local estimating equations

In this paper, we propose augmented inverse probability weighted (AIPW) local estimating equations in dealing with missing data in nonparametric quantile regression context. The missing mechanism here is missing at random. To avoid the problem of misspecification, we adopt nonparametric approach to estimate the propensity score and conditional expectations of estimating functions. The asymptotic properties of our proposed estimator are studied. Majorisation–minimisation algorithm is used to circumvent the nonsmoothness of check function at the origin. When it comes to the choice of bandwidth, the theoretical expression of local optimal bandwidth is derived based on asymptotic properties. Moreover, we apply smoothed bootstrap method to obtain the empirical mean square error and use cross-validation to determine the bandwidth in practice. Simulations are conducted to compare the performance of our proposed methods with other existing methods. Finally, we illustrate our methodology with an analysis of non-insulin-dependent diabetes mellitus data set.

Symmetric smoothed bootstrap methods for ranked set samples

Ranked set sampling (RSS) is a data collection scheme that usually yields more efficient estimators when a measurement of the variable of interest is difficult or expensive to obtain, but sampling units can be ordered easily without actual quantification. We suggest several natural methods for obtaining smoothed bootstrap samples from each row of RSS data, and show the consistency of these methods for a location parameter. Our results hold even if the ranking is imperfect. Furthermore, we propose an optimal bandwidth that minimises the asymptotic mean integrated squared error of the RSS-based kernel cumulative distribution estimator. Then, we use simulations to verify the accuracy of the percentile confidence interval for the population mean for each bootstrap method. Lastly, we apply the smoothed bootstrap method to the test of symmetry.

Estimate Accuracy of the Sample Determination Coefficient R2 and R2 for the Multiple Regression Models

In this paper, we propose estimating standard errors for R2 and R2 and to construct their confidence intervals, using the usual and “smoothed bootstrap methods”, which are accurate measures. It is shown by ”Monte Carlo experiments” that the smoothed bootstrap standard errors are more accurate estimates of usual bootstrap method. It is also shown that although the usual and smoothed bootstrap 0.95 confidence intervals of R2 do not include the true value of the parent coefficient of determination in some particular cases, such a phenomenon does not occur when is used.

Smoothed bootstrap bandwidth selection for nonparametric hazard rate estimation

ABSTRACTA 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.

Estimation and inference in generalized additive coefficient models for nonlinear interactions with high-dimensional covariates

In the low-dimensional case, the generalized additive coefficient model (GACM) proposed by Xue and Yang [Statist. Sinica16 (2006) 1423-1446] has been demonstrated to be a powerful tool for studying nonlinear interaction effects of variables. In this paper, we propose estimation and inference procedures for the GACM when the dimension of the variables is high. Specifically, we propose a groupwise penalization based procedure to distinguish significant covariates for the "large p small n" setting. The procedure is shown to be consistent for model structure identification. Further, we construct simultaneous confidence bands for the coefficient functions in the selected model based on a refined two-step spline estimator. We also discuss how to choose the tuning parameters. To estimate the standard deviation of the functional estimator, we adopt the smoothed bootstrap method. We conduct simulation experiments to evaluate the numerical performance of the proposed methods and analyze an obesity data set from a genome-wide association study as an illustration.

Smoothed and iterated bootstrap confidence regions for parameter vectors

The construction of confidence regions for parameter vectors is a difficult problem in the nonparametric setting, particularly when the sample size is not large. We focus on bootstrap ellipsoidal confidence regions. The bootstrap has shown promise in solving this problem, but empirical evidence often indicates that the bootstrap percentile method has difficulty in maintaining the correct coverage probability, while the bootstrap percentile- t method may be unstable, often resulting in very large confidence regions. This paper considers the smoothed and iterated bootstrap methods to construct the bootstrap percentile method ellipsoidal confidence region. The smoothed bootstrap method is based on a multivariate kernel density estimator. An optimal bandwidth matrix is established for the smoothed bootstrap procedure that reduces the coverage error of the bootstrap percentile method. We also provide an analytical adjustment to the nominal level to reduce the computational cost of the iterated bootstrap method. Simulations demonstrate that the methods can be successfully applied in practice.

Bootstraps for Meta-Analysis with an Application to the Impact of Climate Change

Bootstrap and smoothed bootstrap methods are used to estimate the uncertainty about the total impact of climate change, and to assess the performance of commonly used impact functions. Kernel regression is extended to include restrictions on the functional form. Impact functions do not describe the primary estimates of the economic impacts very well, and monotonic functions do particularly badly. The impacts of climate change do not significantly deviate from zero until 2.5---3.5 $$^{\circ }\hbox {C}$$?C warming. The uncertainty is large, and so is the risk premium. The ambiguity premium is small, however. The certainty equivalent impact is a negative 1.5 % of income for $$2.5\,^{\circ }\hbox {C}$$2.5?C, rising to 15 % (50 %) for $$5.0\,^{\circ }\hbox {C}$$5.0?C for a rate of risk aversion of 1 (2).

Ranking efficiency of Asian container ports: a bootstrapped frontier approach

This paper employs a smoothed homogenous bootstrapped frontier approach to obtain data envelopment analysis (DEA) efficiency estimates for a more reliable efficiency ranking of Asian container ports. The paper applies desirable statistical properties to DEA efficiency estimates pertaining to consistency, unbiasedness, and estimation of confidence intervals. The paper uses a three-pronged analysis, first by identifying outliers using the log-ratio analysis; second, testing hypothesis with regard to global returns to scale employing a bootstrap methodology; and third, implementing the smooth homogenous bootstrap method to derive bias-corrected DEA efficiency estimates and to build confidence intervals. The results of the study enable individual ports to assess whether their production is efficient compared to their counterparts and indicate strategies for efficiency improvement.

Statistical Evaluation of Single-Photon Emission Computed Tomography Image Using Smoothed Bootstrap Method

Many of the neurodegenerative diseases associated with a decrease in regional cerebral blood flow (rCBF) are untreatable, and the appropriate therapeutic strategy is to slow the progression of the disease. Therefore, it is important that a definitive diagnosis is made as soon as possible when such diseases are suspected. Diagnostic imaging methods, such as positron emission tomography (PET) and single-photon emission computed tomography (SPECT), play an important role in such a definitive diagnosis. Since several problems arise when evaluating these images visually, a procedure to evaluate them objectively is necessary, and studies of image analyses using statistical evaluations have been suggested. However, the assumed data distribution in a statistical procedure may occasionally be inappropriate. Therefore, to evaluate the decrease of rCBF, it is important to use a statistical procedure without assumptions about the data distribution. In this study, we propose a new procedure that uses nonparametric or smoothed bootstrap methods to calculate a standardized distribution of the Z-score without assumptions about the data distribution. To test whether the judgment of the proposed procedure is equivalent to that of an evaluation based on the Z-score with a fixed threshold, the procedure was applied to a sample data set whose size was large enough to be appropriate for the assumption of the Z-score. As a result, the evaluations of the proposed procedure were equivalent to that of an evaluation based on the Z-score.

Productivity of travel agencies in Spain: the case of Alicante

The main objective of this study is to analyse the productivity of travel agencies located in Alicante (Spain) in order to obtain results that will complement other previous findings based on efficiency. Malmquist indices and the smoothed bootstrap method have been selected to carry out this analysis. In addition, the Mann–Whitney U test has been adopted to study the relationships between levels of productivity, ownership type, location and experience. In the light of this new perspective, the results do not present an optimistic view of the evolution of the economic activity of the agencies. Furthermore, some of the variables, such as location, have a different effect.

Smoothed nonparametric estimation for current status competing risks data

We study the nonparametric estimation of the cumulative incidence function and the cause-specific hazard function for current status data with competing risks via kernel smoothing. A smoothed naive nonparametric maximum likelihood estimator and a smoothed full nonparametric maximum likelihood estimator are shown to have pointwise asymptotic normality and faster convergence rates than the corresponding unsmoothed nonparametric likelihood estimators. Using the smoothed estimators and the plug-in principle, we can estimate the cause-specific hazard function, which has not been studied previously. We also propose semi-smoothed estimators of the cause-specific hazard as an alternative to the smoothed estimator and demonstrate that neither is uniformly more efficient than the other. Numerical studies show that a smoothed bootstrap method works well for selecting the bandwidths in the smoothed nonparametric estimation. The use of the estimators is exemplified by an application to cumulative incidence and hazard of subtype-specific HIV infection from a sero-prevalence study in injecting drug users in Thailand. Copyright 2013, Oxford University Press.

Travel agencies in Alicante, Spain: a productivity analysis

PurposeThe purpose of this paper is to provide lines of action based on the results aimed at improving the future levels of productivity of the travel agencies in Alicante.Design/methodology/approachEstimates of output‐oriented Malmquist Productivity Indices and the values of their components will be calculated. Furthermore, in order to avoid problems that may arise from these results (which are based on finite samples that are sensitive to sampling variations) bootstrapping techniques (smoothed bootstrap) are used. Lastly, the Mann‐Whitney‐U test is used in order to accept or reject the influence that different factors have on productivity levels.FindingsThis study is a continuation of previous studies analysing the variation productivity of 22 travel agencies located in Alicante (Spain) over the period 2008‐2010. The results obtained from applying the afore‐mentioned methods do not provide substantially different conclusions to those obtained in previous studies. The authors can observe that the technological component (T) is the main factor, which these companies use to sustain their productivity levels and, thanks to this, the negative effects of a poor application of economies of scale are neutralised. In light of this, and for the whole of the period 2008‐2010, the inappropriate management of resources and the inability to obtain higher output levels played an insignificant role. Perhaps, a higher amount of information online about both economics and efficiency could be a good strategy to help these agencies to improve their results.Research limitations/implicationsThe impossibility of increasing the amount of data for carrying out the study, given the scarcity of statistical sources available and the reticence of the travel agencies to provide us with more information.Originality/valueAlthough other studies have been previously carried out using Malmquist indices and bootstrapping techniques to analyse the productivity of travel agencies in other countries or cities, this is the first to be undertaken in Spain using the smoothed bootstrap method based on output‐oriented models, so as to examine this field of study from a new perspective.

Galaxy morphology, luminosity, and environment in the SDSS DR7

We study the influence of the environment on the evolution of galaxies by investigating the luminosity function (LF) of galaxies of different morphological types and colours at different environmental density levels. We construct the LFs separately for galaxies of different morphology (spiral and elliptical) and of different colours (red and blue) using data from the Sloan Digital Sky Survey (SDSS), correcting the luminosities for the intrinsic absorption. We use the global luminosity density field to define different environments, and analyse the environmental dependence of galaxy morphology and colour. The smoothed bootstrap method is used to calculate confidence regions of the derived luminosity functions. We find a strong environmental dependency for the LF of elliptical galaxies. The LF of spiral galaxies is almost environment independent, suggesting that spiral galaxy formation mechanisms are similar in different environments. Absorption by the intrinsic dust influences the bright-end of the LF of spiral galaxies. After attenuation correction, the brightest spiral galaxies are still about 0.5 mag less luminous than the brightest elliptical galaxies, except in the least dense environment, where spiral galaxies dominate the LF at every luminosity. Despite the extent of the SDSS survey, the influence of single rich superclusters is present in the galactic LF of the densest environment.

TWO‐SAMPLE PROBLEMS IN STATISTICAL DATA MODELLING

A common problem in mathematical statistics is to check whether two samples differ from each other. From modelling point of view it is possible to make a statistical test for the equality of two means or alternatively two distribution functions. The second approach allows to represent the two‐sample test graphically. This can be done by adding simultaneous confidence bands to the probability‐probability (P — P) or quantile‐quantile (Q — Q) plots. In this paper we compare empirically the accuracy of the classical two‐sample t‐test, empirical likelihood method and several bootstrap methods. For a real data example both Q — Q and P — P plots with simultaneous confidence bands have been plotted using the smoothed empirical likelihood and smoothed bootstrap methods.

On the Assessment of Regulators' Efficiency: An Application to European Telecommunications

This paper offers a methodology to assess the internal productive efficiency of National Regulatory Authorities (NRAs) based on the performances of regulated markets, measured in terms of the degree of market efficiency (either static or dynamic). The estimation procedure is based on a Data Envelopment Analysis (DEA), along with a smoothed bootstrap method and it is applied to telecommunications sector across 18 European countries, 5 of which are new accession countries, in 2005. After the discussion of several desirable outcomes for a telecom regulator, we construct an ad hoc database containing information about NRAs regulatory inputs and outputs. We run three bootstrapped DEAs in order to rank NRAs according to their efficiency in carrying out their regulatory activities. We find the NRAs in 2004 accession countries are more efficient in pursuing dynamic efficiency goals than the more experienced NRAs, while they perform generally worse when the regulatory outcomes are measured in terms of retail efficiency.

Measures of influence for the functional linear model with scalar response

This paper studies how to identify influential observations in the functional linear model in which the predictor is functional and the response is scalar. Measurement of the effects of a single observation on estimation and prediction when the model is estimated by the principal components method is undertaken. For that, three statistics are introduced for measuring the influence of each observation on estimation and prediction of the functional linear model with scalar response that are generalizations of the measures proposed for the standard regression model by [D.R. Cook, Detection of influential observations in linear regression, Technometrics 19 (1977) 15–18; D. Peña, A new statistic for influence in linear regression, Technometrics 47 (2005) 1–12] respectively. A smoothed bootstrap method is proposed to estimate the quantiles of the influence measures, which allows us to point out which observations have the larger influence on estimation and prediction. The behavior of the three statistics and the quantile estimation bootstrap based method is analyzed via a simulation study. Finally, the practical use of the proposed statistics is illustrated by the analysis of a real data example, which show that the proposed measures are useful for detecting heterogeneity in the functional linear model with scalar response.

A smoothed bootstrap test for independence based on mutual information

A test for independence of multivariate time series based on the mutual information measure is proposed. First of all, a test for independence between two variables based on i.i.d. (time-independent) data is constructed and is then extended to incorporate higher dimensions and strictly stationary time series data. The smoothed bootstrap method is used to estimate the null distribution of mutual information. The experimental results reveal that the proposed smoothed bootstrap test performs better than the existing tests and can achieve high powers even for moderate dependence structures. Finally, the proposed test is applied to assess the actual independence of components obtained from independent component analysis (ICA).

Ranking efficiency units in DEA using bootstrapping an applied analysis for Slovenian farm data

AbstractThis article explores how data envelopment analysis (DEA), along with a smoothed bootstrap method, can be used in applied analysis to obtain more reliable efficiency rankings for farms. The main focus is the smoothed homogeneous bootstrap procedure introduced by Simar and Wilson (1998) to implement statistical inference for the original efficiency point estimates. Two main model specifications, constant and variable returns to scale, are investigated along with various choices regarding data aggregation. The coefficient of separation (CoS), a statistic that indicates the degree of statistical differentiation within the sample, is used to demonstrate the findings. The CoS suggests a substantive dependency of the results on the methodology and assumptions employed. Accordingly, some observations are made on how to conduct DEA in order to get more reliable efficiency rankings, depending on the purpose for which they are to be used. In addition, attention is drawn to the ability of the SLICE MODEL, implemented in GAMS, to enable researchers to overcome the computational burdens of conducting DEA (with bootstrapping).