Parametric Bootstrap Procedure Research Articles

Inference for finite-sample trajectories in dynamic multi-state site-occupancy models using hidden Markov model smoothing

Ecologists and wildlife biologists increasingly use latent variable models to study patterns of species occurrence when detection is imperfect. These models have recently been generalized to accommodate both a more expansive description of state than simple presence or absence, and Markovian dynamics in the latent state over successive sampling seasons. In this paper, we write these multi-season, multi-state models as hidden Markov models to find both maximum likelihood estimates of model parameters and finite-sample estimators of the trajectory of the latent state over time. These estimators are especially useful for characterizing population trends in species of conservation concern. We also develop parametric bootstrap procedures that allow formal inference about latent trend. We examine model behavior through simulation, and we apply the model to data from the North American Amphibian Monitoring Program.

A COMPARISON OF ALTERNATIVE APPROACHES TO SUPREMUM-NORM GOODNESS-OF-FIT TESTS WITH ESTIMATED PARAMETERS

Goodness-of-fit tests based on parametric empirical processes have nonstandard limiting distributions when the null hypothesis is composite — that is, when parameters of the null model are estimated. Several analytic solutions to this problem have been suggested, including the calculation of adjusted critical values for these nonstandard distributions and the transformation of the empirical process such that statistics based on the transformed process are asymptotically distribution-free. The approximation methods proposed by Durbin (1985, Journal of Applied Probability 22(1), 99–122) can be applied to conduct inference for tests based on supremum-norm statistics. The resulting tests have quite accurate size, a fact that has gone unrecognized in the econometrics literature. Some justification for this accuracy lies in the similar features that Durbin’s approximation methods share with the theory of extrema for Gaussian random fields and for Gauss-Markov processes. These adjustment techniques are also related to the transformation methodology proposed by Khmaladze (1981, Theory of Probability and Its Applications26(2), 240–257) through the score function of the parametric model. Simulation experiments suggest that in small samples, Durbin-style adjustments result in tests that have higher power than tests based on transformed processes, and in some cases they have higher power than parametric bootstrap procedures.

Bootstrap Point Optimal Unit Root Tests

In this article, we investigate and compare the behaviour of some bootstrap unit root tests in finite ARMA models with a constant and/or a trend and use them to obtain asymptotic results for the point optimal (hereafter PO) test, in terms of both size and power. We demonstrate the asymptotic validity of bootstrapping the PO test. We provide a feasible method for obtaining approximate critical values for the PO unit root test. Through simulations, we investigate how effective the bootstrap is in different sample sizes, correlative coefficients and close unity autoregressive roots in two different models. Our main objective is to show that the bootstrap PO test can be used in regression models with AR and MA errors and trending regressors. The results reported here provide an analytical investigation of the use of the bootstrap for PO tests with dependent data.The main contribution of this article has two features. First, we choose the PO test and make this powerful but unfeasible procedure both powerful and feasible, by plugging in a consistent estimation of the coefficient structure, and we show that the bootstrap PO test provides asymptotically valid critical values. Second, through simulation, our numerical results suggest that the bootstrap PO test performs well in having the correct size properties and retaining good power in the parametric (and semi-parametric) bootstrap procedure.

On maximum likelihood estimation of the long-memory parameter in fractional Gaussian noise

Approximate normality and unbiasedness of the maximum likelihood estimate (MLE) of the long-memory parameter H of a fractional Brownian motion hold reasonably well for sample sizes as small as 20 if the mean and scale parameter are known. We show in a Monte Carlo study that if the latter two parameters are unknown the bias and variance of the MLE of H both increase substantially. We also show that the bias can be reduced by using a parametric bootstrap procedure. In very large samples, maximum likelihood estimation becomes problematic because of the large dimension of the covariance matrix that must be inverted. To overcome this difficulty, we propose a maximum likelihood method based upon first differences of the data. These first differences form a short-memory process. We split the data into a number of contiguous blocks consisting of a relatively small number of observations. Computation of the likelihood function in a block then presents no computational problem. We form a pseudo-likelihood function consisting of the product of the likelihood functions in each of the blocks and provide a formula for the standard error of the resulting estimator of H. This formula is shown in a Monte Carlo study to provide a good approximation to the true standard error. The computation time required to obtain the estimate and its standard error from large data sets is an order of magnitude less than that required to obtain the widely used Whittle estimator. Application of the methodology is illustrated on two data sets.

Mapping Cancer Risk in Southwestern Ontario with Changing Census Boundaries

Mapping disease risk often involves working with data that have been spatially aggregated to census regions or postal regions, either for administrative reasons or confidentiality. When studying rare diseases, data must be collected over a long time period in order to accumulate a meaningful number of cases. These long time periods can result in spatial boundaries of the census regions changing over time, as is the case with the motivating example of exploring the spatial structure of mesothelioma lung cancer risk in Lambton County and Middlesex County of southwestern Ontario, Canada. This article presents a local-EM kernel smoothing algorithm that allows for the combining of data from different spatial maps, being capable of modeling risk for spatially aggregated data with time-varying boundaries. Inference and uncertainty estimates are carried out with parametric bootstrap procedures, and cross-validation is used for bandwidth selection. Results for the lung cancer study are shown and discussed.

Small area estimation with spatio-temporal Fay–Herriot models

Small area estimation is studied under a spatio-temporal Fay–Herriot model. Model fitting based on restricted maximum likelihood is described and empirical best linear unbiased predictors are derived under the model. A parametric bootstrap procedure is proposed for the estimation of the mean squared error of the small area estimators. The spatio-temporal model is compared with simpler models through simulation experiments, analyzing the gain in efficiency achieved by the use of the more complex model. The performance of the parametric bootstrap estimator of the mean squared error is also assessed. An application with Spanish EU-SILC data is carried out to obtain estimates of poverty indicators for Spanish provinces in 2008, making use of survey data from years 2004–2008.

A gravity model to account for vertical linkages between markets with an application to the cattle/beef sector

A gravity model is developed to explain bilateral trade flows in primary and processed commodities within the same agri-food supply chain. It accounts for vertical production linkages, trade and domestic policies, and supply rigidities at the farm level. Our application focuses on cattle/beef trade flows between 42 countries. The estimated parameters of the model are used to simulate trade flows. We found large differences in the impacts of the full and partial liberalization scenarios. A parametric bootstrap procedure is used to generate confidence intervals around predicted trade liberalization outcomes.

Effects of climate change on an emperor penguin population: analysis of coupled demographic and climate models

Sea ice conditions in the Antarctic affect the life cycle of the emperor penguin (Aptenodytes forsteri). We present a population projection for the emperor penguin population of Terre Adélie, Antarctica, by linking demographic models (stage-structured, seasonal, nonlinear, two-sex matrix population models) to sea ice forecasts from an ensemble of IPCC climate models. Based on maximum likelihood capture-mark-recapture analysis, we find that seasonal sea ice concentration anomalies (SICa ) affect adult survival and breeding success. Demographic models show that both deterministic and stochastic population growth rates are maximized at intermediate values of annual SICa , because neither the complete absence of sea ice, nor heavy and persistent sea ice, would provide satisfactory conditions for the emperor penguin. We show that under some conditions the stochastic growth rate is positively affected by the variance in SICa . We identify an ensemble of five general circulation climate models whose output closely matches the historical record of sea ice concentration in Terre Adélie. The output of this ensemble is used to produce stochastic forecasts of SICa , which in turn drive the population model. Uncertainty is included by incorporating multiple climate models and by a parametric bootstrap procedure that includes parameter uncertainty due to both model selection and estimation error. The median of these simulations predicts a decline of the Terre Adélie emperor penguin population of 81% by the year 2100. We find a 43% chance of an even greater decline, of 90% or more. The uncertainty in population projections reflects large differences among climate models in their forecasts of future sea ice conditions. One such model predicts population increases over much of the century, but overall, the ensemble of models predicts that population declines are far more likely than population increases. We conclude that climate change is a significant risk for the emperor penguin. Our analytical approach, in which demographic models are linked to IPCC climate models, is powerful and generally applicable to other species and systems.

A unified procedure for meta-analytic evaluation of surrogate end points in randomized clinical trials

The meta-analytic approach to evaluating surrogate end points assesses the predictiveness of treatment effect on the surrogate toward treatment effect on the clinical end point based on multiple clinical trials. Definition and estimation of the correlation of treatment effects were developed in linear mixed models and later extended to binary or failure time outcomes on a case-by-case basis. In a general regression setting that covers nonnormal outcomes, we discuss in this paper several metrics that are useful in the meta-analytic evaluation of surrogacy. We propose a unified 3-step procedure to assess these metrics in settings with binary end points, time-to-event outcomes, or repeated measures. First, the joint distribution of estimated treatment effects is ascertained by an estimating equation approach; second, the restricted maximum likelihood method is used to estimate the means and the variance components of the random treatment effects; finally, confidence intervals are constructed by a parametric bootstrap procedure. The proposed method is evaluated by simulations and applications to 2 clinical trials.

Gene-environment interactions on growth trajectories.

It has been suggested that children with larger brains tend to perform better on IQ tests or cognitive function tests. Prenatal head growth and head growth in infancy are two crucial periods for subsequent intelligence. Studies have shown that environmental exposure to air pollutants during pregnancy is associated with fetal growth reduction, developmental delay, and reduced IQ. Meanwhile, genetic polymorphisms may modify the effect of environment on head growth. However, studies on gene-environment or gene-gene interactions on growth trajectories have been quite limited partly due to the difficulty to quantitatively measure interactions on growth trajectories. Moreover, it is known that assessing the significance of gene-environment or gene-gene interactions on cross-sectional outcomes empirically using the permutation procedures may bring substantial errors in the tests. We proposed a score that quantitatively measures interactions on growth trajectories and developed an algorithm with a parametric bootstrap procedure to empirically assess the significance of the interactions on growth trajectories under the likelihood framework. We also derived a Wald statistic to test for interactions on growth trajectories and compared it to the proposed parametric bootstrap procedure. Through extensive simulation studies, we demonstrated the feasibility and power of the proposed testing procedures. We applied our method to a real dataset with head circumference measures from birth to age 7 on a cohort currently being conducted by the Columbia Center for Children's Environmental Health (CCCEH) in Krakow, Poland, and identified several significant gene-environment interactions on head circumference growth trajectories.

Interval estimation of the negative binomial dispersion parameter

Inference concerning the negative binomial dispersion parameter, denoted by c, is important in many biological and biomedical investigations. Properties of the maximum-likelihood estimator of c and its bias-corrected version have been studied extensively, mainly, in terms of bias and efficiency [W.W. Piegorsch, Maximum likelihood estimation for the negative binomial dispersion parameter, Biometrics 46 (1990), pp. 863–867; S.J. Clark and J.N. Perry, Estimation of the negative binomial parameter κ by maximum quasi-likelihood, Biometrics 45 (1989), pp. 309–316; K.K. Saha and S.R. Paul, Bias corrected maximum likelihood estimator of the negative binomial dispersion parameter, Biometrics 61 (2005), pp. 179–185]. However, not much work has been done on the construction of confidence intervals (C.I.s) for c. The purpose of this paper is to study the behaviour of some C.I. procedures for c. We study, by simulations, three Wald type C.I. procedures based on the asymptotic distribution of the method of moments estimate (mme), the maximum-likelihood estimate (mle) and the bias-corrected mle (bcmle) [K.K. Saha and S.R. Paul, Bias corrected maximum likelihood estimator of the negative binomial dispersion parameter, Biometrics 61 (2005), pp. 179–185] of c. All three methods show serious under-coverage. We further study parametric bootstrap procedures based on these estimates of c, which significantly improve the coverage probabilities. The bootstrap C.I.s based on the mle (Boot-MLE method) and the bcmle (Boot-BCM method) have coverages that are significantly better (empirical coverage close to the nominal coverage) than the corresponding bootstrap C.I. based on the mme, especially for small sample size and highly over-dispersed data. However, simulation results on lengths of the C.I.s show evidence that all three bootstrap procedures have larger average coverage lengths. Therefore, for practical data analysis, the bootstrap C.I. Boot-MLE or Boot-BCM should be used, although Boot-MLE method seems to be preferable over the Boot-BCM method in terms of both coverage and length. Furthermore, Boot-MLE needs less computation than Boot-BCM.

Nonparametric Bootstrap for Quasi-Likelihood Ratio Tests

We introduce a nonparametric block bootstrap approach for Quasi-Likelihood Ratio type tests of nonlinear restrictions. Our method applies to extremum estimators, such as quasi-maximum likelihood and generalized method of moments estimators. Unlike existing parametric bootstrap procedures for Quasi-Likelihood Ratio type tests, our procedure constructs bootstrap samples in a fully nonparametric way. We study the higher order properties of our nonparametric block bootstrap and show the asymptotic refinements implied with respect to the standard asymptotic theory. Our approach delivers the same higher order properties of the nonparametric block bootstrap methods introduced in Andrews (2002) and Kim (2003) in relation to Wald and Lagrange Multiplier tests, respectively. Monte Carlo simulations confirm the accuracy of our bootstrap procedure.

Resolving statistical uncertainty in correlation dimension estimation

In this paper we propose a novel method for obtaining standard errors and confidence intervals for the correlation dimension estimated on an observed chaotic time series. This method is based on the U-Statistics theory and an ingenious combination of the moving block and parametric bootstrap procedures. We test the method on the basis of computer simulations for both clean and noisy series. We show that the distribution of the correlation dimension estimate obtained by our method agrees very well with the "true" distribution obtained by the Monte Carlo simulation. One of the main advantage of our method is the ability to estimate the distribution (and hence, the standard error) of the correlation dimension estimate using only one observed time series.

Effect of non‐normality on test statistics for one‐way independent groups designs

The data obtained from one-way independent groups designs is typically non-normal in form and rarely equally variable across treatment populations (i.e., population variances are heterogeneous). Consequently, the classical test statistic that is used to assess statistical significance (i.e., the analysis of variance F test) typically provides invalid results (e.g., too many Type I errors, reduced power). For this reason, there has been considerable interest in finding a test statistic that is appropriate under conditions of non-normality and variance heterogeneity. Previously recommended procedures for analysing such data include the James test, the Welch test applied either to the usual least squares estimators of central tendency and variability, or the Welch test with robust estimators (i.e., trimmed means and Winsorized variances). A new statistic proposed by Krishnamoorthy, Lu, and Mathew, intended to deal with heterogeneous variances, though not non-normality, uses a parametric bootstrap procedure. In their investigation of the parametric bootstrap test, the authors examined its operating characteristics under limited conditions and did not compare it to the Welch test based on robust estimators. Thus, we investigated how the parametric bootstrap procedure and a modified parametric bootstrap procedure based on trimmed means perform relative to previously recommended procedures when data are non-normal and heterogeneous. The results indicated that the tests based on trimmed means offer the best Type I error control and power when variances are unequal and at least some of the distribution shapes are non-normal.

Confidence Intervals for Reliability of Stress-Strength Models in the Normal Case

In this article, we propose some procedures to get confidence intervals for the reliability in stress-strength models. The confidence intervals are obtained either through a parametric bootstrap procedure or using asymptotic results, and are applied to the particular context of two independent normal random variables. The performance of these estimators and other known approximate estimators are empirically checked through a simulation study which considers several scenarios.

Validity of the bootstrap in the critical process with a non-stationary immigration

In the critical branching process with a stationary immigration, the standard parametric bootstrap for an estimator of the offspring mean is invalid. We consider the process with non-stationary immigration, whose mean and variance α(n) and β(n) are finite for each n≥1 and are regularly varying sequences with non-negative exponents α and β, respectively. We prove that if α(n)→∞ and β (n)=o(nα2(n)) as n→∞, then the standard parametric bootstrap procedure leads to a valid approximation for the distribution of the conditional least-squares estimator in the sense of convergence in probability. Monte Carlo and bootstrap simulations for the process confirm the theoretical findings in the paper and highlight the validity and utility of the bootstrap as it mimics the Monte Carlo pivots even when generation size is small.

Fund Rating Model Based on Finite Normal Mixture Distribution

This paper proposes a new method of fund rating based on the cross-sectional distribution of fund performance measured by alpha. This distribution-based fund rating model is more flexible and provides more interesting results than current commercial fund rating method used by Morningstar. Unlike Morningstar's rating, this method does not use preset percentiles to rate funds. It is the distribution of alpha that dictates the number of performance groups in a given fund category and time period. The framework is based on the crucial assumption that the expected fund performance may be different, and the difference of the expected fund performance arises from the segmented market information and/or the differentiated ability of mangers to acquire and analyze information. The multimodal shape and formal normality tests prompt us to model the distribution of alpha by finite normal mixture model. We introduce the parametric bootstrap procedure to determine the number of performance groups in the model. We then employ expectation and maximization (EM) algorithm to estimate the model. Based on the estimated posterior probabilities of the fund, we assign the rating to funds. Our empirical results show that the number of performance groups is not fixed and varies across time and fund categories. We observe a clear tendency of the merging of information sets, which implies that the fund market has become gradually more efficient over time as information was well transmitted and analyzed.

A random effects variance shift model for detecting and accommodating outliers in meta-analysis

BackgroundMeta-analysis typically involves combining the estimates from independent studies in order to estimate a parameter of interest across a population of studies. However, outliers often occur even under the random effects model. The presence of such outliers could substantially alter the conclusions in a meta-analysis. This paper proposes a methodology for identifying and, if desired, downweighting studies that do not appear representative of the population they are thought to represent under the random effects model.MethodsAn outlier is taken as an observation (study result) with an inflated random effect variance. We used the likelihood ratio test statistic as an objective measure for determining whether observations have inflated variance and are therefore considered outliers. A parametric bootstrap procedure was used to obtain the sampling distribution of the likelihood ratio test statistics and to account for multiple testing. Our methods were applied to three illustrative and contrasting meta-analytic data sets.ResultsFor the three meta-analytic data sets our methods gave robust inferences when the identified outliers were downweighted.ConclusionsThe proposed methodology provides a means to identify and, if desired, downweight outliers in meta-analysis. It does not eliminate them from the analysis however and we consider the proposed approach preferable to simply removing any or all apparently outlying results. We do not however propose that our methods in any way replace or diminish the standard random effects methodology that has proved so useful, rather they are helpful when used in conjunction with the random effects model.

A goodness-of-fit test for bivariate extreme-value copulas

It is often reasonable to assume that the dependence structure of a bivariate continuous distribution belongs to the class of extreme-value copulas. The latter are characterized by their Pickands dependence function. In this paper, a procedure is proposed for testing whether this function belongs to a given parametric family. The test is based on a Cramér–von Mises statistic measuring the distance between an estimate of the parametric Pickands dependence function and either one of two nonparametric estimators thereof studied by Genest and Segers [Ann. Statist. 37 (2009) 2990–3022]. As the limiting distribution of the test statistic depends on unknown parameters, it must be estimated via a parametric bootstrap procedure, the validity of which is established. Monte Carlo simulations are used to assess the power of the test and an extension to dependence structures that are left-tail decreasing in both variables is considered.

A Variance-Components Model for Distance-Matrix Phylogenetic Reconstruction

Phylogenetic trees describe evolutionary relationships between related organisms (taxa). One approach to estimating phylogenetic trees supposes that a matrix of estimated evolutionary distances between taxa is available. Agglomerative methods have been proposed in which closely related taxon-pairs are successively combined to form ancestral taxa. Several of these computationally efficient agglomerative algorithms involve steps to reduce the variance in estimated distances. We propose an agglomerative phylogenetic method which focuses on statistical modeling of variance components in distance estimates. We consider how these variance components evolve during the agglomerative process. Our method simultaneously produces two topologically identical rooted trees, one tree having branch lengths proportional to elapsed time, and the other having branch lengths proportional to underlying evolutionary divergence. The method models two major sources of variation which have been separately discussed in the literature: noise, reflecting inaccuracies in measuring divergences, and distortion, reflecting randomness in the amounts of divergence in different parts of the tree. The methodology is based on successive hierarchical generalized least-squares regressions. It involves only means, variances and covariances of distance estimates, thereby avoiding full distributional assumptions. Exploitation of the algebraic structure of the estimation leads to an algorithm with computational complexity comparable to the leading published agglomerative methods. A parametric bootstrap procedure allows full uncertainty in the phylogenetic reconstruction to be assessed. Software implementing the methodology may be freely downloaded from StatTree.