Parametric Bootstrap Procedure Research Articles

Bootstrap test procedure for variance components in nonlinear mixed effects models in the presence of nuisance parameters and a singular Fisher information matrix

Summary We examine the problem of variance component testing in general mixed effects models using the likelihood ratio test. We account for the presence of nuisance parameters, ie, the fact that some untested variances might also be equal to zero. Two main issues arise in this context, leading to a nonregular setting. First, under the null hypothesis, the true parameter value lies on the boundary of the parameter space. Moreover, due to the presence of nuisance parameters, the exact locations of these boundary points are not known, which prevents the use of classical asymptotic theory of maximum likelihood estimation. Then, in the specific context of nonlinear mixed effects models, the Fisher information matrix is singular at the true parameter value. We address these two points by proposing a shrunk parametric bootstrap procedure, which is straightforward to apply even for nonlinear models. We show that the procedure is consistent, solving both the boundary and the singularity issues, and we provide a verifiable criterion for the applicability of our theoretical results. We show through a simulation study that, compared to the asymptotic approach, our procedure has a better small sample performance and is more robust to the presence of nuisance parameters. A real data application on bird growth rates is also provided.

Testing the Presence of Implicit Hiring Quotas with Application to German Universities

Abstract Women are underrepresented in academia in general and economics in particular. I introduce a test to detect an underresearched form of hiring bias: implicit quotas. I derive a test under the null hypothesis of gender-blind hiring that requires no additional information about individual hires and can be used to analyze hiring bias in a variety of other hiring settings. I derive its asymptotic distribution and propose a parametric bootstrap procedure that resamples from the exact distribution. I analyze the distribution of female professors at German universities and find an implicit quota of one or two women on the department level.

A parametric bootstrap approach for computing confidence intervals for genetic correlations with application to genetically determined protein-protein networks

Genetic correlation refers to the correlation between genetic determinants of a pair of traits. When using individual-level data, it is typically estimated based on a bivariate model specification where the correlation between the two variables is identifiable and can be estimated from a covariance model that incorporates the genetic relationship between individuals, e.g., using a pre-specified kinship matrix. Inference relying on asymptotic normality of the genetic correlation parameter estimates may be inaccurate when the sample size is low, when the genetic correlation is close to the boundary of the parameter space, and when the heritability of at least one of the traits is low. We address this problem by developing a parametric bootstrap procedure to construct confidence intervals for genetic correlation estimates. The procedure simulates paired traits under a range of heritability and genetic correlation parameters, and it uses the population structure encapsulated by the kinship matrix. Heritabilities and genetic correlations are estimated using the close-form, method of moment, Haseman-Elston regression estimators. The proposed parametric bootstrap procedure is especially useful when genetic correlations are computed on pairs of thousands of traits measured on the same exact set of individuals. We demonstrate the parametric bootstrap approach on a proteomics dataset from the Jackson Heart Study.

The Reliability Inference for Multicomponent Stress–Strength Model under the Burr X Distribution

The reliability of the multicomponent stress–strength system was investigated under the two-parameter Burr X distribution model. Based on the structure of the system, the type II censored sample of strength and random sample of stress were obtained for the study. The maximum likelihood estimators were established by utilizing the type II censored Burr X distributed strength and complete random stress data sets collected from the multicomponent system. Two related approximate confidence intervals were achieved by utilizing the delta method under the asymptotic normal distribution theory and parametric bootstrap procedure. Meanwhile, point and confidence interval estimators based on alternative generalized pivotal quantities were derived. Furthermore, a likelihood ratio test to infer the equality of both scalar parameters is provided. Finally, a practical example is provided for illustration.

Bootstrap tests for unbiasedness of predictors

In this paper, the analysis is focused on the unbiasedness of prediction based on a linear regression model. The accuracy of the considered predictors is measured by means of a statistical test based on the ex-post prediction variance. A significantly large value of the ratio of ex-post prediction variances and residual variance leads to the rejection of the hypothesis on unbiasedness of prediction. The distribution of the ratio is assessed by means of the parametric bootstrap procedure. Moreover, bootstrap-type tests, attempting to verify the unbiasedness of three predictors of the linear regression, are proposed. The distribution of the linear regression error term is assumed to be normal or t-Student. The power of considered statistical tests is examined based on an extensive simulation analysis.

Heteroscedastic Growth Curve Modeling with Shape-Restricted Splines

Growth curve analysis (GCA) has a wide range of applications in various fields where growth trajectories need to be modeled. Heteroscedasticity is often present in the error term, which can not be handled with sufficient flexibility by standard linear fixed or mixed-effects models. One situation that has been addressed is where the error variance is characterized by a linear predictor with certain covariates. A frequently encountered scenario in GCA, however, is one in which the variance is a smooth function of the mean with known shape restrictions. A naive application of standard linear mixed-effects models would underestimate the variance of the fixed effects estimators and, consequently, the uncertainty of the estimated growth curve. We propose to model the variance of the response variable as a shape-restricted (increasing/decreasing; convex/concave) function of the marginal or conditional mean using shape-restricted splines. A simple iteratively reweighted fitting algorithm that takes advantage of existing software for linear mixed-effects models is developed. For inference, a parametric bootstrap procedure is recommended. Our simulation study shows that the proposed method gives satisfactory inference with moderate sample sizes. The utility of the method is demonstrated using two real-world applications.

Regional pooling in extreme event attribution studies: an approach based on multiple statistical testing

Statistical methods are proposed to select homogeneous regions when analyzing spatial block maxima data, such as in extreme event attribution studies. Here, homogeneitity refers to the fact that marginal model parameters are the same at different locations from the region. The methods are based on classical hypothesis testing using Wald-type test statistics, with critical values obtained from suitable parametric bootstrap procedures and corrected for multiplicity. A large-scale Monte Carlo simulation study finds that the methods are able to accurately identify homogeneous locations, and that pooling the selected locations improves the accuracy of subsequent statistical analyses. The approach is illustrated with a case study on precipitation extremes in Western Europe. The methods are implemented in an R package that allows for easy application in future extreme event attribution studies.

Robust Semi-Parametric Inference for Two-Stage Production Models: A Beta Regression Approach

The data envelopment analysis is related to a non-parametric mathematical tool used to assess the relative efficiency of productive units. In different studies on productive efficiency, it is common to employ semi-parametric procedures in two stages to determine whether any exogenous factors of interest affect the performance of productive units. However, some of these procedures, particularly those based on conventional statistical inference, generate inconsistent estimates when dealing with incoherent data-generating processes. This inconsistency arises due to the efficiency scores being limited to the unit interval, and the estimated scores often exhibit serial correlation and have limited observations. To address such inconsistency, several strategies have been suggested, with the most well-known being an algorithm based on a parametric bootstrap procedure using the truncated normal distribution and its regression model. In this work, we present a modification of this algorithm that utilizes the beta distribution and its regression structure. The beta model allows for better accommodation of asymmetry in the data distribution. Our proposed algorithm introduces inferential characteristics that are superior to the original algorithm, resulting in a more statistically coherent data-generating process and improving the consistency property. We have conducted computational experiments that demonstrate the improved results achieved by our proposal.

Estimating Intra-Regional Inequality with an Application to German Spatial Planning Regions

Abstract Income inequality is a persistent topic of public and political debate. In this context, the focus often shifts from the national level to a more detailed geographical level. In particular, inequality between or within local communities can be assessed. In this article, the estimation of inequality within regions, that is, between households, is considered at a regionally disaggregated level. From a methodological point of view, a small area estimation of the Gini coefficient is carried out using an area-level model linking survey data with related administrative data. Specifically, the Fay-Herriot model is applied using a logit-transformation followed by a bias-corrected back-transformation. The uncertainty of the point estimate is assessed using a parametric bootstrap procedure to estimate the mean squared error. The validity of the methodology is shown in a model-based simulation for the point estimator as well as for the uncertainty measure. The proposed methodology is illustrated by estimating model-based Gini coefficients for spatial planning regions in Germany, using survey data from the Socio-Economic Panel and aggregate data from the 2011 Census. The results show that intra-regional inequality is more diverse than a consideration only between East and West suggests.

Dynamic response analysis for the eastern North Pacific gray whale population: an alternative approach

Gerrodette and DeMaster (1990) conclude that dynamic response analysis indicates that the gray whale population passed through its maximum net productivity level (MNPL, approximately equivalent to MSY level) between 1967 and 1980. Their conclusion is examined using models for population trends which permit a point of inflection; these are fitted globally to the time series of census estimates available up to 1987-88. A cubic and a logistic model are used. The cubic model results indicate with almost 100% confidence that the population passed through MNPL within two years of 1973-74. However, both this conclusion and that of Gerrodette and DeMaster are considered to be unreliable. This is because the curves fitted by both analyses correspond to markedly decreasing population sizes over parts of the periods to which they apply. This is inconsistent with plausible population dynamics behaviour, which is itself an underlying pre-requisite for dynamic response analysis methodology. A suggestion is made as to how applications of dynamic response analysis methodology such as that of Boveng et al. (1988) could be adapted to ensure the necessary respect of such constraints. Results of a parametric bootstrap procedure for confidence interval estimation applied to the logistic model indicate that the probability that the population passed through MNPL during the period of the censuses is not large. The census data are scarcely adequate to allow for reliable estimates of the curvature of the population trajectory to be made. The logistic model dynamic response analysis indicates that there is a somewhat greater likelihood that the gray whale population was below rather than above its MNPL in 1990, given the data available at the time.

A New Two-Sided Class of Lifetime Distributions: Applications to Complete and Right Censored Data

In this article, we first define a new two-sided distribution called the two-sided Kumaraswamy distribution and then we propose a generalized class of lifetime distributions via compounding two-sided Kumaraswamy and a baseline distribution. One of the advantages of this class of new distributions is that they can be unimodal or bimodal. The general model is specified by taking the exponential distribution as the baseline distribution. Some basic properties of the proposed distribution are derived. The model parameters are estimated by means of maximum likelihood method. In addition, parametric and non-parametric bootstrap procedures are used to obtain point estimates and confidence intervals of the parameters of the model. A simulation study has been conducted to examine the bias and the mean square error of the maximum likelihood estimators. We illustrate the performance of the proposed distribution by means of two real data sets (one is complete data set and other is right censored data set) and both the data sets show that the new distribution is more appropriate as compared to Weibull, gamma, weighted exponential, generalized two-sided exponential, generalized transmuted two-sided exponential and generalized exponential distributions.

A goodness-of-fit test for the compound Poisson exponential model

On the basis of bivariate data, assumed to be observations of independent copies of a random vector (S,N), we consider testing the hypothesis that the distribution of (S,N) belongs to the parametric class of distributions that arise with the compound Poisson exponential model. Typically, this model is used in stochastic hydrology, with N as the number of raindays, and S as total rainfall amount during a certain time period, or in actuarial science, with N as the number of losses, and S as total loss expenditure during a certain time period. The compound Poisson exponential model is characterized in the way that a specific transform associated with the distribution of (S,N) satisfies a certain differential equation. Mimicking the function part of this equation by substituting the empirical counterparts of the transform we obtain an expression the weighted integral of the square of which is used as test statistic. We deal with two variants of the latter, one of which being invariant under scale transformations of the S-part by fixed positive constants. Critical values are obtained by using a parametric bootstrap procedure. The asymptotic behavior of the tests is discussed. A simulation study demonstrates the performance of the tests in the finite sample case. The procedure is applied to rainfall data and to an actuarial dataset. A multivariate extension is also discussed.

Rectangular multivariate normal prediction regions for setting reference regions in laboratory medicine

ABSTRACT Reference intervals are among the most widely used medical decision-making tools and are invaluable in the interpretation of laboratory results of patients. Moreover, when multiple biochemical analytes are measured on each patient, a multivariate reference region (MRR) is needed. Such regions are more desirable than separate univariate reference intervals since the latter disregard the cross-correlations among variables. Traditionally, assuming multivariate normality, MRRs have been constructed as ellipsoidal regions, which cannot detect componentwise extreme values. Consequently, MRRs are rarely used in actual practice. In order to address the above drawback of ellipsoidal reference regions, we propose a procedure to construct rectangular MRRs under multivariate normality. The rectangular MRR is computed using a prediction region criterion. However, since the population correlations are unknown, a parametric bootstrap approach is employed for computing the required prediction factor. Also addressed in this study is the computation of mixed reference intervals, which include both two-sided and one-sided prediction limits, simultaneously. Numerical results show that the parametric bootstrap procedure is quite accurate, with estimated coverage probabilities very close to the nominal level. Moreover, the expected volumes of the proposed rectangular regions are substantially smaller than the expected volumes obtained from Bonferroni simultaneous prediction intervals. We also explore the computation of covariate-dependent MRRs in a multivariate regression setting. Finally, we discuss real-life applications of the proposed methods, including the computation of reference ranges for the assessment of kidney function and for components of the insulin-like growth factor system in adults.

Parametric bootstrap procedures for three-factor ANOVA and multiple comparison procedures with unequal group variances

The issue of unmet equal variance assumption in multi-factor ANOVA has been addressed in the literature with several methods, and parametric bootstrap (PB) has been found in the one-way and two-way cases to outperform other methods. We extend previously developed PB procedures for one- and two-way ANOVA, and illustrate with a three-way ANOVA model with unequal group variances (heteANOVA model). We develop a framework for working with these models, analogous to usual multi-factor ANOVA procedures, where F-tests and Tukey’s simultaneous multiple comparison procedures are replaced by PB procedures. Using simulation, we compare these methods to F-tests for each step in model selection, as well as to Tukey’s test for multiple comparison procedures (MCP). The results of our simulations indicate that the PB methods outperform F-tests and Tukey’s test in terms of Type I error when data are unbalanced.

Inferences of the Multicomponent Stress–Strength Reliability for Burr XII Distributions

Multicomponent stress–strength reliability (MSR) is explored for the system with Burr XII distributed components under Type-II censoring. When the distributions of strength and stress variables have Burr XII distributions with common or unequal inner shape parameters, the existence and uniqueness of the maximum likelihood estimators are investigated and established. The associated approximate confidence intervals are obtained by using the asymptotic normal distribution theory along with the delta method and parametric bootstrap procedure, respectively. Moreover, alternative generalized pivotal quantities-based point and confidence interval estimators are developed. Additionally, a likelihood ratio test is presented to diagnose the equivalence of both inner shape parameters or not. Conclusively, Monte Carlo simulations and real data analysis are conducted for illustration.

A framework of tolerance specification for freeform point clouds and capability analysis for reverse engineering processes

The combination of reverse engineering (RE) with additive manufacturing (AM) is widely used for solid freeform fabrication and overcomes limitations with the current capabilities of designers and CAD packages that are restricted to classical model shapes. The shape flexibility offered by RE typically leads to the generated CAD models being represented in point cloud formats or stereolithography files. One resulting challenge is the difficulty with tolerance specification for the RE-generated models, especially for freeform shapes. Geometric metrology for the AM-produced parts to determine their geometric conformation can also be involved. Furthermore, conventional process analyses cannot be performed directly after RE because of the complex geometric data structure. We propose to address all these issues with a new tolerance specification and process analysis framework based on volumetric data analysis. The tolerance zone constructed under our framework, and the corresponding geometric measurements, are consistent with the profile tolerance in geometric dimensioning and tolerancing standards. Reverse engineering's process capability is assessed under our methodology via a parametric bootstrap procedure in the size-and-shape space. The performance and utility of the proposed framework are validated via a process capability study on RE-generated CAD models of multiple, additively manufactured freeform objects.

Population trends from count data: Handling environmental bias, overdispersion and excess of zeroes

The assessment of population trends is a key point in wildlife conservation. Survey data collected over long period may not be comparable due to the presence of environmental biases (i.e. inadequate representation of the variability of environmental covariates in the study area). Moreover, count data may be affected by both overdispersion (i.e. the variance is larger than the mean) and excess of zero counts (potentially leading to zero inflation). The aim of this study was to define a modelling procedure to assess long-term population trends that addressed these three issues and to shed light on the effects of environmental bias, overdispersion, and zero inflation on trend estimates. To test our procedure, we used six bird species whose data were collected in northern Italy from 1992 to 2019. We designed a multi-step approach. First, using generalised additive models (GAMs), we implemented a full factorial design of models (eight models per species) taking or not into account the environmental bias (including or not including environmental covariates, respectively), overdispersion (using a negative binomial distribution or a Poisson distribution, respectively), and zero inflation (using or not using zero-inflated models, respectively). Models were ranked according to the Akaike Information Criterion. Second, annual population indices (median and 95% confidence interval of the number of breeding pairs per point count) were predicted through a parametric bootstrap procedure. Third, long-term population trends were assessed and tested for significance fitting weighted least square linear regression models to the predicted annual indices. To evaluate the effect of environmental bias, overdispersion, and zero inflation on trend estimates, an average discrepancy index was calculated for each model group. The results showed that environmental bias was the most important driver in determining different trend estimates, although overlooking overdispersion and zero inflation could lead to misleading results. For five species, zero-inflated GAMs resulted the best models to predict annual population indices. Our findings suggested a mutual interaction between zero inflation and overdispersion, with overdispersion arising in non-zero-inflated models. Moreover, for species having flocking foraging and/or colonial breeding behaviours, overdispersed and zero-inflated models may be more adequate. In conclusion, properly handling environmental bias, which may affect several data sets coming from long-term monitoring programs, is crucial to obtain reliable estimates of population trends. Furthermore, the extent to which overdispersion and zero inflation may affect trend estimates should be assessed by comparing different models, rather than presumed using statistical assumption.

Penalized angular regression for personalized predictions

AbstractPersonalization is becoming an important aspect of many predictive applications. We introduce a penalized regression method which inherently implements personalization. Personalized angle (PAN) regression constructs regression coefficients that are specific to the covariate vector for which one is producing a prediction, thus personalizing the regression model itself. This is achieved by penalizing the normalized prediction for a given covariate vector. The method therefore penalizes the normalized regression coefficients, or the angles of the regression coefficients in a hyperspherical parametrization, introducing a new angle‐based class of penalties. PAN hence combines two novel concepts: penalizing the normalized coefficients and personalization. For an orthogonal design matrix, we show that the PAN estimator is the solution to a low‐dimensional eigenvector equation. Based on the hyperspherical parametrization, we construct an efficient algorithm to calculate the PAN estimator. We propose a parametric bootstrap procedure for selecting the tuning parameter, and simulations show that PAN regression can outperform ordinary least squares, ridge regression and other penalized regression methods in terms of prediction error. Finally, we demonstrate the method in a medical application.

On Testing the Adequacy of the Inverse Gaussian Distribution

We propose a new class of goodness-of-fit tests for the inverse Gaussian distribution based on a characterization of the cumulative distribution function (CDF). The new tests are of weighted L2-type depending on a tuning parameter. We develop the asymptotic theory under the null hypothesis and under a broad class of alternative distributions. These results guarantee that the parametric bootstrap procedure, which we employ to implement the test, is asymptotically valid and that the whole test procedure is consistent. A comparative simulation study for finite sample sizes shows that the new procedure is competitive to classical and recent tests, outperforming these other methods almost uniformly over a large set of alternative distributions. The use of the newly proposed test is illustrated with two observed data sets.

Estimating Exposure Intensity Based on Odor.

The odor detection threshold for a chemical typically varies widely across population members, and the values can be described by a lognormal distribution. Assuming that the parameters of a chemical's odor detection threshold distribution can be estimated, a method is presented to retrospectively estimate exposure intensity based on detecting the chemical's odor. The context is a single exposure period involving a single air contaminant. In brief, where k out of n identically exposed persons detect the chemical's odor, the best estimate of the chemical's concentration in air corresponds to the k ÷ n fractile of the odor detection threshold distribution. Where n is small and/or k ÷ n is close to zero or one, exact 100×(1-α)% confidence bounds for the fractile estimate can be computed without using the normal distribution assumption. In addition, statistical uncertainty in the parameter estimates of the chemical's odor detection threshold distribution can be considered via a parametric bootstrap procedure, such that an overall 100×(1-α)% confidence interval on the chemical's concentration in air is obtained. The method is illustrated for benzene. Analysis of the available benzene odor detection threshold literature provides best estimates of a population geometric mean equal to 37.8 ppm and geometric standard deviation equal to 1.92. In a hypothetical example in which two out of six workers detect the odor of benzene, the point estimate of the benzene exposure level is 28.5 ppm, and the approximate one-sided lower 95% confidence limit is 8.2 ppm. The present analysis also makes clear that odor does not provide an adequate warning of excessive levels of airborne benzene.