Parametric Bootstrap Research Articles

The asymptotic distribution of robust maximum likelihood estimator with Huber function for the mixed spatial autoregressive model with outliers

There is a wide range of outliers in spatial data, and these potential outliers will have a great impact on parameter estimation and corresponding statistical inference. Relying on the framework of maximum likelihood estimation (MLE), we investigate the asymptotic distribution of robust ML estimator under the mixed spatial autoregressive models with outliers and compare it with that of the ML estimator. Furthermore, based on the asymptotic theoretical result, we conduct the confidence interval of robust MLE and MLE. Similar to the results of MLE, we construct the second-order-corrected robust confidence interval using the parametric and semi-parametric bootstrap method. Simulation studies using Monte Carlo show that the robust estimator with the Huber loss function is more accurate and outperforms the MLE in most sample settings when data is contaminated by outliers. Then the use of the method is demonstrated in the analysis of the Neighborhood Crimes Data and the Boston Housing Price Data. The results further support the eligibility of the robust method in practical situations.

Confidence Intervals for Assessing Non-Inferiority with Assay Sensitivity in a Three-Arm Trial with Normally Distributed Endpoints

Various approaches including hypothesis test and confidence interval (CI) construction have been proposed to assess non-inferiority and assay sensitivity via a known fraction or pre-specified margin in three-arm trials with continuous or discrete endpoints. However, there is little work done on the construction of the non-inferiority margin from historical data and simultaneous generalized CIs (SGCIs) in a three-arm trial with the normally distributed endpoints. Based on the generalized fiducial method and the square-and-add method, we propose two simultaneous CIs for assessing non-inferiority and assay sensitivity in a three-arm trial. For comparison, we also consider the Wald-type Bonferroni simultaneous CI and parametric bootstrap simultaneous CI. An algorithm for evaluating the optimal sample size for attaining the pre-specified power is given. Simulation studies are conducted to investigate the performance of the proposed CIs in terms of their empirical coverage probabilities. An example taken from the mildly asthmatic study is illustrated using the proposed simultaneous CIs. Empirical results show that the proposed generalized fiducial method and the square-and-add method behave better than other two compared CIs.

Asymmetric Randomly Censored Mortality Distribution: Bayesian Framework and Parametric Bootstrap with Application to COVID‐19 Data

This article investigates a survival analysis under randomly censored mortality distribution. From the perspective of frequentist, we derive the point estimations through the method of maximum likelihood estimation. Furthermore, approximate confidence intervals for the parameters are constructed based on the asymptotic distribution of the maximum likelihood estimators. Besides, two parametric bootstraps are implemented to construct the approximate confidence intervals for the unknown parameters. In Bayesian framework, the Bayes estimates of the unknown parameters are evaluated by applying the Markov chain Monte Carlo technique, and highest posterior density credible intervals are also carried out. In addition, the Bayes inference based on symmetric and asymmetric loss functions is obtained. Finally, Monte Carlo simulation is performed to observe the behavior of the proposed methods, and a real data set of COVID‐19 mortality rate is analyzed for illustration.

ЭМПИРИЧЕСКОЕ ИССЛЕДОВАНИЕ МОЩНОСТИ СТАТИСТИЧЕСКОГО КРИТЕРИЯ КОЛМОГОРОВА–СМИРНОВА В ЗАДАЧАХ ПРОВЕРКИ ГИПОТЕЗ О ЗАКОНЕ РАСПРЕДЕЛЕНИЯ

Statistical tests that use A.N. Kolmogorov one-sample statistics and N.V. Smirnov two-sample statistics have been widely used for solving applied problems for almost a hundred years. Those criteria are present in many textbooks and implemented in computer software for data analysis. The purpose of the work is to empirically estimate the power of the Kolmogorov–Smirnov criterion on a set of test problems to check hypotheses about the distribution law, as well as to investigate the properties of estimates of the power of the criterion for various types of resampling. To obtain various estimates of the power of the Kolmogorov–Smirnov criterion in solving test problems, the classical bootstrap, nonparametric bootstrap, parametric bootstrap, bootstrap with a random term, and resampling without returns were used. Based on the results of multiple solution of test problems, medians of p-values, medians of estimates of the power of the Kolmogorov–Smirnov criterion were calculated, and medians of biases of estimates on test problems and medians of all pairwise differences of various estimates of the power of the criterion were found. The distribution laws for the considered Kolmogorov–Smirnov power estimates, for the biases of the power estimates, and for all pairwise differences of the power estimates were investigated. For test problems where real data were considered, calculating power estimates without resampling is impossible, but unbiased power estimates can be predicted as the half-sum of two other power estimates, one obtained with a nonparametric bootstrap and the other obtained with resampling without returns. Methods for restructuring sample data and software for estimating the power of a statistical test developed within the study are universal and can be used for analyzing the properties of power estimates of other one-sample statistical tests, as well as for developing the classical methodology for checking statistical hypotheses.

Improved Point Estimation for the Rayleigh Regression Model

The Rayleigh regression model was recently proposed for modeling amplitude values of synthetic aperture radar (SAR) image pixels. However, inferences from such model are based on the maximum likelihood estimators, which can be biased for small signal lengths. The Rayleigh regression model for SAR images often takes into account small pixel windows, which may lead to inaccurate results. In this letter, we introduce bias-adjusted estimators tailored for the Rayleigh regression model based on: (i) the Cox and Snell's method; (ii) the Firth's scheme; and (iii) the parametric bootstrap method. We present numerical experiments considering synthetic and actual SAR data sets. The bias-adjusted estimators yield nearly unbiased estimates and accurate modeling results.

Control chart using bootstrap method for logistic-exponential percentiles

Lan and Leemis (2008) introduced logistic-exponential (LE) distribution which has varied applications in lifetime modellings. In this article, we consider parametric bootstrap control charts (BCCs) for detecting a shift in the percentile of LE distribution in a process monitoring situation. Four parametric BCCs based on maximum likelihood method, method of least squares, method of Cramer-von-Mises and ` method of maximum product of spacings are used for monitoring percentiles of LE distribution. We perform simulations to see the performances of the proposed four BCCs with respect to average run length. Finally, one data set is analyzed to illustrate our results.

COVIDNearTerm: A simple method to forecast COVID-19 hospitalizations.

COVID-19 has caused tremendous death and suffering since it first emerged in 2019. Soon after its emergence, models were developed to help predict the course of various disease metrics, and these models have been relied upon to help guide public health policy. Here we present a method called COVIDNearTerm to "forecast" hospitalizations in the short term, two to four weeks from the time of prediction. COVIDNearTerm is based on an autoregressive model and utilizes a parametric bootstrap approach to make predictions. It is easy to use as it requires only previous hospitalization data, and there is an open-source R package that implements the algorithm. We evaluated COVIDNearTerm on San Francisco Bay Area hospitalizations and compared it to models from the California COVID Assessment Tool (CalCAT). We found that COVIDNearTerm predictions were more accurate than the CalCAT ensemble predictions for all comparisons and any CalCAT component for a majority of comparisons. For instance, at the county level our 14-day hospitalization median absolute percentage errors ranged from 16 to 36%. For those same comparisons, the CalCAT ensemble errors were between 30 and 59%. COVIDNearTerm is a simple and useful tool for predicting near-term COVID-19 hospitalizations.

Improved profit model for production of MicroLED PCB with asymmetric specifications

Due to strict requirements for high-tech processes, yield is an essential performance measure of quality output and affects production economics significantly. Yield is a performance index that significantly influences the optimal decision on profit and production quantity. For instance, in micro light-emitting diodes (MicroLED) manufacturing, the yield of surface mounting highly affects the production cost. To guarantee the production decision is reliable, accurately evaluating yields is crucial. However, the existing yield evaluation method for asymmetric processes with a very low fraction of defective by process capability index is conservative and may mislead the decisions. In this paper, a revised unbiased estimator is investigated and a weighted parametric bootstrap method for accurate production decision making is provided. With extensive comparisons, the proposed method outperforms the existing analytical method and considerably improves the precision of yield evaluation. We also provided managerial insights for a profit with various yield measure methods. For demonstration, an application of production yield determination for a printed circuit board process for MicroLED is presented.

Inference for the Process Performance Index of Products on the Basis of Power-Normal Distribution

The process performance index (PPI) can be a simple metric to connect the conforming rate of products. The properties of the PPI have been well studied for the normal distribution and other widely used lifetime distributions, such as the Weibull, Gamma, and Pareto distributions. Assume that the quality characteristic of product follows power-normal distribution. Statistical inference procedures for the PPI are established. The maximum likelihood estimation method for the model parameters and PPI is investigated and the exact Fisher information matrix is derived. We discuss the drawbacks of using the exact Fisher information matrix to obtain the confidence interval of the model parameters. The parametric bootstrap percentile and bootstrap bias-corrected percentile methods are proposed to obtain approximate confidence intervals for the model parameters and PPI. Monte Carlo simulations are conducted to evaluate the performance of the proposed methods. One example about the flow width of the resist in the hard-bake process is used for illustration.

Multiple comparisons of precipitation variations in different areas using simultaneous confidence intervals for all possible ratios of variances of several zero-inflated lognormal models.

Flash flooding and landslides regularly cause injury, death, and homelessness in Thailand. An advancedwarning system is necessary for predicting natural disasters, and analyzing the variability of daily precipitation might be usable in this regard. Moreover, analyzing the differences in precipitation data among multiple weather stations could be used to predict variations in meteorological conditions throughout the country. Since precipitation data in Thailand follow a zero-inflated lognormal (ZILN) distribution, multiple comparisons of precipitation variation in different areas can be addressed by using simultaneous confidence intervals (SCIs) for all possible pairwise ratios of variances of several ZILN models. Herein, we formulate SCIs using Bayesian, generalized pivotal quantity (GPQ), and parametric bootstrap (PB) approaches. The results of a simulation study provide insight into the performances of the SCIs. Those based on PB and the Bayesian approach via probability matching with the beta prior performed well in situations with a large amount of zero-inflated data with a large variance. Besides, the Bayesian based on the reference-beta prior and GPQ SCIs can be considered as alternative approaches for small-to-large and medium-to-large sample sizes from large population, respectively. These approaches were applied to estimate the precipitation variability among weather stations in lower southern Thailand to illustrate their efficacies.

One Parameter A (α) Distribution: Different Methods of Estimation

This paper addresses the different methods of estimation of the unknown parameter of one parameter A(α) distribution from the frequentist point of view. We briefly describe different approaches, namely, maximum likelihood estimator, least square and weighted least square estimators, maximum product spacing estimators, Cram´er-von Mises estimator and compare those using extensive numerical simulations. Next, we obtain parametric bootstrap confidence interval of the parameter using frequentist approaches. Finally, one real data set has been analysed for illustrative purposes.

The Generalized DUS Transformed Log-Normal Distribution and Its Applications to Cancer and Heart Transplant Datasets

Many studies have underlined the importance of the log-normal distribution in the modeling of phenomena occurring in biology. With this in mind, in this article we offer a new and motivated transformed version of the log-normal distribution, primarily for use with biological data. The hazard rate function, quantile function, and several other significant aspects of the new distribution are investigated. In particular, we show that the hazard rate function has increasing, decreasing, bathtub, and upside-down bathtub shapes. The maximum likelihood and Bayesian techniques are both used to estimate unknown parameters. Based on the proposed distribution, we also present a parametric regression model and a Bayesian regression approach. As an assessment of the longstanding performance, simulation studies based on maximum likelihood and Bayesian techniques of estimation procedures are also conducted. Two real datasets are used to demonstrate the applicability of the new distribution. The efficiency of the third parameter in the new model is tested by utilizing the likelihood ratio test. Furthermore, the parametric bootstrap approach is used to determine the effectiveness of the suggested model for the datasets.

Decomposing MRI phenotypic heterogeneity in epilepsy: a step towards personalized classification.

In drug-resistant temporal lobe epilepsy, precise predictions of drug response, surgical outcome and cognitive dysfunction at an individual level remain challenging. A possible explanation may lie in the dominant 'one-size-fits-all' group-level analytical approaches that does not allow parsing interindividual variations along the disease spectrum. Conversely, analysing inter-patient heterogeneity is increasingly recognized as a step towards person-centred care. Here, we used unsupervised machine learning to estimate latent relations (or disease factors) from 3 T multimodal MRI features [cortical thickness, hippocampal volume, fluid-attenuated inversion recovery (FLAIR), T1/FLAIR, diffusion parameters] representing whole-brain patterns of structural pathology in 82 patients with temporal lobe epilepsy. We assessed the specificity of our approach against age- and sex-matched healthy individuals and a cohort of frontal lobe epilepsy patients with histologically verified focal cortical dysplasia. We identified four latent disease factors variably co-expressed within each patient and characterized by ipsilateral hippocampal microstructural alterations, loss of myelin and atrophy (Factor 1), bilateral paralimbic and hippocampal gliosis (Factor 2), bilateral neocortical atrophy (Factor 3) and bilateral white matter microstructural alterations (Factor 4). Bootstrap analysis and parameter variations supported high stability and robustness of these factors. Moreover, they were not expressed in healthy controls and only negligibly in disease controls, supporting specificity. Supervised classifiers trained on latent disease factors could predict patient-specific drug response in 76 ± 3% and postsurgical seizure outcome in 88 ± 2%, outperforming classifiers that did not operate on latent factor information. Latent factor models predicted inter-patient variability in cognitive dysfunction (verbal IQ: r = 0.40 ± 0.03; memory: r = 0.35 ± 0.03; sequential motor tapping: r = 0.36 ± 0.04), again outperforming baseline learners. Data-driven analysis of disease factors provides a novel appraisal of the continuum of interindividual variability, which is probably determined by multiple interacting pathological processes. Incorporating interindividual variability is likely to improve clinical prognostics.

Discrete Weibull distribution: different estimation methods under ranked set sampling and simple random sampling

The discrete Weibull (DW) is a discretized version of the well-known Weibull distribution, and, as such can be considered in reliability and survival analyses where the variable of interest involves some kind of count. Furthermore, since the DW distribution can account for both under and overdispersion, it is an alternative to the orthodox Poisson distribution. The main objective of this study is to evaluate the performance of nine different estimation methods, applied to DW distribution, under simple random sampling (SRS) and ranked set sampling (RSS). A comprehensive simulation study was presented for this purpose, including additional simulation based on real data sets. A parametric bootstrap approach was also developed to provide confidence intervals based on RSS samples. The results allowed us to quantify the superiority of RSS over SRS, and provided important insights about the efficiency of the estimation methods when the counts are under or overdispersed.

A note on the coverage behaviour of bootstrap percentile confidence intervals for constrained parameters

The asymptotic behaviour of the commonly used bootstrap percentile confidence interval is investigated when the parameters are subject to linear inequality constraints. We concentrate on the important one- and two-sample problems with data generated from general parametric distributions in the natural exponential family. The focus of this paper is on quantifying the coverage probabilities of the parametric bootstrap percentile confidence intervals, in particular their limiting behaviour near boundaries. We propose a local asymptotic framework to study this subtle coverage behaviour. Under this framework, we discover that when the true parameters are on, or close to, the restriction boundary, the asymptotic coverage probabilities can always exceed the nominal level in the one-sample case; however, they can be, remarkably, both under and over the nominal level in the two-sample case. Using illustrative examples, we show that the results provide theoretical justification and guidance on applying the bootstrap percentile method to constrained inference problems.

Asymptotics for Markov chain mixture detection

Sufficient conditions are provided under which the log-likelihood ratio test statistic fails to have a limiting chi-squared distribution under the null hypothesis when testing between one and two components under a general two-component mixture model, but rather tends to infinity in probability. These conditions are verified when the component densities describe continuous-time, discrete-state-space Markov chains and the results are illustrated via a parametric bootstrap simulation on an analysis of the migrations over time of a set of corporate bonds ratings. The precise limiting distribution is derived in a simple case with two states, one of which is absorbing which leads to a right-censored exponential scale mixture model. In that case, when centred by a function growing logarithmically in the sample size, the statistic has a limiting distribution of Gumbel extreme-value type rather than chi-squared.

Estimating the RMSE of Small Area Estimates without the Tears

Small area estimation (SAE) methods can provide information that conventional direct survey estimation methods cannot. The use of small area estimates based on linear and generalized linear mixed models is still very limited, possibly because of the perceived complexity of estimating the root mean square errors (RMSEs) of the estimates. This paper outlines a study used to determine the conditions under which the estimated RMSEs, produced as part of statistical output (‘plug-in’ estimates of RMSEs) could be considered appropriate for a practical application of SAE methods where one of the main requirements was to use SAS software. We first show that the estimated RMSEs created using an EBLUP model in SAS and those obtained using a parametric bootstrap are similar to the published estimated RMSEs for the corn data in the seminal paper by Battese, Harter and Fuller. We then compare plug-in estimates of RMSEs from SAS procedures used to create EBLUP and EBP estimators against estimates of RMSEs obtained from a parametric bootstrap. For this comparison we created estimates of current smoking in males for 153 local government areas (LGAs) using data from the NSW Population Health Survey in Australia. Demographic variables from the survey data were included as covariates, with LGA-level population proportions, obtained mainly from the Australian Census used for prediction. For the EBLUP, the estimated plug-in estimates of RMSEs can be used, provided the sample size for the small area is more than seven. For the EBP, the plug-in estimates of RMSEs are suitable for all in-sample areas; out-of-sample areas need to use estimated RMSEs that use the parametric bootstrap.

A WILD BOOTSTRAP FOR DEPENDENT DATA

This paper introduces a novel wild bootstrap for dependent data (WBDD) as a means of calculating standard errors of estimators and constructing confidence regions for parameters based on dependent heterogeneous data. The consistency of the bootstrap variance estimator for smooth function of the sample mean is shown to be robust against heteroskedasticity and dependence of unknown form. The first-order asymptotic validity of the WBDD in distribution approximation is established when data are assumed to satisfy a near epoch dependent condition and under the framework of the smooth function model. The WBDD offers a viable alternative to the existing non parametric bootstrap methods for dependent data. It preserves the second-order correctness property of blockwise bootstrap (provided we choose the external random variables appropriately), for stationary time series and smooth functions of the mean. This desirable property of any bootstrap method is not known for extant wild-based bootstrap methods for dependent data. Simulation studies illustrate the finite-sample performance of the WBDD.

Inference for estimators of generalized long memory processes

Surprisingly little is known regarding the asymptotic properties of estimators for cyclical long memory models such as the Gegenbauer autoregressive moving average (GARMA) model. In this paper, we review the GARMA process and study the properties of several estimation methods with an emphasis on inference related to η, the parameter governing the length of long memory cycles. We present extensive simulation evidence to show that both Whittle and constrained sum of squares (CSS) estimators yield satisfactory results in terms of mean bias and RMSE for all parameters, although there are serious inferential concerns. Most notably, under the null of an infinitely long cycle (η = 1), proposed distributional results for the CSS estimator yield very over-sized tests. For the existing semiparametric estimator with more complete distribution theory, there are unresolved practical issues including complicated bandwidth choices that we address. As an alternative approach for applied researchers wishing to employ a parametric estimator, we propose and validate a parametric bootstrap method using a likelihood ratio test statistic for the hypothesis η = 1. We illustrate estimation and inference issues with an application to the US unemployment rate, where evidence of stationary long memory cycles ultimately emerges.

Reliability estimation of [formula omitted]-out-of-[formula omitted] system in a multicomponent stress–strength dependent model based on copula function

In reliability analysis of the traditional stress–strength models, the stress and strength are often assumed to be independent variables. However the case where this two variables are dependent is more realistic in engineering. To evaluate multicomponent system reliability in such case, the stress and strength are assumed to have dependent Kumaraswamy variable and unit Gompertz variable based on Clayton copula. Then the maximum likelihood (ML), least squares (LS), maximum product of spacings (MPS), Cramér–von-Mises and LS (CL) hybrid estimates of the unknown parameters and system reliability are derived using two-step estimation procedure under order sample. Also the asymptotic distribution of the ML estimation and the parametric bootstrap percentile method are used to construct approximate confidence intervals (CI). Moreover, Monte Carlo simulations are implemented to compare the performance of the proposed methods. Finally, one real data set is analyzed for illustrative purposes.