Two-stage Maximum Likelihood Research Articles

Localising two sub-diffraction emitters in 3D using quantum correlation microscopy

The localisation of fluorophores is an important aspect of determining the biological function of cellular systems. Quantum correlation microscopy (QCM) is a promising technique for providing diffraction unlimited emitter localisation that can be used with either confocal or widefield modalities. However, so far, QCM has not been applied to three dimensional localisation problems. Here we show that QCM provides diffraction-unlimited three-dimensional localisation for two emitters within a single diffraction-limited spot. By introducing a two-stage maximum likelihood estimator, our modelling shows that the localisation precision scales as 1/t where t is the total detection time. Diffraction unlimited localisation is achieved using both intensity and photon correlation from Hanbury Brown and Twiss measurements at as few as four measurement locations. We also compare the results of (MC) simulations with the Cramér–Rao lower bound.

Two-Stage Limited-Information Estimation for Structural Equation Models of Round-Robin Variables

We propose and demonstrate a new two-stage maximum likelihood estimator for parameters of a social relations structural equation model (SR-SEM) using estimated summary statistics (Σ^) as data, as well as uncertainty about Σ^ to obtain robust inferential statistics. The SR-SEM is a generalization of a traditional SEM for round-robin data, which have a dyadic network structure (i.e., each group member responds to or interacts with each other member). Our two-stage estimator is developed using similar logic as previous two-stage estimators for SEM, developed for application to multilevel data and multiple imputations of missing data. We demonstrate out estimator on a publicly available data set from a 2018 publication about social mimicry. We employ Markov chain Monte Carlo estimation of Σ^ in Stage 1, implemented using the R package rstan. In Stage 2, the posterior mean estimates of Σ^ are used as input data to estimate SEM parameters with the R package lavaan. The posterior covariance matrix of estimated Σ^ is also calculated so that lavaan can use it to calculate robust standard errors and test statistics. Results are compared to full-information maximum likelihood (FIML) estimation of SR-SEM parameters using the R package srm. We discuss how differences between estimators highlight the need for future research to establish best practices under realistic conditions (e.g., how to specify empirical Bayes priors in Stage 1), as well as extensions that would make 2-stage estimation particularly advantageous over single-stage FIML.

Comparison of paired ordinal data with mis-classification and covariates adjustment

Abstract In this paper, we develop an estimation and testing procedure for comparing matched-pair ordinal outcomes in studies with confounding factors. The classification method for the categories of ordinal outcomes that is accessible for all units may be prone to mis-classification, and thus another error-free classification method that can only be affordable for a fraction of the units are used, resulting in a dataset with partial validation. The distribution of categorical variables is modelled using correlated bivariate Gaussian latent variables, and the confounding factors are adjusted as covariates. The mis-classification of ordinal outcomes is addressed by estimating the mis-classification probabilities through the partial validation structure of the dataset. The mis-classification probabilities and the other parameters are estimated by a two-stage maximum likelihood estimator, and the difference between the matched-pair ordinal outcomes are assessed by a Wald test statistic. Simulation studies were conducted to investigate the accuracy of the estimates of the model parameters, and the type I error rates and power of the proposed testing procedure. The motivating dataset from the Garki Project was analysed to demonstrate the applicability of the proposed approach.

Joint optimization of selective maintenance and repairpersons assignment problem for mission-oriented systems operating under [formula omitted]-dependent competing risks

Mission-oriented systems (MOS) are designed to operate a series of missions seceded by scheduled breaks of limited duration. For MOS, the most appropriate maintenance strategy is the selective maintenance (SM). Under limited maintenance resources, the SM problem (SMP) aims to identify the optimal set of components to maintain in order to meet the required minimum performance for the next missions. However, the majority of the existing SM models merely relies on stochastic independent (s-independent) components. To overcome this restrictive and unrealistic assumption, this work develops a novel SM approach for jointly optimizing SM and repairpersons assignment (RA) problem (RAP) in a MOS operating under s-dependent competing risks. The s-dependent competing risks between components are captured through copula functions whose unknown parameters are estimated by a two-stage maximum likelihood estimation (MLE) method. Furthermore, the MOS operate several missions and scheduled breaks, and multiple repair channels are available. The resulting joint SM and RAP (JSM-RAP) optimization model is formulated as a mixed-integer non-linear program (MINLP) where the objective is to minimize the total maintenance and labor costs for a SM plan that ensures a required minimum reliability threshold for each mission. A genetic algorithm (GA) is used as a solution method. Several numerical experiments are conducted on a ball screw system. The results obtained demonstrate the validity and the add value of the proposed approach, in addition to the robustness and efficiency of the solution method.

Measurement error-filtered machine learning in digital soil mapping

This paper presents a two-stage maximum likelihood framework to deal with measurement errors in digital soil mapping (DSM) when using a machine learning (ML) model. The framework is implemented with random forest and projection pursuit regression to illustrate two different areas of machine learning, i.e. ensemble learning with trees and feature-learning. In our proposed framework, a measurement error variance (MEV) is incorporated as a weight in the log-likelihood function so that measurements with a larger MEV receive less weight when a ML model is calibrated. We evaluate the performance of the error-filtered ML models with an error-filtered regression kriging model, in a comprehensive simulation study and in a real-world case study of Namibian data. From the results we show that prediction accuracy can be increased by using our proposed framework, especially when the MEVs are large and heterogeneous.

Ryu-type extended Marshall-Olkin model with implicit shocks and joint life insurance applications

In this paper we suggest an improvement of the Extended Marshall-Olkin methodology by allowing an implicit effect of the common shocks affecting the elements of the system. Properties of this new model are studied. We propose an empirical application to a sample of censored residual lifetimes of couples of insureds extracted from a data set of annuities contracts of a large Canadian life insurance company. We obtain estimation of the model parameters using a two-stage maximum likelihood technique and discuss the obtained results.

Two-Stage Maximum Likelihood Estimation Procedure for Parallel Constant-Stress Accelerated Degradation Tests

The parallel constant-stress accelerated degradation test (PCSADT) is a popular method used to assess the reliability of highly reliable products in a timely manner. Although the maximum likelihood (ML) method is commonly utilized to estimate the PCSADT parameters, the explicit forms of the ML estimators, and their corresponding Fisher information matrix are usually difficult to obtain. In this article, we propose a two-stage ML (TSML) estimation procedure for a time-transformed model. In the proposed procedure, all the TSML estimators not only have explicit expressions but also possess consistency and asymptotic normality. Hence, this method is tractable for reliability engineers. Furthermore, the TSML estimators can provide constructive information about the unknown accelerated relationship law. The proposed method is also applied to analyze light-emitting diode data and compare the performance of our estimation procedures with the ML method via simulations.

Robust nonlinear structural equation modeling with interaction between exogenous and endogenous latent variables

ABSTRACT A handful of studies have been devoted to nonlinear structural equation modeling (SEM) in the literature. However, they generally overlooked the interactions among exogenous and endogenous latent variables and the interactions among endogenous latent variables. In this study, we propose a maximum likelihood approach for a nonlinear SEM model that incorporates such overlooked interactions. We also propose a two-stage pseudo maximum likelihood approach under the assumption of normal mixture, being computational efficient and robust against distributional misspecification. The simulation study shows that both approaches accurately estimate the unknown parameters if the distribution is correctly specified. However, only the pseudo maximum likelihood approach is robust against distributional misspecification.

Two-stage maximum likelihood approach for item-level missing data in regression

Psychologists use scales comprised of multiple items to measure underlying constructs. Missing data on such scales often occur at the item level, whereas the model of interest to the researcher is at the composite (scale score) level. Existing analytic approaches cannot easily accommodate item-level missing data when models involve composites. A very common practice in psychology is to average all available items to produce scale scores. This approach, referred to as available-case maximum likelihood (ACML), may produce biased parameter estimates. Another approach researchers use to deal with item-level missing data is scale-level full information maximum likelihood (SL-FIML), which treats the whole scale as missing if any item is missing. SL-FIML is inefficient and it may also exhibit bias. Multiple imputation (MI) produces the correct results using a simulation-based approach. We study a new analytic alternative for item-level missingness, called two-stage maximum likelihood (TSML; Savalei & Rhemtulla, Journal of Educational and Behavioral Statistics, 42(4), 405–431. 2017). The original work showed the method outperforming ACML and SL-FIML in structural equation models with parcels. The current simulation study examined the performance of ACML, SL-FIML, MI, and TSML in the context of univariate regression. We demonstrated performance issues encountered by ACML and SL-FIML when estimating regression coefficients, under both MCAR and MAR conditions. Aside from convergence issues with small sample sizes and high missingness, TSML performed similarly to MI in all conditions, showing negligible bias, high efficiency, and good coverage. This fast analytic approach is therefore recommended whenever it achieves convergence. R code and a Shiny app to perform TSML are provided.

Inefficiency and bank failure: A joint Bayesian estimation method of stochastic frontier and hazards models

We propose a Bayesian one-stage approach to estimate the effect of inefficiency on the time to failure (bankruptcy) of U.S. commercial banks. We do so combining stochastic frontier and proportional hazards settings. Most of the existing literature use two-stage methods which may yield inefficient, biased, and inconsistent estimates. Our proposal overcomes these issues, allows computing the marginal distribution of inefficiencies for each observational unit, and facilitates statistical inference of non-linear functions of parameters such as returns to scale. Simulation exercises show that our proposal outperforms the two-stage maximum likelihood approach traditionally used in the literature. In addition, empirical evidence suggests that inefficiency of U.S. commercial banks during the global financial crisis in 2008–2009 played a statistically and economically significant role determining the time to failure.

A Reliability Evaluation Method for System's Dependent Competition Failure and Multi-Parameter Degradation Failure

In order to enhance the reliability of a system that has dependent competition failure, a reliability evaluation method is proposed to evaluate the dependent competition failure and multi-parameter degradation failure. The multi-parameter degradation failure process is described with the Wiener stochastic process and the inverse Gaussian stochastic process. The Copula function is used to model the system's multi-degradation failure process. The two-stage maximum likelihood method is used to estimate the degradation failure parameters. The conditional probability of dependent competition failure in terms of degradation degree is established. The Bayes-Bootstrap method is utilized to correct the dependent competition failure parameters obtained with the maximum likelihood method and to further establish the system's dependent competition failure model. The degradation data of an aero-engine is used as an example to analyze the reliability under competition between dependent competition failure and multi-parameter degradation failure. The analysis results can effectively demonstrate the reliability of an aero-engine's performance and verify the validity of the model, thus having good engineering application values.

Variance Minimization Hedging Analysis Based on a Time-Varying Markovian DCC-GARCH Model

Considering time-varying transition probability (TVTP), this article combines Markov regime switching with a dynamic conditional correlation generalized autoregressive conditional heteroscedasticity (DCC-GARCH) model to construct a new hedging model and study a state-dependent minimum variance hedging ratio. A two-stage maximum likelihood method is constructed to estimate the model parameters. A filtering algorithm is used in an estimation process. Empirical results on commodity futures hedging show that compared with other benchmark models, the proposed one has the best fitting effect. In addition, in terms of hedging effectiveness, the proposed model is superior to other models in most cases, which means that introducing TVTP into a DCC-GARCH model can effectively improve the performance of hedging portfolio. Note to Practitioners —This article deals with a state-dependent minimum variance hedging problem. It combines a time-varying Markov regime switching with dynamic conditional correlation generalized autoregressive conditional heteroscedasticity named DCC-GARCH to construct a new hedging model and estimates a state-dependent hedging ratio. Empirical results from commodity futures hedging show that introducing TVTP into the DCC-GARCH model can effectively reduce portfolio risk and provide better hedging performance than other traditional models, including Markov regime switching DCC-GARCH with a fixed transition probability, DCC-GARCH, ordinary least squares, naive hedging strategies, and unhedged spots. Thus, this article is of guiding significance for hedgers to fully learn the hedging rules of futures market and avoid the spots price risk.

Copula information criterion for model selection with two-stage maximum likelihood estimation

In parametric copula setups, where both the margins and copula have parametric forms, two-stage maximum likelihood estimation, often referred to as inference functions for margins, is used as an attractive alternative to the full maximum likelihood estimation strategy. Exploiting the existing model robust inference of two-stage maximum likelihood estimation, a copula information criterion (CIC) for model selection is developed. In a nutshell, CIC aims for the model that minimizes the Kullback–Leibler divergence from the real data generating mechanism. CIC does not assume that the chosen parametric model captures this true model, unlike what is assumed for AIC. In this sense, CIC is analogous to the Takeuchi Information Criterion (TIC), which is defined for the full maximum likelihood. If the additional assumption that a candidate model is correctly specified is made, then CIC for that model simplifies to AIC. Additionally, CIC can easily be extended to the conditional copula setup where covariates are parametrically linked to the copula model. As a numerical illustration, simulation studies were performed to show that the better model according to CIC also has better prediction performance in general. The result also shows that the bias correction term of CIC penalizes the misspecified model more heavily. This bias correction term has a strong positive relationship with the prediction performance of the model. So, a model with bad prediction performance is being penalized more by CIC. Although this behavior of the bias correction part is an important conceptual advance of CIC, this is not sufficient to make CIC outperform AIC in practice. This is because each misspecified model has the bias correction term and they grow at different speeds, depending on the model. The difference between CIC and AIC becomes minimal as sample size grows because the log-likelihood part outgrows the bias correction part.

Model robust inference with two-stage maximum likelihood estimation for copulas

This article is concerned with inference in parametric copula setups, where both the marginals and the copula have parametric forms. For such models, two-stage maximum likelihood estimation, often referred to as inference function for margins, is used as an attractive alternative to the full maximum likelihood estimation strategy. Previous studies of the two-stage maximum likelihood estimator have largely been based on the assumption that the chosen parametric model captures the true model that generated data. We study the impact of dropping this true model assumption, both theoretically and numerically. We first show that the two-stage maximum likelihood estimator is consistent for a well-defined least false parameter value, different from the analogous least false parameter associated with the full maximum likelihood procedure. Then we demonstrate limiting normality of the full vector of estimators, with concise matrix notation for the variance matrices involved. Along with consistent estimators for these, we have built a model-robust machinery for inference in parametric copula models. The special case where the parametric model is assumed to hold corresponds to situations studied earlier in the literature, with simpler formulas for variance matrices. As a numerical illustration, we perform a set of simulations. We also analyze five-dimensional Norwegian precipitation data. We find that the variance of the copula parameter estimate can both increase and decrease, by dropping the true model assumption. In addition, we observe that the two-stage maximum likelihood estimator is still highly efficient when the true model assumption is dropped and thus the model robust asymptotic variance formulas are used. Additionally, we discover that using highly misspecified models can lead to situations where the asymptotic variance of the two-stage maximum likelihood estimator is lower than that of full maximum likelihood estimator. Our results are also used to analyze the mean squared error properties for both the full and the two-stage maximum likelihood estimators of any focus parameter.

Effective truth discovery and fair reward distribution for mobile crowdsensing

By leveraging the sensing capabilities of consumer mobile devices, mobile crowdsensing (MCS) systems enable a number of new applications for Internet of Things (IoT), such as traffic management, environmental monitoring, and localisation. However, the sensing data collected from the crowd workers are of various qualities, making it difficult to discover the ground truth and maintain the fairness of incentivisation schemes. In this paper, we propose a truth discovery algorithm based on a two-stage Maximum Likelihood Estimator (MLE), which explicitly characterises the heterogeneous sensing capabilities of the crowd and is able to estimate ground truth accurately using only a small amount of data from IoT infrastructures. Moreover, based on the truth discovery algorithm, two reward distribution schemes, LRDS and MRDS, are proposed to ensure fairness of rewarding the crowd according to their effort levels. We evaluate the estimation accuracy of the truth discovery algorithm and the fairness of the reward distribution schemes using both simulations and real-world MCS campaigns. The evaluation results indicate that the proposed methods achieve superior performance compared with state-of-the-art methods in terms of estimation accuracy and fairness of reward distribution.

Structural Equation Modeling of Social Networks: Specification, Estimation, and Application

Psychologists are interested in whether friends and couples share similar personalities or not. However, no statistical models are readily available to test the association between personalities and social relations in the literature. In this study, we develop a statistical model for analyzing social network data with the latent personality traits as covariates. Because the model contains a measurement model for the latent traits and a structural model for the relationship between the network and latent traits, we discuss it under the general framework of structural equation modeling (SEM). In our model, the structural relation between the latent variable(s) and the outcome variable is no longer linear or generalized linear. To obtain model parameter estimates, we propose to use a two-stage maximum likelihood (ML) procedure. This modeling framework is evaluated through a simulation study under representative conditions that would be found in social network data. Its usefulness is then demonstrated through an empirical application to a college friendship network.

Experimental Results on Agent-Based Indoor Localization using WiFi Signaling

This paper discusses experimental results on the possibility of accurately estimating the position of smart devices in known indoor environments using agent technology. Discussed localization approaches are based on WiFi signaling, which can be considered as an ubiquitous technology in the large majority of indoor environments. The use of WiFi signaling ensures that no specific infrastructures nor special on-board sensors are required to support localization. Localization is performed using range estimates from the fixed access points of the WiFi network, which are assumed to have known positions. The performance of two range-based localization algorithms are discussed. The first, called Two-Stage Maximum-Likelihood algorithm, is well-known in the literature, while the second is a recent optimization-based algorithm that uses particle swarm techniques. Results discussed in the last part of the paper show that a proper processing of WiFi-based range estimates allows obtaining accurate position estimates, especially if the optimization-based algorithm is used.

Two-Stage maximum likelihood estimation in the misspecified restricted latent class model.

The maximum likelihood classification rule is a standard method to classify examinee attribute profiles in cognitive diagnosis models (CDMs). Its asymptotic behaviour is well understood when the model is assumed to be correct, but has not been explored in the case of misspecified latent class models. This paper investigates the asymptotic behaviour of a two-stage maximum likelihood classifier under a misspecified CDM. The analysis is conducted in a general restricted latent class model framework addressing all types of CDMs. Sufficient conditions are proposed under which a consistent classification can be obtained by using a misspecified model. Discussions are also provided on the inconsistency of classification under certain model misspecification scenarios. Simulation studies and a real data application are conducted to illustrate these results. Our findings can provide some guidelines as to when a misspecified simple model or a general model can be used to provide a good classification result.

Normal Theory GLS Estimator for Missing Data: An Application to Item-Level Missing Data and a Comparison to Two-Stage ML.

Structural equation models (SEMs) can be estimated using a variety of methods. For complete normally distributed data, two asymptotically efficient estimation methods exist: maximum likelihood (ML) and generalized least squares (GLS). With incomplete normally distributed data, an extension of ML called “full information” ML (FIML), is often the estimation method of choice. An extension of GLS to incomplete normally distributed data has never been developed or studied. In this article we define the “full information” GLS estimator for incomplete normally distributed data (FIGLS). We also identify and study an important application of the new GLS approach. In many modeling contexts, the variables in the SEM are linear composites (e.g., sums or averages) of the raw items. For instance, SEMs often use parcels (sums of raw items) as indicators of latent factors. If data are missing at the item level, but the model is at the composite level, FIML is not possible. In this situation, FIGLS may be the only asymptotically efficient estimator available. Results of a simulation study comparing the new FIGLS estimator to the best available analytic alternative, two-stage ML, with item-level missing data are presented.

Normal Theory Two-Stage ML Estimator When Data Are Missing at the Item Level

In many modeling contexts, the variables in the model are linear composites of the raw items measured for each participant; for instance, regression and path analysis models rely on scale scores, and structural equation models often use parcels as indicators of latent constructs. Currently, no analytic estimation method exists to appropriately handle missing data at the item level. Item-level multiple imputation (MI), however, can handle such missing data straightforwardly. In this article, we develop an analytic approach for dealing with item-level missing data—that is, one that obtains a unique set of parameter estimates directly from the incomplete data set and does not require imputations. The proposed approach is a variant of the two-stage maximum likelihood (TSML) methodology, and it is the analytic equivalent of item-level MI. We compare the new TSML approach to three existing alternatives for handling item-level missing data: scale-level full information maximum likelihood, available-case maximum likelihood, and item-level MI. We find that the TSML approach is the best analytic approach, and its performance is similar to item-level MI. We recommend its implementation in popular software and its further study.