Effect Of Model Misspecification Research Articles

Approximate parameter estimation and mis-specification analysis of degradation model with asymmetric random effects

Asymmetric distributions such as the skew-normal (SN) distribution have been used for modeling the unit-to-unit variation of the degradation rate of products, and its superiority for model fitting and lifetime estimation has been demonstrated in literature. However, there still exist two issues that need to be further studied. That is, how to simply estimate the parameters of the degradation model with asymmetric random effects and whether the asymmetry or skewness affects the remaining useful life (RUL) prediction. Driven by these two issues, the parameter estimation and RUL prediction of the degradation model with general random effects are first introduced and a two-step maximum likelihood estimation (MLE) method for parameter estimation is proposed, which could greatly simplify the parameter estimation of the degradation model with asymmetric random effects. Then, the SN distribution is used as an example to analyze the second issue regarding the effect of model mis-specification on RUL prediction based on regression models and Wiener processes with different asymmetric random effects in theory. Finally, the parameter estimation and mis-specification analysis of degradation models with other asymmetric random effects are conducted through numerical examples and two case studies. The results demonstrate that the proposed two-step MLE method is satisfactory and the effect of asymmetry on RUL prediction is negligible. Therefore, there is no need to consider the asymmetry of random effects for RUL prediction.

The effect of model misspecification of the bounded transformed gamma process on maintenance optimization

The effect of model misspecification of the bounded transformed gamma process on maintenance optimization

On the pitfalls of Gaussian likelihood scoring for causal discovery

Abstract We consider likelihood score-based methods for causal discovery in structural causal models. In particular, we focus on Gaussian scoring and analyze the effect of model misspecification in terms of non-Gaussian error distribution. We present a surprising negative result for Gaussian likelihood scoring in combination with nonparametric regression methods.

Discrimination between accelerated life models via Approximate Bayesian Computation

Accelerated life testing (ALT) is widely used in high-reliability product estimation to get relevant information about an item's performance and its failure mechanisms. To analyse the observed ALT data, reliability practitioners need to select a suitable accelerated life model based on the nature of the stress and the physics involved. A statistical model consists of (i) a lifetime distribution that represents the scatter in product life and (ii) a relationship between life and stress. In practice, several accelerated life models could be used for the same failure mode and the choice of the best model is far from trivial. For this reason, an efficient selection procedure to discriminate between a set of competing accelerated life models is of great importance for practitioners. In this paper, accelerated life model selection is approached by using the Approximate Bayesian Computation (ABC) method and a likelihood-based approach for comparison purposes. To demonstrate the efficiency of the ABC method in calibrating and selecting accelerated life model, an extensive Monte Carlo simulation study is carried out using different distances to measure the discrepancy between the empirical and simulated times of failure data. Then, the ABC algorithm is applied to real accelerated fatigue life data in order to select the most likely model among five plausible models. It has been demonstrated that the ABC method outperforms the likelihood-based approach in terms of reliability predictions mainly at lower percentiles particularly useful in reliability engineering and risk assessment applications. Moreover, it has shown that ABC could mitigate the effects of model misspecification through an appropriate choice of the distance function.

Destructive cure models with proportional hazards lifetimes and associated likelihood inference

In survival analysis, cure models have gained much importance due to rapid advancements in medical sciences. More recently, a subset of cure models, called destructive cure models, have been studied extensively under competing risks scenario wherein initial competing risks undergo a destructive process. In this article, we study destructive cure models by assuming a flexible weighted Poisson distribution (exponentially weighted Poisson, length biased Poisson and negative binomial distributions) for the initial number of competing causes and lifetimes of the susceptible individuals being defined by proportional hazards. The expectation-maximization (EM) algorithm and profile likelihood approach are made use of to estimate the model parameters. An extensive simulation study is carried out under various parameter settings to examine the properties of the models, and accuracy and the robustness of the proposed estimation technique. Effects of model mis-specification on the parameter estimates are also discussed in detail. For further illustration of the proposed methodology, a real-life cutaneous melanoma data set is analyzed.

Evaluating the Impact of Anatomical Partitioning on Summary Topologies Obtained with Bayesian Phylogenetic Analyses of Morphological Data.

Morphological data are a fundamental source of evidence to reconstruct the Tree of Life, and Bayesian phylogenetic methods are increasingly being used for this task. Bayesian phylogenetic analyses require the use of evolutionary models, which have been intensively studied in the past few years, with significant improvements to our knowledge. Notwithstanding, a systematic evaluation of the performance of partitioned models for morphological data has never been performed. Here we evaluate the influence of partitioned models, defined by anatomical criteria, on the precision and accuracy of summary tree topologies considering the effects of model misspecification. We simulated datasets using partitioning schemes, trees, and other properties obtained from two empirical datasets, and conducted Bayesian phylogenetic analyses. Additionally, we reanalyzed 32 empirical datasets for different groups of vertebrates, applying unpartitioned and partitioned models, and, as a focused study case, we reanalyzed a dataset including living and fossil armadillos, testing alternative partitioning hypotheses based on functional and ontogenetic modules. We found that, in general, partitioning by anatomy has little influence on summary topologies analyzed under alternative partitioning schemes with a varying number of partitions. Nevertheless, models with unlinked branch lengths, which account for heterotachy across partitions, improve topological precision at the cost of reducing accuracy. In some instances, more complex partitioning schemes led to topological changes, as tested for armadillos, mostly associated with models with unlinked branch lengths. We compare our results with other empirical evaluations of morphological data and those from empirical and simulation studies of the partitioning of molecular data, considering the adequacy of anatomical partitioning relative to alternative methods of partitioning morphological datasets. [Evolutionary rates; heterogeneity; morphology; Mk model; partition; topology.].

On inference in a class of exponential distribution under imperfect maintenance

This paper deals with statistical inference for lifetime data in presence of imperfect maintenance. For the maintenance model, the Sheu and Griffith model is considered. The lifetime distribution belongs to exponential distribution class. The maximum likelihood estimation procedure of the model parameters is discussed, and confidence intervals are provided using the asymptotic likelihood theory and bootstrap approach. Based on conjugate and discrete priors, Bayesian estimators of the model parameters are developed under symmetric and asymmetric loss functions. The proposed methodologies are applied to simulated data and sensitivity analysis to different parameters and data characteristics is carried out. The effect of model misspecification is also assessed within this class of distributions through a Monte Carlo simulation study. Finally, two datasets are analyzed for demonstrative aims.

Semiparametric control schemes for dynamically monitoring profiles with count data and arbitrary design

Many existing studies on profile monitoring focus on parametric profiles or normally distributed responses, and usually assume that the design points within different profiles are deterministic. In practice, however, profiles with count responses are common, and different profiles often have different within-profile sample sizes and design points. Furthermore, it is difficult to fit models to complex profiles with multiple explanatory variables either parametrically or nonparametrically. This article aims to monitor small-sample size profiles with count response and arbitrary design using a semiparametric model. Two novel control schemes with dynamic control limits are proposed based on the weighted likelihood ratio test and the weighted F test, respectively. Numerical simulations are conducted to investigate the performance of the proposed control charts. The performance between the control chart with constant and dynamic control limits is also compared, and the effect of model misspecification is explored. Finally, a real-data example of automobile warranty claims is presented to illustrate the implementation of the proposed control charts.

Quantifying the effect of probability model misspecification in seismic collapse risk assessment

One of the main steps in probabilistic seismic collapse risk assessment is estimating the fragility function parameters. The maximum likelihood estimation (MLE) approach, which is widely used for this purpose, contains the underlying assumption that the likelihood function is known to follow a specified parametric probability distribution. However, this assumed distribution may not always be consistent with the “true” probability distribution of the collapse data. This paper implements the Information matrix equivalence theorem to identify the presence of model misspecification i.e., if the assumed collapse probability distribution is, in fact, the “true” one. In the presence of model misspecification, the fragility parameter estimates continue to be asymptotically normally distributed but the variance–covariance matrix is no longer equal to the inverse of the Fisher’s Information matrix. To increase the robustness of the variance–covariance matrix, the Huber–White sandwich estimator is implemented. Using collapse data from eight woodframe buildings, the effect of model misspecification on fragility parameter estimates and collapse rate is quantified. For the considered building cases, the parameter estimation uncertainty in the collapse risk did not increase when the “sandwich” estimator was used compared to when probability model misspecification was not considered (i.e., using MLE). The proposed framework should be used to further investigate the issue of probability model misspecification as it relates to fragility parameter estimation since only a single construction type (woodframe buildings) and limit state (collapse) was considered in the current study.

Quantification of model risk with an application to probability of default estimation and stress testing for a large corporate portfolio

This paper addresses the building of obligor-level hazard rate corporate probability of default models for stress testing, departing from the predominant practice in wholesale credit modeling of constructing segment-level models for this purpose. We build models based upon varied financial, credit rating, equity market and macroeconomic factors, using an extensive history of large corporate firms sourced from Moody’s.We develop distance-to-default (DTD) risk factors and design hybrid structural/ Merton reduced-form models as challengers to versions of the models containing other individual variables. We measure the model risk attributed to various modeling assumptions according to the principle of relative entropy and observe that the omitted-variable bias with respect to the DTD risk factor, neglect of interaction effects and incorrect link function specification has the greatest, intermediate and least impact, respectively. Given the sensitive regulatory uses of these models and the concerns raised in the industry about the effect of model misspecification on capital and reserves, our conclusion is that validation methods chosen in the stress testing context should be capable of testing model assumptions. Our research adds to the literature in that it offers state-of-the-art techniques as viable options in the arsenal of model validators, developers and supervisors seeking to manage model risk.

Optimal designs of constant‐stress accelerated life‐tests for one‐shot devices with model misspecification analysis

AbstractThe design of constant‐stress accelerated life‐test (CSALT) is important in reliability estimation. In reliability studies, practitioners usually rely on underlying distribution to design CSALTs. However, model misspecification analysis of optimal designs has not been examined extensively. This paper considers one‐shot device testing data by assuming gamma, Weibull, lognormal and Birnbaum–Saunders (BS) lifetime distributions, which are popular lifetime distributions in reliability studies. We then investigate the effect of model misspecification between these lifetime distributions in the design of optimal CSALTs, in which the asymptotic variance of the estimate of reliability of the device at a specific mission time is minimized subject to a prefixed budget and a termination time of the life‐test. The inspection frequency, number of inspections at each stress level, and allocation of the test devices are determined in optimal design for one‐shot device testing. Finally, a numerical example involving a grease‐based magnetorheological fluids (G‐MRF) data set is used to illustrate the developed methods. Results suggest the assumption of lifetime distribution as Weibull or lognormal to be more robust to model misspecification, while the assumption of gamma lifetime distribution seems to be the most non‐robust (or most sensitive) one.

Inverse Gaussian processes with correlated random effects for multivariate degradation modeling

Many engineering products have more than one failure mode and the evolution of each mode can be monitored by measuring a performance characteristic (PC). It is found that the underlying multi-dimensional degradation often occurs with inherent process stochasticity and heterogeneity across units, as well as dependency among PCs. To accommodate these features, in this paper, we propose a novel multivariate degradation model based on the inverse Gaussian process. The model incorporates random effects that are subject to a multivariate normal distribution to capture both the unit-wise variability and the PC-wise dependence. Built upon this structure, we obtain some mathematically tractable properties such as the joint and conditional distribution functions, which subsequently facilitate the future degradation prediction and lifetime estimation. An expectation-maximization algorithm is developed to infer the model parameters along with the validation tools for model checking. In addition, two simulation studies are performed to assess the performance of the inference method and to evaluate the effect of model misspecification. Finally, the application of the proposed methodology is demonstrated by two illustrative examples.

On the estimation of population size under dependent dual-record system: an adjusted profile-likelihood approach

Motivated by various applications in epidemiology, population studies, criminology, etc., the problem of estimating size of a homogeneous human population based on two-sample capture–recapture experiment is considered in this article. The Lincoln–Petersen estimate, assuming independence between the samples, is widely used though often its relevance is unanimously criticized. Time and behavioural response variation model (denoted by ) is the most suitable here. Effect of model mis-specification on the Lincoln–Petersen estimate is studied in terms of efficiency and robustness. We study the accuracy and efficiency of this estimator under existence of dependence between the samples. This article shows that profile-likelihood and its existing modifications fail to make inference from the model . Therefore, an adjustment over profile likelihood is newly proposed and evaluated through an extensive simulation study. Finally, two real data sets with different characteristics are analyzed as practical illustrations of the proposed method.

A generalized pH acceleration model of nano-sol products and the effects of model misspecification on shelf-life prediction

In existing pH acceleration models, which are used to assess the shelf-life of liquid-phase nano-sol products, a mixture of normal distributions is commonly employed to describe the sizes of particles mixing from two populations. The Gaussian mixture model approach falls short when used to characterize the asymmetric distributions of particle sizes in subgroups. This work considers instead a broader class of a mixture of log-F distributions, to be embedded in a pH acceleration model. This study aims at understanding the impact of the new modeling approach, in the presence of model misspecification of the particle size distribution, on the accuracy and precision for making the shelf-life predictions. This study found that model misspecification indeed significantly affects the shelf-life predictions. The proposed method shows favorable finite sample performance on a simulated data set. Both the quantitative analysis of the impact due to model misspecification and the solution proposed herein could benefit practitioners in the long run.

Likelihood analysis and stochastic EM algorithm for left truncated right censored data and associated model selection from the Lehmann family of life distributions

The Lehmann family of distributions includes Weibull, Gompertz, and Lomax models as special cases, all of which are quite useful for modeling lifetime data. Analyses of left truncated right censored data from the Lehmann family of distributions are carried out in this article. In particular, the special cases of Weibull, Gompertz, and Lomax distributions are considered. Maximum likelihood estimates of the model parameters are obtained. The steps of the stochastic expectation maximization algorithm are developed in this context. Asymptotic confidence intervals for the model parameters are constructed using the missing information principle and two parametric bootstrap approaches. Through extensive Monte Carlo simulations, performances of the inferential methods are examined. The effect of model misspecification is also assessed within this family of distributions through simulations. As it is quite important to select a suitable model for a given data, a study of model selection based on left truncated right censored data is carried out through extensive Monte Carlo simulations. Two examples based on real datasets are provided for illustrating the models and methods of inference discussed here.

Generalized Linear Distributed Lag Models

We propose distributed generalized linear models for the purpose of incorporating lagged effects. The model class provides a more accurate statistical measure of the relationship between the dependent variable and a series of covariates. The estimators from the proposed procedure are shown to be consistent. Simulation studies not only confirm the asymptotic properties of the estimators, but exhibit the adverse effects of model misspecification in terms of accuracy of model estimation and prediction. The application is illustrated by analyzing the presidential election data of 2016.

The random deterioration rate model with measurement error based on the inverse Gaussian distribution

In this paper, we introduce the random deterioration rate model with measurement error in order to incorporate the variability among different components. The motivation behind the random variable model is to capture the randomness in the individual differences across the population. This model incorporates only sample uncertainty of the degradation, and no temporal variability is included. The measurement error models appear to overcome this problem. The random rate analysis is based on repeated measurements of failure sizes generated by a degradation process over time in a components population. Some characteristics of the random deterioration rate model based on the inverse Gaussian distribution and subject to measurement error, are examined. We carry out simulation studies to (i) assess the performance of the maximum likelihood estimates obtained through the Gaussian quadrature along with Quasi-Newton optimization method; and (ii) examine the effects of model misspecification on the model selection criteria’s performance, as well as on the lifetime prediction’s accuracy and precision. The potentiality of the proposed model is illustrated through two real data sets.

Misspecification analysis of two‐phase gamma‐Wiener degradation models

AbstractA degradation model is used to analyze degradation data and product reliability. Owing to model uncertainty, the assumed degradation model may be improper, which can create a difference between the evaluated and actual product reliabilities. Thus, model misspecification analysis is essential. Recent studies on misspecification analysis have mainly considered single‐phase degradation models; two‐phase degradation models have not been investigated intensively. In this study, an approach to model misspecification analysis was developed for a two‐phase gamma‐Wiener process, where the change point was assumed to be known. The gamma and Wiener processes in a two‐phase degradation model may be incorrectly assumed, so five misspecified degradation models were considered: the two‐phase Wiener process, two‐phase gamma process, two‐phase Wiener‐gamma process, one‐phase Wiener process, and one‐phase gamma process. A Monte Carlo simulation was performed to examine the effects of model misspecification on the relative bias and relative variability. A case study was performed to examine the effects of model misspecification on the evaluated mean time to failure and median lifetime of a product for different sample sizes, termination time, and change points. The results clearly demonstrated the effects of model misspecification on the evaluated reliability of a product.

A Model-Ranking Approach for Estimation Based on Accelerated Degradation Test Data

Motivated by an accelerated degradation test (ADT) on the power gain of microwave power amplifiers, in this article we propose a model-ranking approach for the estimation of some important reliability characteristics. Different degradation models and statistical lifetime distributions are applied to model the data obtained from the ADT. We study the effect of model misspecification in estimating the reliability characteristics when the behavior of the degradation process and the underlying degradation-data-generating mechanism are unknown. We then propose a model-ranking approach with weighted estimation procedures when multiple candidate models are under consideration. Through a Monte Carlo simulation study, we show that the proposed approach is robust and insensitive to model misspecification. Finally, the ADT data from the motivating example are used to illustrate the proposed methodologies.

Incorporating multi-level taste heterogeneity in route choice modeling: From disaggregated behavior analysis to aggregated network loading

The aggregated traffic flow forecasting problem with multi-level mixed logit based route choice model is discussed in this paper. Multi-level unobserved taste heterogeneity in route choice is further divided to two parts, the OD pair specific and observation specific. With the proposed model, the observations between the same OD pair, as well as the random utilities between alternative routes are correlated because of sharing the same OD pair specific taste heterogeneity. According to the theoretical analysis, these correlations result in the forecasting errors of aggregated link traffic flow that cannot be ignored as the increasing of number of trips. Numerical studies are carried out to illustrate the quantitative effects of incorporating multi-level taste heterogeneity on disaggregated route choice prediction and aggregated network loading, as well as the effects of model mis-specification on parameter recovery and prediction performance.